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BEGIN:VEVENT
UID:news-1655@dmi.unibas.ch
DTSTAMP:20240402T130832
DTSTART;TZID=Europe/Zurich:20240419T110000
DTEND;TZID=Europe/Zurich:20240419T120000
SUMMARY:Seminar in Numerical Analysis: Barbara Verfürth (University of Bonn)
LOCATION:
DESCRIPTION:In the last years, there has been an increasing interest in tim
e-modulated materials to obtain enhanced properties. As mathema
tical model, we study the classical wave equation with time-dep
endent coefficient, which may also include spatial multiscale f
eatures. Based on joint work with Bernhard Maier, we present a
numerical multiscale method for spatially multiscale, (slowly)
time-evolving coefficients. The method is inspired by the Local
ized Orthogonal Decomposition (LOD) and entails time-dependent
multiscale spaces. We provide a rigorous a priori error analysi
s for the considered setting. Numerical examples illustrate the
theoretical findings and investigate an adaptive approach for
the computation of the time-dependent basis functions. On the o
ther hand, we will also briefly discuss the setting of spatiall
y homogeneous, temporal multiscale coefficients. (Higher-order)
multiscale expansions may help to interpret effective physical
material properties and are numerically illustrated.\r\nFor fu
rther information about the seminar, please visit this webpage
[https://dmi.unibas.ch/de/forschung/mathematik/seminar-in-numer
ical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1550@dmi.unibas.ch
DTSTAMP:20231204T091610
DTSTART;TZID=Europe/Zurich:20231215T110000
DTEND;TZID=Europe/Zurich:20231215T120000
SUMMARY:Seminar in Numerical Analysis: Caroline Geiersbach (WIAS Berlin)
LOCATION:
DESCRIPTION:Many problems in shape optimization involve constraints in the
form of one or more partial differential equations. In practice
, the material properties of the underlying shape on which a PD
E is defined are not known exactly; it is natural to use a prob
ability distribution based on empirical measurements and incorp
orate this information when designing an optimal shape. Additio
nally, one might wish to obtain a shape that is robust in its r
esponse to certain external inputs, such as forces. It is helpf
ul to view shape optimization problems subject to uncertainty t
hrough the lens of stochastic optimization, where a wealth of t
heory and algorithms already exist for finite-dimensional probl
ems. The focus will be on the algorithmic handling of these pro
blems in the case of a high stochastic dimension. Stochastic ap
proximation, which dynamically samples from the stochastic spac
e over the course of iterations, is favored in this case, and w
e show how these methods can be applied to shape optimization.
We study the classical stochastic gradient method, which was in
troduced in 1951 by Robbins and Monro and is widely used in mac
hine learning. In particular, we investigate its application to
infinite-dimensional shape manifolds. Further, we present nume
rical examples showing the performance of the method, also in c
ombination with the augmented Lagrangian method for problems wi
th geometric constraints. \r\nJoint work with: Kathrin Welker,
Estefania Loayza-Romero, Tim Suchan\r\n \r\nFor further infor
mation about the seminar, please visit this webpage [https://dm
i.unibas.ch/de/forschung/mathematik/seminar-in-numerical-analys
is/].
END:VEVENT
BEGIN:VEVENT
UID:news-1570@dmi.unibas.ch
DTSTAMP:20231127T102321
DTSTART;TZID=Europe/Zurich:20231208T110000
DTEND;TZID=Europe/Zurich:20231208T120000
SUMMARY:Seminar in Numerical Analysis: Martin Vohralik (Inria Paris)
LOCATION:
DESCRIPTION:A posteriori estimates enable to certify the error committed in
a numerical simulation. In particular, the equilibrated flux r
econstruction technique yields a guaranteed error upper bound,
where the flux, obtained by a local postprocessing, is of indep
endent interest since it is always locally conservative. In thi
s talk, we tailor this methodology to model nonlinear and time-
dependent problems to obtain estimates that are robust, i.e., o
f quality independent of the strength of the nonlinearities and
the final time. These estimates include, and build on, common
iterative linearization schemes such as Zarantonello, Picard, N
ewton, or M- and L-ones. We first consider steady problems and
conceive two settings: we either augment the energy difference
by the discretization error of the current linearization step,
or we design iteration-dependent norms that feature weights giv
en by the current iterate. We then turn to unsteady problems. H
ere we first consider the linear heat equation and finally move
to the Richards one, that is doubly nonlinear and exhibits bot
h parabolic–hyperbolic and parabolic–elliptic degeneracies.
Robustness with respect to the final time and local efficiency
in both time and space are addressed here. Numerical experimen
ts illustrate the theoretical findings all along the presentati
on. Details can be found in [1-4].\r\nA. Ern, I. Smears, M. Voh
ralík, Guaranteed, locally space-time efficient, and polynomia
l-degree robust a posteriori error estimates for high-order dis
cretizations of parabolic problems, SIAM J. Numer. Anal. 55 (20
17), 2811–2834.\r\nA. Harnist, K. Mitra, A. Rappaport, M. Voh
ralík, Robust energy a posteriori estimates for nonlinear elli
ptic problems, HAL Preprint 04033438, 2023.\r\nK. Mitra, M. Vo
hralík, A posteriori error estimates for the Richards equation
, Math. Comp. (2024), accepted for publication.\r\nK. Mitra, M.
Vohralík, Guaranteed, locally efficient, and robust a posteri
ori estimates for nonlinear elliptic problems in iteration-depe
ndent norms. An orthogonal decomposition result based on iterat
ive linearization, HAL Preprint 04156711, 2023.\r\n \r\nFor f
urther information about the seminar, please visit this webpage
[https://dmi.unibas.ch/de/forschung/mathematik/seminar-in-nume
rical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1544@dmi.unibas.ch
DTSTAMP:20230922T165627
DTSTART;TZID=Europe/Zurich:20231110T110000
DTEND;TZID=Europe/Zurich:20231110T120000
SUMMARY:Seminar in Numerical Analysis: Larisa Beilina (University of Göteborg)
LOCATION:
DESCRIPTION:An adaptive finite element/finite difference domain decompositi
on method for solution of time-dependent Maxwell's equations f
or electric field in conductive media will be presented. This m
ethod is applied for reconstruction of dielectric permittivity
and conductivity functions using time-dependent scattered data
of electric field at the boundary of the domain.\r\nAll recons
truction algorithms are based on optimization approach for find
ing of stationary point of the Lagrangian. Derivation of a post
eriori error estimates for the regularized solution and Tikhono
v functional will be presented. Based on these estimates adap
tive reconstruction algorithms are developed. Computational t
ests will show robustness of proposed algorithms in 3D.\r\n \r
\nFor further information about the seminar, please visit this
webpage [https://dmi.unibas.ch/de/forschung/mathematik/seminar-
in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1583@dmi.unibas.ch
DTSTAMP:20231024T091747
DTSTART;TZID=Europe/Zurich:20231027T110000
DTEND;TZID=Europe/Zurich:20231027T120000
SUMMARY:Seminar in Numerical Analysis: Carsten Gräser (FAU Erlangen-Nürnberg)
LOCATION:
DESCRIPTION:We consider the regularization of a supervised learning proble
m by partial differential equations (PDEs). For the resulting
regularized problem we derive error bounds in terms of a PDE e
rror term and a data error term. These error contributions qua
ntify the accuracy of the PDE model used for regularization an
d the data coverage. Furthermore, the discretization of the PD
E-regularized learning problem by generalized Galerkin methods
including finite elements and neural networks approaches is
investigated. A nonlinear version of Céa's lemma allows to de
rive errors bounds for both classes of discretizations and giv
es first insights into error analysis of variational neural ne
twork discretizations of PDEs.\r\n \r\nFor further informatio
n about the seminar, please visit this webpage [https://dmi.uni
bas.ch/de/forschung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1537@dmi.unibas.ch
DTSTAMP:20231013T095041
DTSTART;TZID=Europe/Zurich:20231020T110000
DTEND;TZID=Europe/Zurich:20231020T120000
SUMMARY:Seminar in Numerical Analysis: Marco Zank (U Wien)
LOCATION:
DESCRIPTION:For the discretization of time-dependent partial differential e
quations, the standard approaches are explicit or implicit time
-stepping schemes together with finite element methods in space
. An alternative approach is the usage of space-time methods, w
here the space-time domain is discretized and the resulting glo
bal linear system is solved at once. In this talk, some recent
developments in space-time finite element methods are reviewed.
For this purpose, the heat equation and the wave equation serv
e as model problems. First, for both model problems, space-time
variational formulations and their unique solvability in space
-time Sobolev spaces are discussed, where a modified Hilbert tr
ansformation is used such that ansatz and test spaces are equal
. Second, conforming discretization schemes, using piecewise po
lynomial, globally continuous functions, are introduced. Solvab
ility and stability of these numerical schemes are discussed. N
ext, we investigate efficient direct solvers for the occurring
huge linear systems. The developed solvers are based on the Bar
tels--Stewart method and on the Fast Diagonalization method, wh
ich result in solving a sequence of spatial subproblems. The so
lver based on the Fast Diagonalization method allows solving th
ese spatial subproblems in parallel, leading to a full parallel
ization in time. In the last part of the talk, numerical exampl
es are shown and discussed.\r\n \r\nFor further information ab
out the seminar, please visit this webpage [https://dmi.unibas.
ch/de/forschung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1562@dmi.unibas.ch
DTSTAMP:20230926T153730
DTSTART;TZID=Europe/Zurich:20231013T110000
DTEND;TZID=Europe/Zurich:20231013T120000
SUMMARY:Seminar in Numerical Analysis: Markus Weimar (Julius-Maximilians-Universität Würzburg)
LOCATION:
DESCRIPTION:As a rule of thumb in approximation theory, the asymptotic spee
d of convergence of numerical algorithms is governed by the reg
ularity of the objects we like to approximate. Besides classica
l isotropic Sobolev smoothness, in the last decades the notion
of so-called dominating- mixed regularity of functions turned o
ut to be an important concept in numerical analysis. Indeed, it
naturally arises in high-dimensional real-world applications,
e.g., related to the electronic Schrödinger equation. Although
optimal approximation rates for embeddings within the scales
of isotropic or dominating-mixed Lp-Sobolev spaces are well-u
nderstood, not that much is known for embeddings across those
scales (break-of-scale embeddings).\r\nIn this lecture, we fir
st review the Fourier analytic approach towards by now well-est
ablished (Besov and Triebel-Lizorkin) scales of distribution sp
aces that measure either isotropic or dominating-mixed regula
rity. In addition, we introduce new function spaces of hybrid s
moothness which are able to simultaneously capture both types
of regularity at the same time. As a further generalization of
the aforementioned scales, they particularly include standard
Sobolev spaces on domains. On the other hand, our new spaces yi
eld an appropri- ate framework to study break-of-scale embeddin
gs by means of harmonic analysis. We shall present (non-)adapti
ve wavelet-based multiscale algorithms that approximate such em
bed- dings at optimal dimension-independent rates of convergenc
e. Important special cases cover the approximation of functions
having dominating-mixed Sobolev smoothness w.r.t. Lp in the
norm of the (isotropic) energy space H1.\r\nThe talk is based
on a recent paper [1] which represents the first part of a join
t work with Glenn Byrenheid (FSU Jena), Markus Hansen (PU Marbu
rg), and Janina Hübner (RU Bochum).\r\nReferences:\r\n[1] G. B
yrenheid, J. Hübner, and M. Weimar. Rate-optimal sparse approx
imation of compact break-of-scale embeddings. Appl. Comput. Har
mon. Anal. 65:40–66, 2023 (arXiv:2203.10011).\r\nFor further
information about the seminar, please visit this webpage [http
s://dmi.unibas.ch/de/forschung/mathematik/seminar-in-numerical-
analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1538@dmi.unibas.ch
DTSTAMP:20230915T113109
DTSTART;TZID=Europe/Zurich:20230929T110000
DTEND;TZID=Europe/Zurich:20230929T120000
SUMMARY:Seminar in Numerical Analysis: Rüdiger Kempf (U Bayreuth)
LOCATION:
DESCRIPTION:Reproducing kernel Hilbert spaces (RKHSs) and the closely rela
ted kernel methods are well-established and well-studied tool
s in classical approximation theory. More recently, they see ma
ny uses in other problems in applied and numerical analysis.\r\
nIn machine learning, support vector machines heavily rely on R
KHSs. For neural networks Barron spaces are connected to certai
n RKHSs and offer a possibility for a theoretical analysis of t
hese networks.\r\nAnother application of RKHSs is in high(er)-d
imensional approximation. For instance in the field of quasi Mo
nte-Carlo methods, kernel-techniques are used to derive an erro
r analysis for high-dimensional quadrature rules. We also devel
oped a novel kernel-based approximation method for higher-dimen
sional meshfree function reconstruction, based on Smolyak opera
tors.\r\nIn this talk I will provide an introduction into the t
heory of RKHSs, their kernels and associated kernel methods. In
particular, I will focus on a multiscale approximation scheme
for rescaled radial basis functions. This method will then be u
sed to derive the new tensor product multilevel method for hi
gher- dimensional meshfree approximation, which I will discuss
in detail.\r\n \r\nFor further information about the seminar,
please visit this webpage [https://dmi.unibas.ch/de/forschung/m
athematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1536@dmi.unibas.ch
DTSTAMP:20230904T192551
DTSTART;TZID=Europe/Zurich:20230922T110000
DTEND;TZID=Europe/Zurich:20230922T120000
SUMMARY:Seminar in Numerical Analysis: Robert Gruhlke (FU Berlin)
LOCATION:
DESCRIPTION:Ensemble methods have become ubiquitous for solving Bayesian in
ference problems, in particular the efficient sampling from pos
terior densities. State-of-the-art subclasses of Markow-Chain-
Monte-Carlo methods rely on gradient information of the log-den
sity including Langevin samplers such as Ensemble Kalman Sampl
er (EKS) and Affine Invariant Langevin Dynamics (ALDI). These d
ynamics are described by stochastic differential equations (SD
Es) with time homogeneous drift terms. \r\nIn this talk we pre
sent enhancement strategies of such ensemble methods based on s
ample enrichment and homotopy formalism, that ultimately lead t
o time-dependent drift terms that possible assimilate a larger
class of target distributions while providing faster mixing ti
mes. \r\nFurthermore, we present an alternative route to const
ruct time-inhomogeneous drift terms based on reverse Diffusion
processes that are popular in state-of-the-art Generative Mod
elling such as Diffusion maps. Here, we propose learning these
log-densities by propagation of the target distribution throug
h an Ornstein-Uhlenbeck process. For this, we solve the associa
ted Hamilton-Jabobi-Bellman equation through an adaptive explic
it Euler discretization using low-rank compression such as func
tional Tensor Trains for the spatial discretization.\r\n \r\nF
or further information about the seminar, please visit this web
page [https://dmi.unibas.ch/de/forschung/mathematik/seminar-in-
numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1464@dmi.unibas.ch
DTSTAMP:20230509T105935
DTSTART;TZID=Europe/Zurich:20230512T110000
DTEND;TZID=Europe/Zurich:20230512T120000
SUMMARY:Seminar in Numerical Analysis: Martin Eigel (WIAS Berlin)
LOCATION:
DESCRIPTION:Weighted least squares methods have been examined thouroughly t
o obtain quasi-optimal convergence results for a chosen (polyno
mial) basis of a linear space. A focus in the analysis lies on
the construction of optimal sampling measures and the derivatio
n of a sufficient sample complexity for stable reconstructions.
When considering holomorphic functions such as solutions of co
mmon parametric PDEs, the anisotropic sparsity they exhibit can
be exploited to achieve improved results adapted to the consid
ered problem. In particular, the sparsity of the data transfers
to the solution sparsity in terms of polynomial chaos coeffici
ents. When using nonlinear model classes, it turns out that the
known results cannot be used directly. To obtain comparable a
priori rates, we introduce a new weighted version of Stechkin's
lemma. This enables to obtain optimal complexity results for a
model class of low-rank tensor trains. We also show that the s
olution sparsity results in sparse component tensors and sketch
how this can be realised in practical algorithms. A nice appli
cation is the reconstruction of Galerkin solutions for parametr
ic PDEs. With this, a provably converging a posteriori adaptive
algorithm can be derived for linear model PDEs with non-affine
coefficients.\r\n \r\nFor further information about the semin
ar, please visit this webpage [https://dmi.unibas.ch/de/forschu
ng/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1500@dmi.unibas.ch
DTSTAMP:20230427T100910
DTSTART;TZID=Europe/Zurich:20230505T110000
DTEND;TZID=Europe/Zurich:20230505T120000
SUMMARY:Seminar in Numerical Analysis: Elena Moral Sánchez (Max-Planck Institute for Plasma Physics)
LOCATION:
DESCRIPTION:The cold-plasma wave equation describes the propagation of an e
lectromagnetic wave in a magnetized plasma, which is an inhomog
eneous, dispersive and anisotropic medium. The thermal effects
are assumed to be negligible, which leads to a linear partial d
ifferential equation. Besides, we assume that the electromagnet
ic field of the propagating wave is in the time-harmonic regime
. This model has applications in magnetic confinement fusion de
vices, like the Tokamak. Namely, electromagnetic waves are used
to heat up the plasma (Electron cyclotron resonance heating (E
CRH)) or for interferometry and reflectometry diagnostics (to m
easure plasma density and position, etc.). In the first part o
f this talk, we introduce the cold-plasma model, together with
a qualitative study of the plasma modes which expose the comple
xity of the problem. In the second part, we describe the proble
m and the simplifications we carry out, which yield the indefin
ite Helmholtz equation. It is solved with B-Spline Finite Eleme
nts provided by the Psydac library and some results are shown.
Lastly, we discuss the performance and potential ways of precon
ditioning.\r\n \r\nFor further information about the seminar,
please visit this webpage [https://dmi.unibas.ch/de/forschung/m
athematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1472@dmi.unibas.ch
DTSTAMP:20230417T092542
DTSTART;TZID=Europe/Zurich:20230428T110000
DTEND;TZID=Europe/Zurich:20230428T120000
SUMMARY:Seminar in Numerical Analysis: Frédéric Nataf (CNRS — Université Pierre et Marie Curie)
LOCATION:
DESCRIPTION:We introduce a scalable adaptive element-based domain decomposi
tion (DD) method for solving saddle point problems defined as a
block two by two matrix. The algorithm does not require any kn
owledge of the constrained space. We assume that all sub matric
es are sparse and that the diagonal blocks are spectrally equiv
alent to a sum of positive semi definite matrices. The latter a
ssumption enables the design of adaptive coarse space for DD me
thods that extends the GenEO theory to saddle point problems. N
umerical results on three dimensional elasticity problems for s
teel-rubber structures discretized by a finite element with con
tinuous pressure are shown for up to one billion degrees of fre
edom along with comparisons to Algebraic Multigrid Methods.\r\n
\r\nFor further information about the seminar, please visit t
his webpage [https://dmi.unibas.ch/de/forschung/mathematik/semi
nar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1480@dmi.unibas.ch
DTSTAMP:20230418T090618
DTSTART;TZID=Europe/Zurich:20230421T110000
DTEND;TZID=Europe/Zurich:20230421T120000
SUMMARY:Seminar in Numerical Analysis: Omar Lakkis (University of Sussex)
LOCATION:
DESCRIPTION:Least-squares finite element recovery-based methods provide a s
imple and practical way to approximate linear elliptic PDEs in
nondivergence form where standard variational approach either f
ails or requires technically complex modifications.\r\nThis ide
a allows the creation of efficient solvers for fully nonlinear
elliptic equations, the linearization of which leaves us with a
n equation in nondivergence form. An important class of fully n
onlinear elliptic PDEs can be written in Hamilton--Jacobi--Bell
man (Dynamic Programming) form, i.e., as the supremum of a coll
ection of linear operators acting on the unkown.\r\nThe least-s
quares FEM approach, a variant of the nonvariational finite ele
ment method, is based on gradient or Hessian recovery and allow
s the use of FEMs of arbitrary degree. The price to pay for usi
ng higher order FEMs is the loss of discrete-level monotonicity
(maximum principle), which is valid for the PDE and crucial in
proving the convergence of many degree one FEM and finite diff
erence schemes.\r\nSuitable functional spaces and penalties in
the least-squares's cost functional must be carefully crafted i
n order to ensure stability and convergence of the scheme with
a good approximation of the gradient (or Hessian) under the Cor
des condition on the family of linear operators being optimized
.\r\nFurthermore, the nonlinear operator which is not necessari
ly everywhere differentiable, must be linearized in appropriate
functional spaces using semismooth Newton or Howard's policy i
teration method. A crucial contribution of our work, is the pro
of of convergence of the semismooth Newton method at the contin
uum level, i.e., on infinite dimesional functionals spaces. Thi
s allows an easy use of our non-monotone schemes which provides
convergence rates as well as a posteriori error estimates.\r\n
\r\nFor further information about the seminar, please visit t
his webpage [https://dmi.unibas.ch/de/forschung/mathematik/semi
nar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1455@dmi.unibas.ch
DTSTAMP:20230403T123724
DTSTART;TZID=Europe/Zurich:20230414T110000
DTEND;TZID=Europe/Zurich:20230414T120000
SUMMARY:Seminar in Numerical Analysis: Vesa Kaarnioja (FU Berlin)
LOCATION:
DESCRIPTION:We describe a fast method for solving elliptic PDEs with uncert
ain coefficients using kernel-based interpolation over a rank-1
lattice point set [1]. By representing the input random field
of the system using a model proposed by Kaarnioja, Kuo, and Slo
an [2], in which a countable number of independent random varia
bles enter the random field as periodic functions, it is shown
that the kernel interpolant can be constructed for the PDE solu
tion (or some quantity of interest thereof) as a function of th
e stochastic variables in a highly efficient manner using fast
Fourier transform. The method works well even when the stochast
ic dimension of the problem is large, and we obtain rigorous er
ror bounds which are independent of the stochastic dimension of
the problem. We also outline some techniques that can be used
to further improve the approximation error and computational co
mplexity of the method [3].\r\n \r\nReferences:\r\n[1] V. Kaar
nioja, Y. Kazashi, F. Y. Kuo, F. Nobile, and I. H. Sloan. Fast
approximation by periodic kernel-based lattice-point interpolat
ion with application in uncertainty quantification. Numer. Math
. 150:33-77, 2022.\r\n[2] V. Kaarnioja, F. Y. Kuo, and I. H. Sl
oan. Uncertainty quantification using periodic random variables
. SIAM J. Numer. Anal. 58(2):1068-1091, 2020.\r\n[3] V. Kaarnio
ja, F. Y. Kuo, and I. H. Sloan. Lattice-based kernel approximat
ion and serendipitous weights for parametric PDEs in very high
dimensions. Preprint 2023, arXiv:2303.17755 [math.NA].\r\n \r\
nFor further information about the seminar, please visit this w
ebpage [https://dmi.unibas.ch/de/forschung/mathematik/seminar-i
n-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1468@dmi.unibas.ch
DTSTAMP:20230403T123812
DTSTART;TZID=Europe/Zurich:20230317T110000
DTEND;TZID=Europe/Zurich:20230317T120000
SUMMARY:Seminar in Numerical Analysis: Marc Dambrine (Université de Pau et des Pays de l'Adour)
LOCATION:
DESCRIPTION:As it is often the case in optimization, the solution of a shap
e problem is sensitive to the parameters of the problem. For ex
ample, the loading of a structure to be optimised is known in a
n imprecise way. In this talk, I will present the different app
roaches that have been recently proposed to incorporate these u
ncertainties in the definition of the objective. I will present
numerical illustrations from structural optimization.\r\n \r\
nFor further information about the seminar, please visit this w
ebpage [https://dmi.unibas.ch/de/forschung/mathematik/seminar-i
n-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1402@dmi.unibas.ch
DTSTAMP:20220920T084344
DTSTART;TZID=Europe/Zurich:20221216T110000
DTEND;TZID=Europe/Zurich:20221216T120000
SUMMARY:Seminar in Numerical Analysis: Christophe Geuzaine (Université de Liège)
LOCATION:
DESCRIPTION:This talk is devoted to non-overlapping Schwarz domain decompos
ition methods for the resolution of high frequency flow acousti
cs problems of industrial relevance. First, we will present rec
ent advances on non-reflecting boundary techniques that provide
local approximations to the Dirichlet-to-Neumann operator for
convected and heterogeneous time-harmonic wave propagation prob
lems [1]. Then we will show how to adapt a generic domain decom
position framework to flow acoustics, based on these newly desi
gned transmission conditions, and highlight the benefit of the
approach on the simulation of three-dimensional noise radiation
of a high by-pass ratio turbofan engine intake [2]. [1] March
ner, P., Antoine, X., Geuzaine, C., & Bériot, H. (2022). Const
ruction and numerical assessment of local absorbing boundary co
nditions for heterogeneous time-harmonic acoustic problems. SIA
M Journal on Applied Mathematics, 82(2), 476-501. [2] Lieu, A.
, Marchner, P., Gabard, G., Beriot, H., Antoine, X., & Geuzaine
, C. (2020). A non-overlapping Schwarz domain decomposition met
hod with high-order finite elements for flow acoustics. Compute
r Methods in Applied Mechanics and Engineering, 369, 113223.\r\
nFor further information about the seminar, please visit this w
ebpage [https://dmi.unibas.ch/de/forschung/mathematik/seminar-i
n-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1410@dmi.unibas.ch
DTSTAMP:20221130T142441
DTSTART;TZID=Europe/Zurich:20221209T110000
DTEND;TZID=Europe/Zurich:20221209T120000
SUMMARY:Seminar in Numerical Analysis: Patrick Ciarlet (ENSTA Paris)
LOCATION:
DESCRIPTION:Variational formulations are a popular tool to analyse linear P
DEs (eg. neutron diffusion, Maxwell equations, Stokes equations
...), and it also provides a convenient basis to design numeri
cal methods to solve them. Of paramount importance is the inf-s
up condition, designed by Ladyzhenskaya, Necas, Babuska and Bre
zzi in the 1960s and 1970s. As is well-known, it provides sharp
conditions to prove well-posedness of the problem, namely exis
tence and uniqueness of the solution, and continuous dependence
with respect to the data. Then, to solve the approximate, or d
iscrete, problems, there is the (uniform) discrete inf-sup cond
ition, to ensure existence of the approximate solutions, and co
nvergence of those solutions to the exact solution. Often, the
two sides of this problem (exact and approximate) are handled s
eparately, or at least no explicit connection is made between t
he two.\r\nIn this talk, I will focus on an approach that is co
mpletely equivalent to the inf-sup condition for problems set i
n Hilbert spaces, the T-coercivity approach. This approach reli
es on the design of an explicit operator to realize the inf-sup
condition. If the operator is carefully chosen, it can provide
useful insight for a straightforward definition of the approxi
mation of the exact problem. As a matter of fact, the derivatio
n of the discrete inf-sup condition often becomes elementary, a
t least when one considers conforming methods, that is when the
discrete spaces are subspaces of the exact Hilbert spaces. In
this way, both the exact and the approximate problems are consi
dered, analysed and solved at once.\r\nIn itself, T-coercivity
is not a new theory, however it seems that some of its strength
s have been overlooked, and that, if used properly, it can be a
simple, yet powerful tool to analyse and solve linear PDEs. In
particular, it provides guidelines such as, which abstract too
ls and which numerical methods are the most “natural” to an
alyse and solve the problem at hand. In other words, it allows
one to select simply appropriate tools in the mathematical, or
numerical, toolboxes. This claim will be illustrated on classic
al linear PDEs, and for some generalizations of those models.\r
\nFor further information about the seminar, please visit this
webpage [https://dmi.unibas.ch/de/forschung/mathematik/seminar-
in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1409@dmi.unibas.ch
DTSTAMP:20220928T143806
DTSTART;TZID=Europe/Zurich:20221202T110000
DTEND;TZID=Europe/Zurich:20221202T120000
SUMMARY:Seminar in Numerical Analysis: Sébastien Imperiale (Inria — LMS, Ecole Polytechnique, CNRS — Université Paris-Saclay, MΞDISIM)
LOCATION:
DESCRIPTION:The objective of this work is to propose and analyze numerical
schemes to solve transient wave propagation problems that are e
xponentially stable (i.e. the solution decays to zero exponenti
ally fast). Applications are in data assimilation strategies or
the discretisation of absorbing boundary conditions. More prec
isely the aim of our work is to propose a discretization proces
s that enables to preserve the exponential stability at the dis
crete level as well as a high order consistency when using a hi
gh-order finite element approximation. The main idea is to add
to the wave equation a stabilizing term which damps the high-fr
equency oscillating components of the solutions such as spuriou
s waves. This term is built from a discrete multiplier analysis
that proves the exponential stability of the semi-discrete pro
blem at any order without affecting the order of convergence.\r
\nFor further information about the seminar, please visit this
webpage [https://dmi.unibas.ch/de/forschung/mathematik/seminar-
in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1408@dmi.unibas.ch
DTSTAMP:20221110T132504
DTSTART;TZID=Europe/Zurich:20221118T110000
DTEND;TZID=Europe/Zurich:20221118T120000
SUMMARY:Seminar in Numerical Analysis: Alexey Chernov (Universität Oldenburg)
LOCATION:
DESCRIPTION:We investigate a class of parametric elliptic eigenvalue proble
ms where the coefficients (and hence the solution) may depend o
n a parameter y. Understanding the regularity of the solution a
s a function of y is important for construction of efficient nu
merical approximation schemes. Several approaches are available
in the existing literature, e.g. the complex-analytic argumen
t by Andreev and Schwab (2012) and the real-variable argument b
y Gilbert et al. (2019+). The latter proof strategy is more exp
licit, but, due to the nonlinear nature of the problem, leads t
o slightly suboptimal results. In this talk we close this gap a
nd (as a by-product) extend the analysis to the more general cl
ass of coefficients.\r\nFor further information about the semin
ar, please visit this webpage [https://dmi.unibas.ch/de/forschu
ng/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1412@dmi.unibas.ch
DTSTAMP:20221021T141230
DTSTART;TZID=Europe/Zurich:20221104T110000
DTEND;TZID=Europe/Zurich:20221104T120000
SUMMARY:Seminar in Numerical Analysis: Naomi Schneider (Universität Siegen)
LOCATION:
DESCRIPTION:Both the approximation of the gravitational potential via the d
ownward continuation of satellite data and of wave velocities v
ia the travel time tomography using earthquake data are geoscie
ntific ill- posed inverse problems. To monitor certain aspects
of the system Earth, like the mass transport or its geomagnetic
field, it is, however, important to tackle these challenges. T
raditionally, an approximation of such a linear(ized) inverse p
roblem is obtained in one, a-priori chosen basis system: either
a global one, e.g. spherical harmonics or polynomials on the b
all, or a local one, e.g. radial basis functions and wavelets o
n the sphere or finite elements on the ball. In the Geomathemat
ics Group Siegen, we developed methods that enable us to combin
e different types of trial functions for such an approximation.
The idea is to make the most of the benefits of different type
s of available trial functions. The algorithms are called the (
Learning) Inverse Problem Matching Pursuits (LIPMPs). They cons
truct an approximation iteratively from an intentionally overco
mplete set of trial functions, the dictionary, such that the Ti
khonov functional is reduced. Due to the learning add-on, the d
ictionary can very well be infinite. Moreover, the computationa
l costs are usually decreased. In this talk, we give details on
the LIPMPs and show some current numerical results.\r\nFor fur
ther information about the seminar, please visit this webpage [
https://dmi.unibas.ch/de/forschung/mathematik/seminar-in-numeri
cal-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1352@dmi.unibas.ch
DTSTAMP:20220516T103313
DTSTART;TZID=Europe/Zurich:20220520T110000
DTEND;TZID=Europe/Zurich:20220520T120000
SUMMARY:Seminar in Numerical Analysis: Jens Saak (Max Planck Institute for Dynamics of Complex Technical Systems)
LOCATION:
DESCRIPTION:Optimal control problems subject to constraints given by partia
l differential equations are a powerful tool for the improvemen
t of many tasks in science an technology. Classic optimization
today is applicable on various problems and tackling nonlinear
equations and inclusion of box constraints on the solutions is
flexible. However, especially for non-stationary problems, smal
l perturbations along the trajectories can easily lead to large
deviations in the desired solutions. Consequently, optimality
may be lost just as easily. On the other hand, the linear-quadr
atic regulator problem in system theory is an approach to make
a dynamical system react to perturbation via feedback controls
that can be expressed by the solutions of matrix Riccati equati
ons. It’s applicability is limited by the linearity of the dy
namical system and the efficient solvability of the quadratic m
atrix equation. In this talk, we discuss how certain classes of
non-stationary PDEs can be reformulated (after spatial semi-di
scretization) into structured linear dynamical systems that all
ow the Riccati feedback to be computed. This allows us to combi
ne both approaches and thus steer solutions of perturbed PDEs b
ack to the optimized trajectories. The key to efficient solvers
for the Riccati equations is the usage of the specific structu
re in the problems and the fact that the Riccati solutions usua
lly feature a strong singular value decay, and thus good low-ra
nk approximability.\r\nFor further information about the semina
r, please visit this webpage [https://dmi.unibas.ch/de/forschun
g/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1351@dmi.unibas.ch
DTSTAMP:20220419T163823
DTSTART;TZID=Europe/Zurich:20220422T110000
DTEND;TZID=Europe/Zurich:20220422T120000
SUMMARY:Seminar in Numerical Analysis: Matthias Voigt (FernUni Schweiz)
LOCATION:
DESCRIPTION:We introduce a model reduction approach for linear time-invaria
nt second-order systems based on positive real balanced truncat
ion. Our method guarantees to preserve passivity of the reduced
-order model as well as the positive definiteness of the mass a
nd stiffness matrices and admits an a priori gap metric error b
ound. Our construction of the second-order reduced model is bas
ed on the consideration of an internal symmetry structure and t
he invariant zeros of the system and their sign-characteristics
for which we derive a normal form. The results are available i
n [1].\r\n[1] I. Dorschky, T. Reis, and M. Voigt. Balanced trun
cation model reduction for symmetric second order systems - a p
assivity-based approach. SIAM J. Matrix Anal. Appl., 42(4):1602
--1635, 2021.\r\nFor further information about the seminar, ple
ase visit this webpage [https://dmi.unibas.ch/de/forschung/math
ematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1333@dmi.unibas.ch
DTSTAMP:20220315T174614
DTSTART;TZID=Europe/Zurich:20220408T110000
DTEND;TZID=Europe/Zurich:20220408T120000
SUMMARY:Seminar in Numerical Analysis: Ralf Hiptmair (ETH Zürich)
LOCATION:
DESCRIPTION:We consider scalar-valued shape functionals on sets of shapes w
hich are small perturbations of a reference shape. The shapes a
re described by parameterizations and their closeness is induce
d by a Hilbert space structure on the parameter domain. We just
ify a heuristic for finding the best low-dimensional parameter
subspace with respect to uniformly approximating a given shape
functional. We also propose an adaptive algorithm for achieving
a prescribed accuracy when representing the shape functional w
ith a small number of shape parameters.\r\nFor further informat
ion about the seminar, please visit this webpage [https://dmi.u
nibas.ch/de/forschung/mathematik/seminar-in-numerical-analysis/
].
END:VEVENT
BEGIN:VEVENT
UID:news-1334@dmi.unibas.ch
DTSTAMP:20220325T150305
DTSTART;TZID=Europe/Zurich:20220401T110000
DTEND;TZID=Europe/Zurich:20220401T120000
SUMMARY:Seminar in Numerical Analysis: Johannes Pfefferer (Technische Universität München)
LOCATION:
DESCRIPTION:Many areas of science and engineering involve optimal control o
f processes that are modeled through partial differential equat
ions. This talk will introduce the theoretical foundation and n
umerical methods based on finite elements for solving PDE const
rained optimal control problems. We will discuss different disc
retization concepts and corresponding discretization error esti
mates. The discussion will include the consideration of optimal
control problems with control constraints as well as with stat
e constraints.\r\nFor further information about the seminar, pl
ease visit this webpage [https://dmi.unibas.ch/de/forschung/mat
hematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1340@dmi.unibas.ch
DTSTAMP:20220321T111220
DTSTART;TZID=Europe/Zurich:20220325T110000
DTEND;TZID=Europe/Zurich:20220325T120000
SUMMARY:Seminar in Numerical Analysis: Stepan Shakhno (Ivan Franko National University of Lviv)
LOCATION:
DESCRIPTION:In this talk, one- and two-step methods for solving nonlinear e
quations with nondifferentiable operators are proposed. These m
ethods are based on two methods: using derivatives and using di
vided differences. The local and semi-local convergence of the
proposed methods is studied and the order of their convergence
is established. We apply our results to the numerical solving o
f systems of nonlinear equations.\r\nFor further information ab
out the seminar, please visit this webpage [https://dmi.unibas.
ch/de/forschung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1335@dmi.unibas.ch
DTSTAMP:20220310T115442
DTSTART;TZID=Europe/Zurich:20220318T110000
DTEND;TZID=Europe/Zurich:20220318T120000
SUMMARY:Seminar in Numerical Analysis: Markus Bachmayr (Universität Mainz)
LOCATION:
DESCRIPTION:We consider the computational complexity of approximating ellip
tic PDEs with random coefficients by sparse product polynomial
expansions. Except for special cases (for instance, when the sp
atial discretisation limits the achievable overall convergence
rate), previous approaches for a posteriori selection of poly
nomial terms and corresponding spatial discretizations do not g
uarantee optimal complexity in the sense of computational costs
scaling linearly in the number of degrees of freedom. We show
that one can achieve optimality of an adaptive Galerkin scheme
for discretizations by spline wavelets in the spatial variable
when a multiscale representation of the affinely parameterized
random coefficients is used. \r\nM. Bachmayr and I. Voulis,
An adaptive stochastic Galerkin method based on multilevel expa
nsions of random fields: Convergence and optimality, arXiv:210
9:09136 [https://arxiv.org/abs/2109.09136]\r\nFor further infor
mation about the seminar, please visit this webpage [https://dm
i.unibas.ch/de/forschung/mathematik/seminar-in-numerical-analys
is/].
END:VEVENT
BEGIN:VEVENT
UID:news-1277@dmi.unibas.ch
DTSTAMP:20211204T191942
DTSTART;TZID=Europe/Zurich:20211217T110000
DTEND;TZID=Europe/Zurich:20211217T120000
SUMMARY:Seminar in Numerical Analysis: Eliane Bécache (POEMS, CNRS, INRIA, ENSTA Paris, Institut Polytechnique de Paris)
LOCATION:
DESCRIPTION:The PML method is one of the most widely used for the numerical
simulation of wave propagation problems set in unbounded domai
ns. However difficulties arise when the exterior domain has s
ome complexity which prevents from using classical approaches.
For instance, it is well-known that PML may be unstable for tim
e-domain eslastodynamic waves in some anisotropic materials. Mo
re recently, is has also been noticed that standard PML cannot
work in presence of some dispersive materials. In some cases,
new stable PMLs have been designed.\r\nIn this talk, we addres
s the questions of well-posedness, stability and convergence of
standard and new models of PMLs in the context of electromagne
tic waves for non-dispersive and dispersive materials.\r\nFor f
urther information about the seminar, please visit this webpage
[https://dmi.unibas.ch/de/forschung/mathematik/seminar-in-nume
rical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1276@dmi.unibas.ch
DTSTAMP:20211204T143630
DTSTART;TZID=Europe/Zurich:20211210T110000
DTEND;TZID=Europe/Zurich:20211210T120000
SUMMARY:Seminar in Numerical Analysis: Mike Botchev (Keldysh Institute of Applied Mathematics)
LOCATION:
DESCRIPTION:An efficient Krylov subspace algorithm for computing actions of
the phi matrix function for large matrices is proposed. This m
atrix function is widely used in exponential time integration,
Markov chains, and network analysis and many other applications
. Our algorithm is based on a reliable residual based stopping
criterion and a new efficient restarting procedure. We analyze
residual convergence and prove, for matrices with numerical ra
nge in the stable complex half-plane, that the restarted method
is guaranteed to converge for any Krylov subspace dimension. N
umerical tests demonstrate efficiency of our approach for solvi
ng large scale evolution problems resulting from discretized in
space time-dependent PDEs, in particular, diffusion and convec
tion-diffusion problems.\r\nFor further information about the s
eminar, please visit this webpage [https://dmi.unibas.ch/de/for
schung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1258@dmi.unibas.ch
DTSTAMP:20211025T123428
DTSTART;TZID=Europe/Zurich:20211203T110000
DTEND;TZID=Europe/Zurich:20211203T120000
SUMMARY:Seminar in Numerical Analysis: Larisa Beilina (Chalmers tekniska högskola)
LOCATION:
DESCRIPTION:We will discuss how to apply an adaptive finite element metho
d (AFEM) for numerical solution of an electromagnetic volume i
ntegral equation. The problem of solution of this equation is
formulated as an optimal control problem for minimizing of the
Tikhonov's regularization functional. A posteriori error est
imates for the error in the obtained finite element reconstruc
tion and error in the Tikhonov's functional will be presented.\
r\nBased on these estimates, different adaptive finite element
algorithms are formulated. Numerical examples will show effic
iency of the proposed adaptive algorithms to improve quality o
f 3D reconstruction of target during the process of microwave
thermometry which is used in cancer therapies. This is joint w
ork with the group of Biomedical Imaging at the Department of
Electrical Engineering at CTH, Chalmers.\r\nFor further inform
ation about the seminar, please visit this webpage [https://dmi
.unibas.ch/de/forschung/mathematik/seminar-in-numerical-analysi
s/].
END:VEVENT
BEGIN:VEVENT
UID:news-1260@dmi.unibas.ch
DTSTAMP:20211109T170252
DTSTART;TZID=Europe/Zurich:20211119T110000
DTEND;TZID=Europe/Zurich:20211119T120000
SUMMARY:Seminar in Numerical Analysis: Jaap van der Vegt (Universiteit Twente)
LOCATION:
DESCRIPTION:In the numerical solution of partial differential equations, it
is frequently necessary to ensure that certain variables, e.g.
, density, pressure, or probability density distribution, remai
n within strict bounds. Strict observation of these bounds is c
rucial, otherwise unphysical solutions will be obtained that mi
ght result in the failure of the numerical algorithm. Bounds on
certain variables are generally ensured in discontinuous Galer
kin (DG) discretizations using positivity preserving limiters,
which locally modify the solution to ensure that the constraint
s are satisfied, while preserving higher order accuracy. In pra
ctice this approach is mostly limited to DG discretizations com
bined with explicit time integration methods. The combination o
f (positivity preserving) limiters in DG discretizations and im
plicit time integration methods results, however, in serious pr
oblems. Many positivity preserving limiters are not easy to app
ly in time-implicit DG discretizations and have a non-smooth fo
rmulation, which hampers the use of standard Newton methods to
solve the nonlinear algebraic equations resulting from the time
-implicit DG discretization. This often results in poor converg
ence.\r\nIn this presentation, we will discuss a different appr
oach to ensure that a higher order accurate DG solution satisfi
es the positivity constraints. Instead of using a limiter, we i
mpose the positivity constraints directly on the algebraic equa
tions resulting from a higher order accurate time-implicit DG d
iscretization using techniques from mathematical optimization t
heory. This approach ensures that the positivity constraints
are satisfied and does not affect the higher order accuracy of
the time-implicit DG discretization. The resulting algebraic eq
uations are then solved using a specially designed semi-smooth
Newton method that is well suited to deal with the resulting no
nlinear complementarity problem. We will demonstrate the algori
thm on several parabolic model problems.\r\nFor further informa
tion about the seminar, please visit this webpage [https://dmi.
unibas.ch/de/forschung/mathematik/seminar-in-numerical-analysis
/].
END:VEVENT
BEGIN:VEVENT
UID:news-1257@dmi.unibas.ch
DTSTAMP:20211025T121842
DTSTART;TZID=Europe/Zurich:20211112T110000
DTEND;TZID=Europe/Zurich:20211112T120000
SUMMARY:Seminar in Numerical Analysis: Rolf Krause (Università della Svizzera italiana)
LOCATION:
DESCRIPTION:Non-convex minimization problems show up in manifold applicatio
ns: non-linear elasticity, phase field models, fracture propaga
tion, or the training of neural networks. Traditional multilev
el decompositions are the basic ingredient of the most effici
ent class of solution methods for linear systems, i.e. of multi
grid methods, which allow to solve certain classes of linear sy
stems with optimal complexity. The transfer of these concepts t
o non-linear problems, however, is not straightforward, neither
in terms of the design of the multilevel decomposition nor in
terms of convergence properties. In this talk, we will discuss
multilevel decompositions for convex, non-convex and possibly
non-smooth minimization problems. We will discuss in detail ho
w multilevel optimization methods can be constructed and anal
yzed and we will illustrate the sometimes significant gain in
performance, which can be achieved by multilevel minimization
techniques. Examples from mechanics, geophysics, and machine le
arning will illustrate our discussion.\r\nFor further informati
on about the seminar, please visit this webpage [https://dmi.un
ibas.ch/de/forschung/mathematik/seminar-in-numerical-analysis/]
.
END:VEVENT
BEGIN:VEVENT
UID:news-1256@dmi.unibas.ch
DTSTAMP:20211025T121402
DTSTART;TZID=Europe/Zurich:20211029T110000
DTEND;TZID=Europe/Zurich:20211029T120000
SUMMARY:Seminar in Numerical Analysis: Jochen Garke (Rheinische Friedrich-Wilhelms-Universität Bonn)
LOCATION:
DESCRIPTION:We present a conceptual framework that helps to bridge the know
ledge gap between the two individual communities from machine
learning and numerical simulation to identify potential combin
ed approaches and to promote the development of hybrid systems
. We give examples of different types of combinations using ex
emplary approaches of simulation-assisted machine learning and
machine-learning assisted simulation. We also discuss an adva
nced pairing where we see particular further potential for hyb
rid systems.\r\nFor further information about the seminar, plea
se visit this webpage [https://dmi.unibas.ch/de/forschung/mathe
matik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1157@dmi.unibas.ch
DTSTAMP:20210627T202913
DTSTART;TZID=Europe/Zurich:20210604T110000
DTEND;TZID=Europe/Zurich:20210604T120000
SUMMARY:Seminar in Numerical Analysis: Ivan Dokmanić (Universität Basel)
LOCATION:
DESCRIPTION:This talk will be an overview of my group's research between de
ep learning and inverse problems. I will first describe the cur
rent (?) state of the field and then present a medley of our re
sults, including 1) a neural network architecture for wave-base
d inverse problems derived from Fourier integral operators; 2)
an approach to nonlinear traveltime tomography based on neural
priors; and 3) provably injective neural networks that are univ
ersal approximators of probability measures supported on low-di
mensional manifolds. My secret hope is to spark discussions tha
t could evolve to collaborations.\r\nFor further information ab
out the seminar, please visit this webpage [https://dmi.unibas.
ch/de/forschung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1158@dmi.unibas.ch
DTSTAMP:20210627T202830
DTSTART;TZID=Europe/Zurich:20210507T110000
DTEND;TZID=Europe/Zurich:20210507T120000
SUMMARY:Seminar in Numerical Analysis: Erik Burman (University College London)
LOCATION:
DESCRIPTION:In many applications both in medical science and in the geoscie
nces the accurate approximation of solutions to wave equations
is an important component for optimisation or inverse identifi
cation. Examples include thermoacoustic imaging or high frequen
cy ultrasound treatments in medicine (HIFU) or fault slip anal
ysis in seismology. These problems have in common the need for
computational solution of an inverse problem where the forward
problem is set in a heterogeneous domain. Indeed typically the
sound speed in the bulk domain jumps over material interfaces.
Sometimes there is even a need for coupling of the acoustic and
elastodynamic equations in the presence of liquid inclusions.
In this talk we will give a snapshot of our ongoing work in the
se topics, motivated by two such applications: HIFU and the pro
pagation of seismic waves. After a brief introduction of the ap
plications we will first discuss the analysis of some approxima
tion methods for inverse initial value problems subject to the
wave equation. We will then consider a hybrid high order method
for the approximation of wave propagation in heterogeneous med
ia, using cut element techniques to avoid meshing of interfaces
. Finally we will discuss some open problems that remain in ord
er to understand the approximation of the inverse initial value
problem in heterogeneous media using high order methods.\r\nFo
r further information about the seminar, please visit this webp
age [https://dmi.unibas.ch/de/forschung/mathematik/seminar-in-n
umerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1184@dmi.unibas.ch
DTSTAMP:20210627T202737
DTSTART;TZID=Europe/Zurich:20210430T110000
DTEND;TZID=Europe/Zurich:20210430T120000
SUMMARY:Seminar in Numerical Analysis: Pieter Barendrecht (KAUST)
LOCATION:
DESCRIPTION:In this talk, we're going to take a closer look at the basics o
f both univariate and bivariate splines, including Bézier- and
B-spline curves, box splines and subdivision surfaces. Next, w
e'll shift our focus to applications of smooth spline surfaces
of arbitrary manifold topology within the realm of computer gra
phics. Finally, a couple of aspects and applications of splines
in the context of numerical methods will be discussed. Expect
more illustrations than equations, and in addition a couple of
(interactive) live software demos!\r\nFor further information a
bout the seminar, please visit this webpage [https://dmi.unibas
.ch/de/forschung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1167@dmi.unibas.ch
DTSTAMP:20210627T202614
DTSTART;TZID=Europe/Zurich:20210423T110000
DTEND;TZID=Europe/Zurich:20210423T120000
SUMMARY:Seminar in Numerical Analysis: Markus Melenk (TU Wien)
LOCATION:
DESCRIPTION:We consider the Helmholtz equation with piecewise analytic coef
ficients at large wavenumber k > 0. The interface where the coe
fficients jump is assumed to be analytic. We develop a k-explic
it regularity theory for the solution that takes the form of a
decomposition into two components: the first component is a pie
cewise analytic, but highly oscillatory function and the second
one has finite regularity but features wavenumber-independent
bounds. This decomposition generalizes earlier decompositions o
f [MS10, MS11, EM11, MSP12], which considered the Helmholtz equ
ation with constant coefficients, to the case of (piecewise) an
alytic coefficients. This regularity theory allows to show for
high order Galerkin discretizations (hp-FEM) of the Helmholtz e
quation that quasi-optimality is reached if (a) the approximati
on order p is selected as p = O(log k) and (b) the mesh size h
is such that kh/p is sufficiently small. This extends the resul
ts of [MS10, MS11, EM11, MSP12] about the onset of quasi-optima
lity of hp-FEM for the Helmholtz equation to the case of the he
terogeneous Helmholtz equation.\r\nJoint work with: Maximilian
Bernkopf (TU Wien), Théophile Chaumont-Frelet (Inria).\r\nRefe
rences [EM11] S. Esterhazy and J.M. Melenk, On stability
of discretizations of the Helmholtz equation, in: Numerical An
alysis of Multiscale Problems, Graham et al., eds, Springer 201
2 [MS10] J.M. Melenk and S. Sauter, Convergence Analysis f
or Finite Element Discretizations of the Helmholtz equation wit
h Dirichlet-to-Neumann boundary conditions, Math. Comp. 79:1871
–1914, 2010 [MS11] J.M. Melenk and S. Sauter, Wavenumber
explicit convergence analysis for finite element discretizatio
ns of the Helmholtz equation, SIAM J. Numer. Anal., 49:1210–1
243, 2011 [MSP12] J.M. Melenk, S. Sauter, A. Parsania, Generali
zed DG-methods for highly indefinite Helmholtz problems, J. Sci
. Comp. 57:536–581, 2013\r\nFor further information about the
seminar, please visit this webpage [https://dmi.unibas.ch/de/f
orschung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1154@dmi.unibas.ch
DTSTAMP:20210627T202454
DTSTART;TZID=Europe/Zurich:20210416T110000
DTEND;TZID=Europe/Zurich:20210416T120000
SUMMARY:Seminar in Numerical Analysis: Guy Gilboa (Technion - Israel Institute of Technology)
LOCATION:
DESCRIPTION:Recent studies on nonlinear eigenvalue problems show surprising
analogies to harmonic analysis (e.g. Fourier or wavelets). In
this talk we first show the total-variation (TV) transform, ba
sed on the TV gradient flow, and its application in image proce
ssing. We then present new results on analyzing gradient flows
of homogeneous nonlinear operators (such as the p-Laplacian). O
ur framework allows a thorough investigation of Dynamic-Mode-De
composition (DMD), a central dimensionality reduction method fo
r time series data. We present analytic solutions of simple non
linear cases, reveal shortcomings of DMD and propose improved d
ecomposition methods.\r\nFor further information about the semi
nar, please visit this webpage [https://dmi.unibas.ch/de/forsch
ung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1161@dmi.unibas.ch
DTSTAMP:20210627T202314
DTSTART;TZID=Europe/Zurich:20210409T110000
DTEND;TZID=Europe/Zurich:20210409T120000
SUMMARY:Seminar in Numerical Analysis: Barbara Kaltenbacher (Universität Klagenfurt)
LOCATION:
DESCRIPTION:High intensity (focused) ultrasound HIFU is used in numerous me
dical and industrial applications ranging from litotripsy and t
hermotherapy via ultrasound cleaning and welding to sonochemist
ry. In this talk, we will highlight two computational aspects
related to the relevant nonlinear acoustic phenomena, namely a
bsorbing boundary conditions for the treatment of open domain p
roblems; optimization tasks for ultrasound focusing. Strictly
speaking, acoustic sound propagation takes place in full space
or at least in a domain that is typically much larger than the
region of interest Ω. To restrict attention to a bounded domai
n Ω, e.g, for computational purposes, artificial reflections o
n the boundary ∂Ω have to be avoided. This can be done by i
mposing so-called absorbing boundary conditions ABC that induce
dissipation of outgoing waves. Here it will turn out to be cru
cial to take into account nonlinearity of the PDE also in these
ABC. This is joint work with Igor Shevchenko (Imperial Colleg
e London). In the context of applications in HIFU, focusing of
nonlinearly propagating waves amounts to optimization problems
. The design of ultrasound excitation via piezoelectric transdu
cers leads to a boundary control problem; focusing high intensi
ty ultrasound by a silicone lens requires shape optimization. F
or both problem classes, we will discuss the derivation of grad
ient information in order to formulate optimality conditions an
d drive numerical optimization methods. This is joint work wit
h Christian Clason (University of Duisburg-Essen), Vanja Nikoli
ć (TU München), and Gunther Peichl (University of Graz). Fi
nally we will provide an outlook on imaging with nonlinearly ac
oustic waves, which amounts to identifying spatially varying
coefficients (sound speed and/or coefficient of nonlinearity) i
n the Westervelt equation. This is recent joint work with Masa
hiro Yamamoto (University of Tokyo) and William Rundell (Texas
A&M University).\r\nFor further information about the seminar,
please visit this webpage [https://dmi.unibas.ch/de/forschung/m
athematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1127@dmi.unibas.ch
DTSTAMP:20210623T193706
DTSTART;TZID=Europe/Zurich:20201211T110000
DTEND;TZID=Europe/Zurich:20201211T120000
SUMMARY:Seminar in Numerical Analysis: Heiko Gimperlein (Heriot-Watt University)
LOCATION:
DESCRIPTION:Diffusion processes beyond Brownian motion have recently attrac
ted significant interest from different communities in mathemat
ics, the physical and biological sciences. They are described b
y partial differential equations involving nonlocal operators w
ith singular non-integrable kernels, such as fractional Laplaci
ans. This talk discusses the challenges of their approximation
by finite elements and discusses our recent results on the a pr
iori analysis of h, p and hp-versions for the integral fraction
al Laplacian, as well as their preconditioning. \r\nFor furthe
r information about the seminar, please visit this webpage [htt
ps://dmi.unibas.ch/de/forschung/mathematik/seminar-in-numerical
-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1098@dmi.unibas.ch
DTSTAMP:20210623T193612
DTSTART;TZID=Europe/Zurich:20201204T110000
DTEND;TZID=Europe/Zurich:20201204T120000
SUMMARY:Seminar in Numerical Analysis: Bastian von Harrach (Goethe-Universität Frankfurt)
LOCATION:
DESCRIPTION:We derive a simple criterion that ensures uniqueness, Lipschitz
stability and global convergence of Newton's method for the f
inite dimensional zero-finding problem of a continuously diffe
rentiable, pointwise convex and monotonic function. Our criter
ion merely requires to evaluate the directional derivative of
the forward function at finitely many evaluation points and fo
r finitely many directions.\r\nWe then demonstrate that this re
sult can be used to prove uniqueness, stability and global con
vergence for an inverse coefficient problem with finitely many
measurements. We consider the problem of determining an unkno
wn inverse Robin transmission coefficient in an elliptic PDE. U
sing a relation to monotonicity and localized potentials techn
iques, we show that a piecewise-constant coefficient on an a-p
riori known partition with a-priori known bounds is uniquely d
etermined by finitely many boundary measurements and that it c
an be uniquely and stably reconstructed by a globally converge
nt Newton iteration. We derive a constructive method to identi
fy these boundary measurements, calculate the stability consta
nt and give a numerical example.\r\n For further information ab
out the seminar, please visit this webpage [https://dmi.unibas.
ch/de/forschung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1103@dmi.unibas.ch
DTSTAMP:20210623T193501
DTSTART;TZID=Europe/Zurich:20201120T110000
DTEND;TZID=Europe/Zurich:20201120T120000
SUMMARY:Seminar in Numerical Analysis: Gabriel Lord (Radboud University Nijmegen)
LOCATION:
DESCRIPTION:We examine how time step adaptivity can be used to control pot
ential instability arising from non-Lipschitz terms for st
ochastic partial differential equations (SPDEs). I will give
a brief introduction to SPDEs and illustrate the stability issu
e with the standard uniform step Euler method to motivate the
adaptive method. I will present a strong convergence result
and outline the steps of the proof. To illustrate the method
we examine the stochastic Allen-Cahn, Swift-Hohenberg, Kur
amoto-Sivashinsky equations and finally will discuss a potentia
l use of the adaptivity for the deterministic system. This is j
oint work with Stuart Campbell.
END:VEVENT
BEGIN:VEVENT
UID:news-1102@dmi.unibas.ch
DTSTAMP:20210623T193410
DTSTART;TZID=Europe/Zurich:20201113T110000
DTEND;TZID=Europe/Zurich:20201113T120000
SUMMARY:Seminar in Numerical Analysis: Gilles Vilmart (Université de Genève)
LOCATION:
DESCRIPTION:
END:VEVENT
BEGIN:VEVENT
UID:news-1101@dmi.unibas.ch
DTSTAMP:20210623T193336
DTSTART;TZID=Europe/Zurich:20201030T110000
DTEND;TZID=Europe/Zurich:20201030T120000
SUMMARY:Seminar in Numerical Analysis: Alfio Borzi (Universität Würzburg)
LOCATION:
DESCRIPTION:The Liouville equation is the fundamental building block of mod
els that govern the evolution of density functions of multi-pa
rticle systems. These models include different Fokker-Planck a
nd Boltzmann equations that arise in many application fields r
anging from gas dynamics to pedestrians' motion where the need
arises to control these systems.\r\nThis talk provides an intr
oduction to the formulation and solution of optimal control pr
oblems governed by the Liouville equation and related models.
The purpose of this framework is the design of robust controls
to steer the motion of particles, pedestrians, etc., where th
ese agents are represented in terms of density functions. For
this purpose, expected-value cost functionals are considered t
hat include attracting potentials and different costs of the co
ntrols, whereas the control mechanism in the governing models
is part of the drift or is included in a collision term.\r\nIn
this talk, theoretical and numerical results concerning ensemb
le optimal control problems with Liouville, Fokker-Planck and
linear Boltzmann equations are presented.\r\n For further info
rmation about the seminar, please visit this webpage [https://d
mi.unibas.ch/de/forschung/mathematik/seminar-in-numerical-analy
sis/].
END:VEVENT
BEGIN:VEVENT
UID:news-1099@dmi.unibas.ch
DTSTAMP:20210623T193201
DTSTART;TZID=Europe/Zurich:20201016T110000
DTEND;TZID=Europe/Zurich:20201016T120000
SUMMARY:Seminar in Numerical Analysis: Jürgen Dölz (Universität Bonn)
LOCATION:
DESCRIPTION:We propose an efficient algorithm for the treatment of Volterra
integral operators based on H2-matrix compression techniques.
The algorithm is built in an evolutionary manner, and therefore
, is well suited for the problems, where the right hand side de
pends on the solution itself and is not known for all time step
s a priori. The resulting algorithm is of linear complexity O(N
) w.r.t. to the number of time steps, and requires O(N) active
memory. The memory consumption can be reduced to O(log N) for t
he kernels of convolution type using the Laplace inversion tech
niques introduced by Lubich et al; the connection to the FOCQ a
lgorithm is drawn. We demonstrate the effectiveness of our algo
rithm on a series of numerical examples.\r\n For further inform
ation about the seminar, please visit this webpage [https://dm
i.unibas.ch/de/forschung/mathematik/seminar-in-numerical-analys
is/].
END:VEVENT
BEGIN:VEVENT
UID:news-939@dmi.unibas.ch
DTSTAMP:20191125T124030
DTSTART;TZID=Europe/Zurich:20191220T110000
DTEND;TZID=Europe/Zurich:20191220T120000
SUMMARY:Seminar in Numerical Analysis: Kristin Kirchner (ETH Zürich)
LOCATION:
DESCRIPTION:Many models in spatial statistics are based on Gaussian Matérn
fields. Motivated by the relation between this class of Gaussi
an random fields and stochastic partial differential equations
(PDEs), we consider the numerical solution of fractional-order
elliptic stochastic PDEs with additive spatial white noise on a
bounded Euclidean domain.We propose an approximation supported
by a rigorous error analysis which shows different notions of
convergence at explicit and sharp rates. We furthermore discuss
the computational complexity of the proposed method. Finally,
we present several numerical experiments, which attest the theo
retical outcomes, as well as a statistical application where we
use the method for inference, i.e., for parameter estimation g
iven data, and for spatial prediction.\r\nFor further informati
on about the seminar, please visit this webpage [https://dmi.u
nibas.ch/de/forschung/mathematik/seminar-in-numerical-analysis/
].
END:VEVENT
BEGIN:VEVENT
UID:news-931@dmi.unibas.ch
DTSTAMP:20191210T105146
DTSTART;TZID=Europe/Zurich:20191213T110000
DTEND;TZID=Europe/Zurich:20191213T120000
SUMMARY:Seminar in Numerical Analysis: Théophile Chaumont-Frelet (INRIA, Nice/Sophia Antipolis)
LOCATION:
DESCRIPTION:The Helmholtz equation models the propagation of a time-harmoni
c wave. It has received much attention since it is widely emplo
yed in applications, but still challenging to numerically simul
ate in the high-frequency regime. In this seminar, we focus on
acoustic waves for the sake of simplicity and consider finite
element discretizations. The main goal of the presentation is t
o highlight the improved performance of high order methods (as
compared to linear finite elements) when the frequency is large
. We will very briefly cover the zero-frequency case, that cor
responds to the well-studied Poisson equation. We take advantag
e of this classical setting to recall central concepts of the f
inite element theory such as quasi-optimality and interpolation
error. The second part of the seminar is devoted to the high-
frequency case. We show that without restrictive assumptions on
the mesh size, the finite element method is unstable, and quas
i-optimality is lost. We provide a detailed analysis, as well a
s numerical examples, which show that higher order methods are
less affected by this phenomena, and thus more suited to discre
tize high-frequency problems. Before drawing our main conclusi
ons, we briefly discuss advanced topics, such as the use of unf
itted meshes in highly heterogeneous media and mesh refinements
around re-entrant corners.\r\nFor further information about th
e seminar, please visit this webpage [https://dmi.unibas.ch/de
/forschung/mathematik/seminar-in-numerical-analysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-926@dmi.unibas.ch
DTSTAMP:20191130T081127
DTSTART;TZID=Europe/Zurich:20191206T110000
DTEND;TZID=Europe/Zurich:20191206T120000
SUMMARY:Seminar in Numerical Analysis: Ira Neitzel (Universität Bonn)
LOCATION:
DESCRIPTION:joint work with Dominik Hafemeyer, Florian Mannel and Boris Vex
ler\r\n We consider a convex optimal control problem governed
by a partial differential equation in one space dimension whic
h is controlled by a right-hand-side living in the space of fu
nctions with bounded variation. These functions tend to favor o
ptimal controls that are piecewise constant with often finitel
y many jump poins. We are interested in deriving finite elemen
t discretization error estimates for the controls when the sta
te ist discretized with usual piecewise linear finite elements,
and the controls is either variationally discrete or piecwise
constant. Due to the structure of the objective function, usu
al techniques for estimating the control error cannot be appli
ed. Instead, these have to be derived from (suboptimal) error e
stimates for the state, which can later be improved. \r\nFor fu
rther information about the seminar, please visit this webpage
[https://dmi.unibas.ch/de/forschung/mathematik/seminar-in-nume
rical-analysis/archive-list/].
END:VEVENT
BEGIN:VEVENT
UID:news-930@dmi.unibas.ch
DTSTAMP:20191107T104152
DTSTART;TZID=Europe/Zurich:20191115T110000
DTEND;TZID=Europe/Zurich:20191115T120000
SUMMARY:Seminar in Numerical Analysis: Michael Multerer (USI Lugano)
LOCATION:
DESCRIPTION:
END:VEVENT
BEGIN:VEVENT
UID:news-920@dmi.unibas.ch
DTSTAMP:20191104T081039
DTSTART;TZID=Europe/Zurich:20191108T110000
DTEND;TZID=Europe/Zurich:20191108T120000
SUMMARY:Seminar in Numerical Analysis: Omar Lakkis (University of Sussex)
LOCATION:
DESCRIPTION:Aposteriori error estimates provide a rigorous foundation for t
he derivation of efficient adaptive algorithms for the approxim
ation of solutions of partial differential equations (PDEs).
While the literature is rich with results for the approximation
of elliptic and parabolic PDEs, it is much less developed for
the hyperbolic equations such as the acoustic or elastic wave e
quations. In this talk, I will review some of the "standard"
aposteriori results for the scalar linear wave equation, includ
ing those of [1] and [2], and present recent improvements and f
urther developments to lower order Sobolev norms based on Baker
’s Trick [3] for backward Euler schemes. Subsequent focus w
ill be given to practically relevant methods such as Verlet, Co
sine, or Newmark methods, a popular example of which is the Lea
p-frog method [4].\r\nNotes: This is based on joint work with E
.H. Georgoulis, C. Makridakis and J.M. Virtanen.\r\nReferences:
\r\n[1] W. Bangerth and R. Rannacher, J. Comput. Acoust. 9(2):5
75–591, 2001.[2] C. Bernardi and E. Süli, Math. Models Metho
ds Appl. Sci. 15(2):199--225, 2005.[3] E. H. Georgoulis, O. Lak
kis, and C. Makridakis. IMA J. Numer. Anal., 33(4):1245–1264,
2013, http://arxiv.org/abs/1003.3641[4] E. H. Georgoulis, O. L
akkis, C. Makridakis, and J. M. Virtanen. SIAM J. Numer. Anal.,
54(1), 2016, http://arxiv.org/abs/1411.7572 \r\nFor further in
formation about the seminar, please visit this webpage [https:
//dmi.unibas.ch/de/forschung/mathematik/seminar-in-numerical-an
alysis/].
END:VEVENT
BEGIN:VEVENT
UID:news-932@dmi.unibas.ch
DTSTAMP:20191023T141150
DTSTART;TZID=Europe/Zurich:20191101T110000
DTEND;TZID=Europe/Zurich:20191101T120000
SUMMARY:Seminar in Numerical Analysis: Giacomo De Souza (EPFL)
LOCATION:
DESCRIPTION:Traditional explicit Runge--Kutta schemes, though computational
ly inexpensive, are inefficient for the integration of stiff or
dinary differential equations due to stability issues. Converse
ly, implicit schemes are stable but can be overly expensive due
to the solution of possibly large nonlinear systems with Newto
n-like methods, whose convergence is neither guaranteed for lar
ge time steps. Explicit stabilized schemes such as the Runge--K
utta--Chebyshev method (RKC) represent a viable compromise, a
s the width of their stability domain grows quadratically with
respect to the number of function evaluations, thus presenting
enhanced stability properties with a reasonable computational c
ost. These methods are particularly efficient for systems arisi
ng from the space discretization of parabolic partial different
ial equations (PDEs).The efficiency of these methods deteriorat
es as the system becomes stiffer, even if stiffness is induced
by only few degrees of freedom. In the framework of discretized
parabolic PDEs, the number of function evaluations has to be c
hosen inversely proportional to the smallest element size in or
der to achieve stability, thus largely wasting computational re
sources on locally-refined meshes. We first tackle this issue b
y replacing the right hand side of the PDE with an averaged for
ce, which is obtained by damping the high modes down using the
dissipative effect of the equation itself and which is cheap to
evaluate. Combining RKC methods with the averaged force we giv
e rise to multirate RKC schemes, for which the number of expens
ive function evaluations is independent of the small elements'
size.The stability properties of our method are demonstrated on
a model problem and numerical experiments confirm that the sta
bility bottleneck caused by a few of fine mesh elements can be
overcome without sacrificing accuracy.\r\nFor further informati
on about the seminar, please visit this webpage [https://dmi.u
nibas.ch/de/forschung/mathematik/seminar-in-numerical-analysis/
].
END:VEVENT
BEGIN:VEVENT
UID:news-887@dmi.unibas.ch
DTSTAMP:20190514T111137
DTSTART;TZID=Europe/Zurich:20190525T110000
DTEND;TZID=Europe/Zurich:20190524T120000
SUMMARY:Seminar in Numerical Analysis: Florian Faucher (Université de Pau)
LOCATION:
DESCRIPTION:We study the inverse problem associated with the propagation of
time-harmonic waves. In the seismic context, the available mea
surements correspond with partial reflection data, obtained fro
m one side illumination (only from the Earth surface). The inve
rse problem aims at recovering the subsurface Earth medium para
meters and we employ the Full Waveform Inversion (FWI) method,
which relies on an iterative minimization algorithm of the diff
erence between the measurement and simulation. \r\n\r\nWe inves
tigate the deployment of new devices developed in the acoustic
setting: the dual-sensors, which are able to capture both the p
ressure field and the vertical velocity of the waves. For solvi
ng the inverse problem, we define a new cost function, adapted
to these two types of data and based upon the reciprocity. We f
irst note that the stability of the problem can be shown to be
Lipschitz, assuming piecewise linear parameters. In addition, r
eciprocity waveform inversion allows a separation between the o
bservational and numerical acquisitions. In fact, the numerical
sources do not have to coincide with the observational ones, o
ffering new possibilities to create adapted computational acqui
sitions, consequently reducing the numerical cost. We illustrat
e our approach with three-dimensional medium reconstructions, w
here we start with minimal information on the target models. We
also extend the methodology for elasticity. \r\n\r\nEventually
, if time allows, we shall explore the model representation in
numerical seismic inversion, where the adaptive eigenspace meth
od appears as a promising approach to have a compromise between
number of unknowns and resolution. \r\n\r\nRefe
rences \r\n\r\n[1] G. Alessandrini, M. V. de Hoop, F. Faucher
, R. Gaburro and E. Sincich, Inverse problem for the Helmholtz
equation with Cauchy data: reconstruction with conditional well
-posedness driven iterative regularization, ESAIM: M2AN (2019).
\r\n\r\n[2] E. Beretta, M. V. De Hoop, F. Faucher, and O. Sc
herzer, Inverse boundary value problem for the Helmholtz equati
on: quantitative conditional Lipschitz stability estimates. SIA
M Journal on Mathematical Analysis, 48(6), pp.3962-3983 (2016).
\r\n\r\n[3] M. J. Grote, M. Kray, and U. Nahum, Adaptive eige
nspace method for inverse scattering problems in the frequency
domain. Inverse Problems, 33(2), 025006 (2017). \r\n\r\n[4] H
. Barucq, F. Faucher, and O. Scherzer, Eigenvector Model Descri
ptors for Solving an Inverse Problem of Helmholtz Equation. arX
iv preprint arXiv:1903.08991 (2019).For further information abo
ut the seminar, please visit this <link de forschung mathematik
seminar-in-numerical-analysis internal link in current>webpage
.
END:VEVENT
BEGIN:VEVENT
UID:news-885@dmi.unibas.ch
DTSTAMP:20190507T193740
DTSTART;TZID=Europe/Zurich:20190517T110000
DTEND;TZID=Europe/Zurich:20190517T120000
SUMMARY:Seminar in Numerical Analysis: Thomas Wihler (Universität Bern)
LOCATION:
DESCRIPTION:A wide variety of (fixed-point) iterative methods for the solut
ion of nonlinear equations (in Hilbert spaces) exists. In many
cases, such schemes can be interpreted as iterative local linea
rization methods, which can be obtained by applying a suitable
linear preconditioning operator to the original (nonlinear) equ
ation. Based on this observation, we will derive a unified abst
ract framework which recovers some prominent iterative schemes.
Furthermore, in the context of numerical solutions methods for
nonlinear partial differential equations, we propose a combina
tion of the iterative linearization approach and the classical
Galerkin discretization method, thereby giving rise to the so-c
alled iterative linearization Galerkin (ILG) methodology. Moreo
ver, still on an abstract level, based on elliptic reconstructi
on techniques, we derive a posteriori error estimates which sep
arately take into account the discretization and linearization
errors. Furthermore, we propose an adaptive algorithm, which pr
ovides an efficient interplay between these two effects. In add
ition, some iterative methods and numerical computations in the
specific context of finite element discretizations of quasilin
ear stationary conservation laws will be presented.For further
information about the seminar, please visit this <link de forsc
hung mathematik seminar-in-numerical-analysis internal link in
current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-861@dmi.unibas.ch
DTSTAMP:20190423T105445
DTSTART;TZID=Europe/Zurich:20190503T110000
DTEND;TZID=Europe/Zurich:20190503T120000
SUMMARY:Seminar in Numerical Analysis: Rémi Abgrall (Universität Zürich)
LOCATION:
DESCRIPTION:Since the work of B. Wendroff and P. Lax, we know what should b
e the correct form of the numerical approximation of conservat
ion law. We also know, after Hou and Le Floch, what kind of pr
oblems we are facing when the flux form is not respected. How
ever, this is not the end of the story. All these works use a o
ne dimensional way of thinking: the main player is the normal
flux across cell interfaces. In addition there are several exc
ellent numerical methods that do not fit the form of the lax W
endroff theorem. In that talk, I will introduce a more general
setting and show that any reasonable scheme for conservation
law can be put in that framework. In addition, I will show tha
t an equivalent flux formulation, with a suitable definition o
f what is a flux, can be explicitly constructed (and compute
d), so that any reasonable scheme can be put in a finite volum
e form. I will end the talk by showing some applications: how
to systematically construct entropy stable scheme, or starting
from the non conservative form of a system-say the Euler eq
uations-, how to construct a suitable discretisation. And more
. This is a joint work with P. Bacigaluppi (now postdoc at ETH
) and S. Tokareva (now at Los Alamos).For further information
about the seminar, please visit this <link de forschung mathema
tik seminar-in-numerical-analysis internal link in current>webp
age.
END:VEVENT
BEGIN:VEVENT
UID:news-857@dmi.unibas.ch
DTSTAMP:20190402T151711
DTSTART;TZID=Europe/Zurich:20190412T110000
DTEND;TZID=Europe/Zurich:20190412T120000
SUMMARY:Seminar in Numerical Analysis: Chris Stolk (University of Amsterdam)
LOCATION:
DESCRIPTION:In this talk I discuss a recently developed finite difference d
iscretisation of the Helmholtz equation and some solution metho
ds for the resulting linear systems. In high-frequency Helmholt
z problems, pollution errors, due to numerical dispersion, are
a main source of error. We will show that such errors can be st
rongly reduced compared to other schemes, including high-order
finite elements, by selecting coefficients for the discrete sys
tem that maximise the accuracy of geometrical optics phases and
amplitudes. Such low dispersion schemes are of interest by the
mselves, but can also be used to improve the efficiency of mult
igrid schemes. Computation times for a solver combining a multi
grid method with domain decomposition compare well to those of
alternative methods.For further information about the seminar,
please visit this <link de forschung mathematik seminar-in-nume
rical-analysis internal link in current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-844@dmi.unibas.ch
DTSTAMP:20190320T092425
DTSTART;TZID=Europe/Zurich:20190329T110000
DTEND;TZID=Europe/Zurich:20190329T120000
SUMMARY:Seminar in Numerical Analysis: Robert Scheichl (Universität Heidelberg)
LOCATION:
DESCRIPTION:Sample-based multilevel uncertainty quantification tools, such
as multilevel Monte Carlo, multilevel quasi-Monte Carlo or mul
tilevel stochastic collocation, have recently gained huge popu
larity due to their potential to efficiently compute robust es
timates of quantities of interest (QoI) derived from PDE model
s that are subject to uncertainties in the input data (coeffic
ients, boundary conditions, geometry, etc). Especially for pro
blems with low regularity, they are asymptotically optimal in
that they can provide statistics about such QoIs at (asymptoti
cally) the same cost as it takes to compute one sample to the
target accuracy. However, when the data uncertainty is localise
d at random locations, such as for manufacturing defects in co
mposite materials, the cost per sample can be reduced signific
antly by adapting the spatial discretisation individually for
each sample. Moreover, the adaptive process typically produces
coarser approximations that can be used directly for the mult
ilevel uncertainty quantification. In this talk, I will presen
t two novel developments that aim to exploit these ideas. In t
he first part I will present Continuous Level Monte Carlo (CLM
C), a generalisation of multilevel Monte Carlo (MLMC) to a con
tinuous framework where the level parameter is a continuous var
iable. This provides a natural framework to use sample-wise ad
aptive refinement strategy, with a goal-oriented error estimat
or as our new level parameter. We introduce a practical CLMC e
stimator (and algorithm) and prove a complexity theorem showin
g the same rate of complexity as for MLMC. Also, we show that
it is possible to make the CLMC estimator unbiased with respec
t to the true quantity of interest. Finally, we provide two nu
merical experiments which test the CLMC framework alongside a
sample-wise adaptive refinement strategy, showing clear gains
over a standard MLMC approach with uniform grid hierarchies. In
the second part, I will show how to extend the sample-adaptiv
e strategy to multilevel stochastic collocation (MLSC) methods
providing a complexity estimate and numerical experiments for
a MLSC method that is fully adaptive in the dimension, in the
polynomial degrees and in the spatial discretisation. This is
joint work with Gianluca Detommaso (Bath), Tim Dodwell (Exete
r) and Jens Lang (Darmstadt).For further information about the
seminar, please visit this <link de forschung mathematik semina
r-in-numerical-analysis internal link in current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-829@dmi.unibas.ch
DTSTAMP:20190225T091300
DTSTART;TZID=Europe/Zurich:20190301T110000
DTEND;TZID=Europe/Zurich:20190301T120000
SUMMARY:Seminar in Numerical Analysis: Markus Zimmermann (Technische Universität München)
LOCATION:
DESCRIPTION:Solution spaces are sets of engineering solutions, i.e., design
s that satisfy all engineering requirements. Seeking solution s
paces rather than just one possibly optimal solution is numeric
ally challenging, but it can significantly simplify the develop
ment of systems in the presence of uncertainty and complexity.
For different system components, solution spaces are decomposed
into independent target regions that enable distributed develo
pment work and encompass uncertainty without particular underly
ing uncertainty model. A basic stochastic algorithm to maximize
so-called box-shaped solution spaces is presented. Two recent
extensions are discussed: first, representations as Cartesian p
roduct of two- and higher-dimensional spaces and, second, so-ca
lled solution-compensation spaces, where design variables are g
rouped according to the order in which they need to be specifie
d. Applications to vehicle development for crash and driving dy
namics are presented.For further information about the seminar,
please visit this <link de forschung mathematik seminar-in-num
erical-analysis internal link in current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-823@dmi.unibas.ch
DTSTAMP:20190214T222417
DTSTART;TZID=Europe/Zurich:20190222T110000
DTEND;TZID=Europe/Zurich:20190222T120000
SUMMARY:Seminar in Numerical Analysis: Edwin Mai (Universität der Bundeswehr München)
LOCATION:
DESCRIPTION:With an increasing range of applications, Shape Optimisation pr
oblems receive more and more interest in the engineering commun
ity, while solving such problems is still a demanding task. In
this talk the example of a Stokes channel flow with the objecti
ve to reduce the energy dissipation is considered, on which an
optimise-then-discretize approach shall be applied. Starting wi
th a gradient descent method, based on the analytical shape der
ivative and the adjoint approach, an initial optimisation proce
dure is discussed and differences in the shape derivative repre
sentation and their numerical implications are highlighted. Sub
sequently a possible way to derive shape hessian information in
a so-called tangent-on-reverse method, i.e. combining the adjo
int and sensitivity approach, is introduced. The shape hessian
is utilised in a reduced SQP method for the equally constrained
channel flow problem comprising of the objective, PDE and addi
tional geometric constraints. In contrary to a one-shot approac
h the reduced approach requires the state and adjoint equations
to be solved exactly for each optimisation step. Finally, some
features of the numerical implementation using the finite elem
ent software package FEniCS and the obtained results are presen
ted to show superiority of using hessian information.For furthe
r information about the seminar, please visit this <link de for
schung mathematik seminar-in-numerical-analysis internal link i
n current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-412@dmi.unibas.ch
DTSTAMP:20181206T092622
DTSTART;TZID=Europe/Zurich:20181214T110000
DTEND;TZID=Europe/Zurich:20181214T120000
SUMMARY:Seminar in Numerical Analysis: Martin Burger (Universität Erlangen)
LOCATION:
DESCRIPTION:In this talk we will discuss nonlinear spectral decom
positions in Banach spaces, which shed a new light on
multiscale methods in imaging and open new possibilities of
filtering techniques. We provide a novel geometric
interpretation of nonlinear eigenvalue problems in Banach
spaces and provide conditions under which gradient f
lows for norms or seminorms yield a spectral decompos
ition. We will see that under these conditions standa
rd variational schemes are equivalent to the gradient
flows for arbitrary large time step, recovering prev
ious results e.g. for the one dimensional total varia
tion flow as special cases. \r\n\r\nFor further information abo
ut the seminar, please visit this <link de forschung mathematik
seminar-in-numerical-analysis internal link in current>webpage
.
END:VEVENT
BEGIN:VEVENT
UID:news-409@dmi.unibas.ch
DTSTAMP:20181203T104240
DTSTART;TZID=Europe/Zurich:20181207T110000
DTEND;TZID=Europe/Zurich:20181207T120000
SUMMARY:Seminar in Numerical Analysis: Zakaria Belhachmi (LMIA - Université de Haute-Alsace)
LOCATION:
DESCRIPTION:We present some ideas on modelling with diffusion operators so
me PDEs based geometry inpainting problems. The objective is to
provide a closed loop continuous to discrete models. The loop
consists of an initial family of simple PDEs depending on som
e parameters selected at the discrete level from a posteriori
informations. The choice of these parameters modify dynamicall
y the system of equations and the resulting models converge (i
n the Gamma-convergence sense) to a limit -continuous- model th
at capture the jump set of the restaured image. We also discus
s the compression-based inpainting within this approach.For fu
rther information about the seminar, please visit this <link de
forschung mathematik seminar-in-numerical-analysis external-li
nk-new-window internal link in current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-393@dmi.unibas.ch
DTSTAMP:20181116T170413
DTSTART;TZID=Europe/Zurich:20181123T110000
DTEND;TZID=Europe/Zurich:20181123T120000
SUMMARY:Seminar in Numerical Analysis: Assyr Abdulle (EPFL)
LOCATION:
DESCRIPTION:In this talk we discuss several challenges that arise in Bayesi
an inference for ordinary and partial differential equations.
The numerical solvers used to compute the forward model of suc
h problems induce a propagation of the discretization error in
to the posterior measure for the parameters of interest. This
uncertainty originating from the numerical approximation error
can be accounted for using probabilistic numerical methods. N
ew probabilistic numerical methods for ordinary differential e
quations that share geometric properties of the true solution
will be presented in the first part of this talk. In the se
cond part of the talk, we will discuss a Bayesian approach for
inverse problems involving elliptic partial differential equat
ions with multiple scales. Computing repeated forward problems
in a multiscale context is computationnally too expensive and
we propose a new strategy based on the use of "effective"
forward models originating from homogenization theory. Conver
gence of the true posterior distribution for the parameters of
interest towards the homogenized posterior is established via
G-convergence for the Hellinger metric. A computational appro
ach based on numerical homogenization and reduced basis method
s is proposed for an efficient evaluation of the forward model
in a Markov Chain Monte-Carlo procedure. References: A.
Abdulle, G. Garegnani, Random time step probabilistic methods f
or uncertainty quantification in chaotic and geometric numeric
al integration, Preprint (2018), submitted for publication. A
. Abdulle, A. Di Blasio, Numerical homogenization and model ord
er reduction for multiscale inverse problems, to appear in SIA
M MMS. A. Abdulle, A. Di Blasio, A Bayesian numerical homogeni
zation method for elliptic multiscale inverse problems, Prepri
nt (2018), submitted for publication. For further information
about the seminar, please visit this <link de forschung mathema
tik seminar-in-numerical-analysis internal link in current>webp
age.
END:VEVENT
BEGIN:VEVENT
UID:news-307@dmi.unibas.ch
DTSTAMP:20181024T123048
DTSTART;TZID=Europe/Zurich:20181116T113000
DTEND;TZID=Europe/Zurich:20181116T123000
SUMMARY:Seminar in Numerical Analysis: Ludovic Métivier (Université Grenoble Alpes)
LOCATION:
DESCRIPTION:Full waveform inversion (FWI) is a powerful high resolution sei
smic imaging method, used in the academy for global and region
al scale imaging, and in the oil & gas industry for exploratio
n purposes. It can be understood as a PDE constrained optimiza
tion problem: the misfit between recorded seismic data and syn
thetic seismic data computed as the solution of a wave propaga
tion problem is reduced over a space of parameters controlling
the wave propagation. One of the main limitation of FWI is it
s dependency on the accuracy of the initial guess of the solut
ion. This limitation is due to the non-convexity of the standa
rd least-squares misfit function used to measure the discrepan
cy between recorded and synthetic data, and the use of local o
ptimization techniques to reduce this misfit. In recent studie
s, we have studied the interest for using a misfit function ba
sed on an optimal transport distance to mitigate this issue. T
he convexity of this distance with respect to shifted patterns
is the main reason why we are interested in this distance, as
it can be seen as a proxy for the convexity with respect to t
he wave velocities we want to reconstruct. In this talk, we
will give an overview of this work, starting by introducing ba
sic concepts on optimal transport, before detailing the diffic
ulties for using the optimal transport distance in the framewo
rk of FWI, and reviewing the solutions we have proposed.\r\n\r\
nFor further information about the seminar, please visit this <
link de forschung mathematik seminar-in-numerical-analysis exte
rnal-link-new-window internal link in current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-359@dmi.unibas.ch
DTSTAMP:20181031T192501
DTSTART;TZID=Europe/Zurich:20181109T110000
DTEND;TZID=Europe/Zurich:20181109T120000
SUMMARY:Seminar in Numerical Analysis: Stefan Sauter (Universität Zürich)
LOCATION:
DESCRIPTION:In our talk we consider the Maxwell equations in the frequency
domain, discretized by Nédélec-hp-finite elements. We develop
a stability and convergence analysis which is explicit with r
espect to the wave number k, the mesh size h, and the local po
lynomial degree p. It turns out that, for the choice p>=log(k)
, the discretization does not suffer from the so-called pollut
ion effect. This is known for high-frequency acoustic scatterin
g. However, the analysis of Maxwell equations requires the dev
elopment of twelve additional theoretical tools which we call "
the twelve apostels". In our talk, we explain these "apostels"
and how they are needed to prove the stability and convergence
of our method.\r\n\r\nThis talk comprises joint work with Prof
. Markus Melenk, TU Wien. \r\n\r\nFor further information about
the seminar, please visit this <link de forschung mathematik s
eminar-in-numerical-analysis internal link in current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-350@dmi.unibas.ch
DTSTAMP:20181023T144703
DTSTART;TZID=Europe/Zurich:20181102T110000
DTEND;TZID=Europe/Zurich:20181102T120000
SUMMARY:Seminar in Numerical Analysis: Maryna Kachanovska (ENSTA ParisTech)
LOCATION:
DESCRIPTION:In this work we consider the problem of the sound propagation
in a bronchial network. Asymptotically, this phenomenon can be
modelled by a weighted wave equation posed on a fractal (i.e.
self-similar) 1D tree. The principal difficulty for the nume
rical resolution of the problem is the 'infiniteness' of the g
eometry. To deal with this issue, we will present transparent
boundary conditions, used to truncate the computational domai
n to a finite subtree.\r\n\r\nThe construction of such transpa
rent conditions relies on the approximation of the Dirichlet-t
o-Neumann (DtN) operator, whose symbol is a meromorphic functi
on that satisfies a certain non-linear functional equation. W
e present two approaches to approximate the DtN in the time do
main, alternative to the low-order absorbing boundary condition
s, which appear inefficient in this case.\r\n\r\n The first ap
proach stems from the use of the convolution quadrature (cf. [
Lubich 1988], [Banjai, Lubich, Sayas 2016]), which consists in
constructing an exact DtN for a semi-discretized in time probl
em. In this case the combination of the explicit leapfrog meth
od for the volumic terms and the implicit trapezoid rule for t
he boundary terms leads to a second-order scheme stable under
the classical CFL condition.\r\n\r\nThe second approach is mot
ivated by the Engquist-Majda ABCs (cf. [Engquist, Majda 1977])
, and consists in approximating the DtN by local operators, obt
ained from the truncation of the meromorphic series which repr
esents the symbol of the DtN. We show how the respective error
can be controlled and provide some complexity estimates.\r\n\
r\nThis is a joint work with Patrick Joly (INRIA, France) and A
drien Semin (TU Darmstadt, Germany). \r\n\r\nFor further inform
ation about the seminar, please visit this <link de forschung m
athematik seminar-in-numerical-analysis internal link in curren
t>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-344@dmi.unibas.ch
DTSTAMP:20181012T175433
DTSTART;TZID=Europe/Zurich:20181019T110000
DTEND;TZID=Europe/Zurich:20181019T120000
SUMMARY:Seminar in Numerical Analysis: Martin Rumpf (Universität Bonn)
LOCATION:
DESCRIPTION:We investigate a generalization of cubic splines to Riemannian
manifolds. Spline curves are defined as minimizers of the splin
e energy - a combination of the Riemannian path energy and the
time integral of the squared covariant derivative of the path v
elocity - under suitable interpolation conditions. A variationa
l time discretization for the spline energy leads to a constrai
ned optimization problem over discrete paths on the manifold. E
xistence of continuous and discrete spline curves is establishe
d using the direct method in the calculus of variations. Furthe
rmore, the convergence of discrete spline paths to a continuous
spline curve follows from the Γ-convergence of the discrete t
o the continuous spline energy. Finally, selected example setti
ngs are discussed, including splines on embedded finite-dimensi
onal manifolds, on a high-dimensional manifold of discrete shel
ls with applications in surface processing, and on the infinite
-dimensional shape manifold of viscous rods. This is based on j
oint work with Behrend Heeren and Benedikt Wirth.\r\n\r\nFor fu
rther information about the seminar, please visit this <link de
forschung mathematik seminar-in-numerical-analysis internal li
nk in current>webpage.
END:VEVENT
BEGIN:VEVENT
UID:news-223@dmi.unibas.ch
DTSTAMP:20180716T181333
DTSTART;TZID=Europe/Zurich:20180601T110000
DTEND;TZID=Europe/Zurich:20180601T120000
SUMMARY:Seminar in Numerical Analysis: Kenneth Duru (Ludwig-Maximilians-Universität München)
LOCATION:
DESCRIPTION:High order accurate and explicit time-stable solvers are well s
uited for hyperbolic wave propagation problems. However, becau
se of the complexities of real geometries, internal interfaces,
nonlinear boundary/interface conditions and the presence of di
sparate spatial and temporal scales present in real media and s
ources, discontinuities and sharp wave fronts become fundamenta
l features of the solutions. Thus, high order accuracy, geometr
ically flexible and adaptive numerical algorithms are critical
for high fidelity and efficient simulations of wave phenomena i
n many applications. I will present a physics-based numerical f
lux suitable for inter-element and boundary conditions in disco
ntinuous Galerkin approximations of first order hyperbolic PDEs
. Using this physics-based numerical penalty-flux, we will deve
lop a provably energy-stable discontinuous Galerkin approximati
ons of the elastic waves in complex and discontinuous media. B
y construction the numerical flux is upwind and yields a discre
te energy estimate analogous to the continuous energy estimate.
The discrete energy estimates hold for conforming and non-conf
orming curvilinear elements. The ability to handle non-conformi
ng curvilinear meshes allows for flexible adaptive mesh refinem
ent strategies. The numerical scheme have been implemented in E
xaHyPE, a simulation engine for hyperbolic PDEs on adaptive str
uctured meshes, for exa-scale supercomputers. I will show 3D nu
merical experiments demonstrating stability and high order accu
racy. Finally, we present a large scale geophysical regional wa
ve propagation problem in a heterogeneous Earth model with geol
ogically constrained media heterogeneity and geometrically comp
lex free-surface topography.
END:VEVENT
BEGIN:VEVENT
UID:news-222@dmi.unibas.ch
DTSTAMP:20180716T180942
DTSTART;TZID=Europe/Zurich:20180525T113000
DTEND;TZID=Europe/Zurich:20180525T123000
SUMMARY:Seminar in Numerical Analysis: Ludovic Métivier (Université Grenoble Alpes)
LOCATION:
DESCRIPTION:Full waveform inversion (FWI) is a powerful high resolution sei
smic imaging method, used in the academy for global and region
al scale imaging, and in the oil & gas industry for exploratio
n purposes. It can be understood as a PDE constrained optimiza
tion problem: the misfit between recorded seismic data and syn
thetic seismic data computed as the solution of a wave propaga
tion problem is reduced over a space of parameters controlling
the wave propagation. One of the main limitation of FWI is it
s dependency on the accuracy of the initial guess of the solut
ion. This limitation is due to the non-convexity of the standar
d least-squares misfit function used to measure the discrepanc
y between recorded and synthetic data, and the use of local op
timization techniques to reduce this misfit. In recent studies
, we have studied the interest for using a misfit function bas
ed on an optimal transport distance to mitigate this issue. Th
e convexity of this distance with respect to shifted patterns
is the main reason why we are interested in this distance, as
it can be seen as a proxy for the convexity with respect to th
e wave velocities we want to reconstruct. In this talk, we wi
ll give an overview of this work, starting by introducing basic
concepts on optimal transport, before detailing the difficult
ies for using the optimal transport distance in the framework
of FWI, and reviewing the solutions we have proposed.
END:VEVENT
BEGIN:VEVENT
UID:news-221@dmi.unibas.ch
DTSTAMP:20180716T175831
DTSTART;TZID=Europe/Zurich:20180518T110000
DTEND;TZID=Europe/Zurich:20180518T120000
SUMMARY:Seminar in Numerical Analysis: Holger Fröning (Universität Heidelberg)
LOCATION:
DESCRIPTION:We are observing a continuous increase in concurrency and heter
ogeneity for computing systems of any scale, ranging from small
mobile devices to huge datacenters, and driven by a steady dem
and for more computing power. One of the prime examples for an
application with virtually unlimited computational requirements
is machine learning, in particular deep neural networks (DNN).
At the level of data-centers, DNN training has already led to
a ubiquitous use of graphics processing units (GPUs), forming a
prime example for specialization for computational improvement
. Still, this application is strongly hindered by insufficient
compute power and by scalability limitations. Contrary, mobile
architectures for DNN inference are still nascent, and a large
amount of proposals have been published in the recent years. Bo
th applications, training and inference, can furthermore benefi
t a lot from algorithmic optimizations to reduce the computatio
nal requirements. This talk presents a short introduction of th
e application, a summary of our observations, and our own resea
rch on reduced precision by extreme forms of quantizations. Fin
ally, this talk will offer some opinions on anticipated researc
h problems.
END:VEVENT
BEGIN:VEVENT
UID:news-220@dmi.unibas.ch
DTSTAMP:20180716T175646
DTSTART;TZID=Europe/Zurich:20180504T110000
DTEND;TZID=Europe/Zurich:20180504T120000
SUMMARY:Seminar in Numerical Analysis: Jan Hamaekers (Fraunhofer SCAI)
LOCATION:
DESCRIPTION:In this talk, we introduce a new scheme for the efficient numer
ical treatment of the electronic Schrödinger equation for mole
cules. It is based on the combination of a many-body expansion,
which corresponds to the so-called bond order dissection Anova
approach, with a hierarchy of basis sets of increasing order.
Here, the energy is represented as a finite sum of contribution
s associated to subsets of nuclei and basis sets in a telescopi
ng sum like fashion. Under the assumption of data locality of t
he electronic density (nearsightedness of electronic matter), t
he terms of this expansion decay rapidly and higher terms may b
e neglected. We further extend the approach in a dimension-adap
tive fashion to generate quasi-optimal approximations, i.e. a s
pecific truncation of the hierarchical series such that the tot
al benefit is maximized for a fixed amount of costs. This way,
we are able to achieve substantial speed up factors compared to
conventional first principles methods depending on the molecul
ar system under consideration. In particular, the method can de
al efficiently with molecular systems which include only a smal
l active part that needs to be described by accurate but expens
ive models. Finally, we discuss to apply such a multi-level man
y-body decomposition in the context of machine learning for man
y-body systems.
END:VEVENT
BEGIN:VEVENT
UID:news-219@dmi.unibas.ch
DTSTAMP:20180716T175418
DTSTART;TZID=Europe/Zurich:20180427T110000
DTEND;TZID=Europe/Zurich:20180427T120000
SUMMARY:Seminar in Numerical Analysis: Pierre-Henri Tournier (UPMC - University Pierre and Marie Curie)
LOCATION:
DESCRIPTION:This work deals with preconditioning the time-harmonic Maxwell
equations with absorption, where the preconditioner is construc
ted using two-level overlapping Additive Schwarz Domain Decompo
sition, and the PDE is discretised using finite-element methods
of fixed, arbitrary order. The theory shows that if the absorp
tion is large enough, and if the subdomain and coarse mesh diam
eters are chosen appropriately, then classical two-level overla
pping Additive Schwarz Domain Decomposition preconditioning per
forms optimally – in the sense that GMRES converges in a wave
number-independent number of iterations – for the problem wit
h absorption. This work is an extension of the theory proposed
in [1] for the Helmholtz equation. Numerical experiments illust
rate this theoretical result and also (i) explore replacing the
PEC boundary conditions on the subdomains by impedance boundar
y conditions, and (ii) show that the preconditioner for the pro
blem with absorption is also an effective preconditioner for th
e problem with no absorption. The numerical results include exa
mples arising from applications: a problem with absorption aris
ing from medical imaging shows the robustness of the preconditi
oner against heterogeneity, and a scattering problem by the COB
RA cavity shows good scalability of the preconditioner with up
to 3000 processors. Finally, additional numerical results for t
he elastic wave equation are presented for benchmarks in seismi
c inversion.\r\n\r\n[1] I. G. Graham, E. A. Spence, and E. Vain
ikko. Domain decomposition preconditioning for high-frequency H
elmholtz problems with absorption. Mathematics of Computation,
86(307):2089–2127, 2017.
END:VEVENT
BEGIN:VEVENT
UID:news-218@dmi.unibas.ch
DTSTAMP:20180716T180010
DTSTART;TZID=Europe/Zurich:20180413T110000
DTEND;TZID=Europe/Zurich:20180413T120000
SUMMARY:Seminar in Numerical Analysis: Philipp Morgenstern (Leibniz Universität Hannover)
LOCATION:
DESCRIPTION:We introduce a mesh refinement algorithm for the Adaptive Isoge
ometric Method using multivariate T-splines. We investigate lin
ear independence of the T-splines, nestedness of the T-spline s
paces, and linear complexity in the sense of a uniform upper bo
und on the ratio of generated and marked elements, which is cru
cial for a later proof of rate-optimality of the method. Altoge
ther, this work paves the way for a provably rate-optimal Adapt
ive Isogeometric Method with T-splines in any space dimension.\
r\n\r\nAs an outlook to future work, we outline an approach for
the handling of zero knot intervals and multiple lines in the
interior of the domain, which are used in CAD applications for
controlling the continuity of the spline functions, and we also
sketch basic ideas for the local refinement of two-dimensional
meshes that do not have tensor-product structure.
END:VEVENT
BEGIN:VEVENT
UID:news-217@dmi.unibas.ch
DTSTAMP:20180716T174831
DTSTART;TZID=Europe/Zurich:20180323T110000
DTEND;TZID=Europe/Zurich:20180323T120000
SUMMARY:Seminar in Numerical Analysis: Gregor Gantner (TU Wien)
LOCATION:
DESCRIPTION:Since the advent of isogeometric analysis (IGA) in 2005, the fi
nite element method (FEM) and the boundary element method (BEM)
with splines have become an active field of research. The cent
ral idea of IGA is to use the same functions for the approximat
ion of the solution of the considered partial differential equa
tion (PDE) as for the representation of the problem geometry in
computer aided design (CAD). Usually, CAD is based on tensor-p
roduct splines. To allow for adaptive refinement, several exten
sions of these have emerged, e.g., hierarchical splines, T-spli
nes, and LR-splines. In view of geometry induced generic singul
arities and the fact that isogeometric methods employ higher-or
der ansatz functions, the gain of adaptive refinement (resp. lo
ss for uniform refinement) is huge.\r\n\r\nIn this talk, we fir
st consider an adaptive FEM with hierarchical splines of arbitr
ary degree for linear elliptic PDE systems of second order with
Dirichlet boundary condition for arbitrary dimension d≥2. We
assume that the problem geometry can be parametrized over the
d-dimensional unit cube. We propose a refinement strategy to ge
nerate a sequence of locally refined meshes and corresponding d
iscrete solutions. Adaptivity is driven by some weighted-residu
al a posteriori error estimator. In [1], we proved linear conve
rgence of the error estimator with optimal algebraic rate.\r\n\
r\nNext, we consider an adaptive BEM with hierarchical splines
of arbitrary degree for weakly-singular integral equations of t
he first kind that arise from the solution of linear elliptic P
DE systems of second order with constant coefficients and Diric
hlet boundary condition. We assume that the boundary of the geo
metry is the union of surfaces that can be parametrized over th
e unit square. Again, we propose a refinement strategy to gener
ate a sequence of locally refined meshes and corresponding disc
rete solutions, where adaptivity is driven by some weighted-res
idual a posteriori error estimator. In [2], we proved linear co
nvergence of the error estimator with optimal algebraic rate. I
n contrast to prior works, which are restricted to the Laplace
model problem, our analysis allows for arbitrary elliptic PDE o
perators of second order with constant coefficients.\r\n\r\nFin
ally, for one-dimensional boundaries, we investigate an adaptiv
e BEM with standard splines instead of hierarchical splines. We
modify the corresponding algorithm so that it additionally use
s knot multiplicity increase which results in local smoothness
reduction of the ansatz space. In [3], we proved linear converg
ence of the employed weighted-residual error estimator with opt
imal algebraic rate.\r\n\r\nREFERENCES\r\n[1] G. Gantner, D. Ha
berlik, and Dirk Praetorius, Adaptive IGAFEM with optimal conve
rgence rates: Hierarchical B-splines. Math. Mod. Meth. in Appl.
S., Vol. 27, 2017.\r\n\r\n[2] G. Gantner, Optimal adaptivity f
or splines in finite and boundary element methods, PhD thesis,
TU Wien, 2017.\r\n\r\n[3] Michael Feischl, Gregor Gantner, Alex
ander Haberl, and Dirk Praetorius. Adaptive 2D IGA boundary ele
ment methods. Eng. Anal. Bound. Elem., Vol. 62, 2016.
END:VEVENT
BEGIN:VEVENT
UID:news-224@dmi.unibas.ch
DTSTAMP:20180716T181506
DTSTART;TZID=Europe/Zurich:20171110T110000
DTEND;TZID=Europe/Zurich:20171110T120000
SUMMARY:Seminar in Numerical Analysis: Martin Eigel (WIAS Berlin)
LOCATION:
DESCRIPTION:The Stochastic Galerkin FEM (SGFEM) is a common method to numer
ically solve PDEs with random data with the aim to obtain a fu
nctional representation of the stochastic solution. As with an
y spectral method, the curse of dimensionality renders the app
roach very challenging whenever the randomness depends on a la
rge or even infinite set of parameters. This makes function sp
ace adaptation and model reduction strategies a necessity. We
review adaptive SGFEM based on reliable a posteriori error est
imators for the affine and the lognormal cases. As an alternat
ive to a sparse discretisation, the representation in a hierar
chical tensor format is examined. Moreover, as an application o
f the result, we present an adaptive method for explicit sampl
ing-free Bayesian inversion.
END:VEVENT
BEGIN:VEVENT
UID:news-225@dmi.unibas.ch
DTSTAMP:20180716T181648
DTSTART;TZID=Europe/Zurich:20171103T110000
DTEND;TZID=Europe/Zurich:20171103T120000
SUMMARY:Seminar in Numerical Analysis: Herbert Egger (TU Darmstadt)
LOCATION:
DESCRIPTION:We consider the simulation of gas transport through pipe networ
ks. An appropriate mixed variational formulation is proposed to
take into account the coupling conditions at pipe junctions au
tomatically. This allows to obtain energy stable and mass conse
rving discretization schemes by Galerkin projection. A mixed fi
nite element method is briefly discussed as a particular choice
. We also discuss the preservation of further structural proper
ties, like uniform exponential stability and the correct approx
imation of asymptotic regimes.
END:VEVENT
BEGIN:VEVENT
UID:news-226@dmi.unibas.ch
DTSTAMP:20180716T181833
DTSTART;TZID=Europe/Zurich:20171006T110000
DTEND;TZID=Europe/Zurich:20171006T120000
SUMMARY:Seminar in Numerical Analysis: Stephan Schmidt (Universität Würzburg)
LOCATION:
DESCRIPTION:Many PDE constrained optimization problems fall into the catego
ry of shape optimization, meaning the geometry of the domain i
s the unknown to be found. Most natural applications are drag
minimization in fluid dynamics, but many tomography and image
reconstruction problems also fall into this category.\r\n\r\nT
he talk introduces shape optimization as a special sub-class o
f PDE constraint optimization problems. The main focus here wi
ll be on generating Newton-like methods for large scale applic
ations. The key for this endeavor is the derivation of the shap
e Hessian, that is the second directional derivative of a cost
functional with respect to geometry changes in a weak form ba
sed on material derivatives instead of classical local shape d
erivatives. To avoid human errors, a computer aided derivation
system is also introduced.\r\n\r\nThe methodologies are tested
on problem from fluid dynamics and geometric inverse problems.
END:VEVENT
BEGIN:VEVENT
UID:news-227@dmi.unibas.ch
DTSTAMP:20180716T202427
DTSTART;TZID=Europe/Zurich:20170512T110000
DTEND;TZID=Europe/Zurich:20170512T120000
SUMMARY:Seminar in Numerical Analysis: Anatole von Lilienfeld (Universität Basel)
LOCATION:
DESCRIPTION:Many of the most relevant chemical properties of matter depend
explicitly on atomistic and electronic details, rendering a fi
rst principles approach to chemistry mandatory. Alas, even whe
n using high-performance computers, brute force high-throughpu
t screening of compounds is beyond any capacity for all but th
e simplest systems and properties due to the combinatorial nat
ure of chemical space, i.e. all compositional, constitutional,
and conformational isomers. Consequently, efficient explorati
on algorithms need to exploit all implicit redundancies presen
t in chemical space. I will discuss recently developed statist
ical learning approaches for interpolating quantum mechanical
observables in compositional and constitutional space.
END:VEVENT
BEGIN:VEVENT
UID:news-228@dmi.unibas.ch
DTSTAMP:20180716T202644
DTSTART;TZID=Europe/Zurich:20170505T110000
DTEND;TZID=Europe/Zurich:20170505T120000
SUMMARY:Seminar in Numerical Analysis: Sébastien Imperiale (INRIA)
LOCATION:
DESCRIPTION:In this talk we present an approach for local space-time discre
tisation of linear wave equations. Assuming that high order Ga
lerkin discontinuous method or spectral finite elements are us
ed we propose some (high order) time discretisation that can b
e chosen independently in each region of the domain of interest
. Each time discretisation is adapted to the mesh or physics c
onstraints. The different obtained schemes obtained are then c
oupled in a stable way by writing transmission conditions. A c
onvergence analysis will be presented, it is based upon energy
analysis.
END:VEVENT
BEGIN:VEVENT
UID:news-229@dmi.unibas.ch
DTSTAMP:20180716T202938
DTSTART;TZID=Europe/Zurich:20170421T110000
DTEND;TZID=Europe/Zurich:20170421T120000
SUMMARY:Seminar in Numerical Analysis: Martin Hanke-Bourgeois (Universität Mainz)
LOCATION:
DESCRIPTION:We reconsider the impact of small volume perturbations of the c
onductivity coefficient of second order elliptic equations in d
ivergence form. The asymptotic expansion of the associated Neum
ann-Dirichlet operators on bounded domains allows the developme
nt and analysis of sophisticated algorithms to solve correspond
ing inverse boundary value problems of impedance tomography. Ex
amples of such algorithms are the MUSIC scheme and the topologi
cal derivative. Novel applications include the incorporation of
discrete electrode models and the exploitation of multiple dri
ving frequencies.
END:VEVENT
BEGIN:VEVENT
UID:news-230@dmi.unibas.ch
DTSTAMP:20180716T203249
DTSTART;TZID=Europe/Zurich:20170407T110000
DTEND;TZID=Europe/Zurich:20170407T120000
SUMMARY:Seminar in Numerical Analysis: Stefan Kurz (TU Darmstadt)
LOCATION:
DESCRIPTION:Superconducting cavities are standard components of particle a
ccelerators. Their design is typically described by parametrize
d ellipses and determined by mathematical optimization. The si
mulation model is subject to demanding requirements, such as a
relative accuracy of 10−9 for the resonance frequency of th
e accelerating mode. Since the geometry and the electromagneti
c fields are smooth, an approach in the gist of isogeometric a
nalysis (IGA) suggests itself. The geometry is modeled by a NU
RBS mapping, while the electromagnetic fields are discretized
by the B-spline de Rahm complex [2]. An IGA finite element met
hod (FEM) for the Maxwell eigenvalue problem was investigated
and showed promising results [3]. For the same accuracy, the n
umber of required degrees of freedom was reduced by a factor 3
. . . 9 compared to classical FEM. However, CAD systems featur
e surface descriptions only, so the volumetric spline model ha
d to be created manually.\r\n\r\nTo live up to the promises of
IGA, namely closing the gap bewteen design and analysis, we s
uggest an IGA boundary element method (BEM). We will review th
e state-of-the-art of all relevant building blocks. We will ad
dress the B-spline de Rham complex on a boundary manifold, the
Galerkin discretization of the electric field integral equatio
n, and present a convergence result. We will discuss a recent
contour integral method [1] to solve the resulting non-linear
eigenvalue problem. Aspects of integrating so-called ”fast m
ethods” will also be presented, in particular Adaptive Cross
Approximation [5] and Calderón preconditioning [4].\r\n\r\n[
1] W.-J. Beyn. An integral method for solving nonlinear eigenva
lue problems. Linear Algebra Appl, 436(10):3839–3863, 2012.\r
\n\r\n[2] A. Buffa, G. Sangalli, and R. Vázquez. Isogeometric
analysis in electromagnetics: B-splines approximation. Comput
Method Appl M, 199:1143–1152, 2010.\r\n\r\n[3] J. Corno, C.
de Falco, H. De Gersem, and S. Schöps. Isogeometric simulatio
n of Lorentz detuning in superconducting accelerator cavities.
Comput Phys Commun, 201:1–7, February 2016.\r\n\r\n[4] J. Li
, D. Dault, B. Liu, Y. Tong, and B. Shanker. Subdivision based
isogeometric analysis technique for electric field integral e
quations for simply connected structures. J Comput Phys, 319:14
5–162, 2016.\r\n\r\n[5] B. Marussig, J. Zechner, G. Beer, an
d T.-P. Fries. Fast isogeometric boundary element method based
on independent field approximation. Comput Method Appl M, 284
:458–488, 2015.\r\n\r\nThe work of Stefan Kurz is supported b
y the ’Excellence Initiative’ of the German Federal and St
ate Governments and the Graduate School of Computational Engin
eering at Technische Universität Darmstadt.
END:VEVENT
BEGIN:VEVENT
UID:news-231@dmi.unibas.ch
DTSTAMP:20180716T203426
DTSTART;TZID=Europe/Zurich:20170324T110000
DTEND;TZID=Europe/Zurich:20170324T120000
SUMMARY:Seminar in Numerical Analysis: Jens Oettershagen (Universität Bonn)
LOCATION:
DESCRIPTION:In this talk, we discuss algorithms for multivariate integratio
n in reproducing kernel Hilbert spaces (RKHS). Here, we study
optimally weighted Monte Carlo integration in Sobolev spaces w
ith dominating mixed smoothness. Moreover, we consider integra
tion in spaces of analytic functions. To this end, we construc
t optimally weighted univariate quadrature rules with carfully
selected points and employ them within a generalized sparse g
rid to the problem of multivariate integration. Applications a
re given in econometrics and parametric differential equations
with affine linear diffusion coefficients.
END:VEVENT
BEGIN:VEVENT
UID:news-232@dmi.unibas.ch
DTSTAMP:20180716T203725
DTSTART;TZID=Europe/Zurich:20161216T110000
DTEND;TZID=Europe/Zurich:20161216T120000
SUMMARY:Seminar in Numerical Analysis: Christian Rieger (Universität Bonn)
LOCATION:
DESCRIPTION:In this talk, we will briefly discuss a general methodology of
approximation algorithms based on reproducing kernels and thei
r associated Hilbert spaces. We will outline how reproducing k
ernels naturally arise in many reconstruction problems.\r\n\r\
nFurthermore, we will present a deterministic a priori (often
exponential) convergence analysis via sampling inequalities whi
ch can be employed to analyze a large class of regularized rec
onstruction schemes.\r\n\r\nSuch an analysis enables us to der
ive a priori couplings of various discretization and regulariz
ation parameters. Such parameters can range from iteration num
bers in numerical linear algebra, numerical evaluation of inpu
t parameters to rounding errors.\r\n\r\nAn important issue is t
he choice of the reproducing kernel. We will discuss some impl
ications of such choices and address the problem of approximat
ing the solution of a parametric partial differential equation
using problem adapted kernels.\r\n\r\nThis is partly based on
joint work with M. Griebel and B. Zwicknagl (both Bonn Universi
ty).
END:VEVENT
BEGIN:VEVENT
UID:news-233@dmi.unibas.ch
DTSTAMP:20180716T203922
DTSTART;TZID=Europe/Zurich:20161209T110000
DTEND;TZID=Europe/Zurich:20161209T120000
SUMMARY:Seminar in Numerical Analysis: Daniel Peterseim (Universität Bonn)
LOCATION:
DESCRIPTION:This talk presents a variational approach for the numerical ho
mogenization of elliptic partial differential equations with ar
bitrary rough diffusion coefficients. The trial and test space
in this (Petrov-)Galerkin method are derived from linear fini
te elements on a coarse mesh of width H by local fine-scale co
rrection. The correction is based on the pre-computation of ce
ll problems on patches of diameter H log(1/H). The moderate ov
erlap of the patches suffices to prove O(H) convergence of the
method without any pre-asymptotic effects. The key step in th
e error analysis is the proof of the exponential decay of the
so-called fine-scale Green's function, i.e., the impulse respon
se of the variational equation in the absence of coarse-scale
finite element functions. The method allows the characterizati
on of effective coefficients on a given target scale of numeri
cal resolution. Among further applications of the approach are
pollution-free high-frequency scattering and explicit time st
epping on spatially adaptive meshes.
END:VEVENT
BEGIN:VEVENT
UID:news-234@dmi.unibas.ch
DTSTAMP:20180716T204059
DTSTART;TZID=Europe/Zurich:20161202T110000
DTEND;TZID=Europe/Zurich:20161202T120000
SUMMARY:Seminar in Numerical Analysis: Xavier Antoine (Université de Lorraine)
LOCATION:
DESCRIPTION:The aim of this talk is to introduce nonoverlapping Schwarz dom
ain decomposition methods for time harmonic waves (acoustics,
electromagnetism). In particular, we will focus on the constru
ction of rational Padé transmission boundary conditions to
get fast converging solvers for prospecting high frequency pro
blems. Some numerical simulations will illustrate the theoreti
cal developments. The methods have been implemented in a freel
y available software called GetDDM.
END:VEVENT
BEGIN:VEVENT
UID:news-235@dmi.unibas.ch
DTSTAMP:20180716T204408
DTSTART;TZID=Europe/Zurich:20161118T110000
DTEND;TZID=Europe/Zurich:20161118T120000
SUMMARY:Seminar in Numerical Analysis: Peter Zaspel (Universität Heidelberg / HITS)
LOCATION:
DESCRIPTION:Hierarchical matrices approximate specific types of dense matri
ces, e.g., from discretized integral equations, kernel-based a
pproximation and Gaussian process regression, leading to log-l
inear time complexity in dense matrix-vector products. To be a
ble to solve large-scale applications, H-matrix algorithms hav
e to be parallelized. A special kind of parallel hardware are
many-core processors, e.g. graphics processing units (GPUs). T
he parallelization of H-matrices on many-core processors is di
fficult due to the complex nature of the underlying algorithms
that need to be mapped to rather simple parallel operations.\r
\n\r\nWe are interested to use these many-core processors for t
he full H-matrix construction and application process. A motiv
ation for this interest lies in the well-known claim that futu
re standard processors will evolve towards many-core hardware,
anyway. In order to be prepared for this development, we want
to discuss many-core parallel formulations of classical H-mat
rix algorithms and adaptive cross approximations.\r\n\r\nIn the
presentation, the use of H-matrices is motivated by the model
application of kernel-based approximation for the solution of
parametric PDEs, e.g. PDEs with stochastic coefficients. The m
ain part of the talk will be dedicated to the challenges of H-
matrix parallelizations on many-core hardware with the specifi
c model hardware of GPUs. We propose a set of parallelization
strategies which overcome most of these challenges. Benchmarks
of our implementation are used to explain the effect of diffe
rent parallel formulations of the algorithms.
END:VEVENT
BEGIN:VEVENT
UID:news-236@dmi.unibas.ch
DTSTAMP:20180716T204738
DTSTART;TZID=Europe/Zurich:20161104T110000
DTEND;TZID=Europe/Zurich:20161104T120000
SUMMARY:Seminar in Numerical Analysis: Thorsten Hohage (Universität Göttingen)
LOCATION:
DESCRIPTION:Inverse problems usually consist in finding causes for observed
effects. The essential difficulty in solving inverse problems
is ill-posedness: causes typically do not depend continuously
on their effects although vice versa effects typically depend
continuously on causes. To avoid infinite amplification of me
asurement errors, regularization methods have to be employed t
o solve inverse problems numerically. The aim of regularizatio
n theory is to analyze the convergence and speed of convergenc
e of such methods as the noise level tends to 0.\r\n\r\nOver t
he last years Variational Source Conditions (VSCs) have become
a standard assumption for the analysis of these methods. Compa
red to spectral source conditions they have a number of advant
ages: They can be used for general nonquadratic penalty and da
ta fidelity terms, lead to simpler proofs, are often not only
sufficient, but even necessary for certain convergence rates,
and they do not involve the derivative of the forward operator
(and hence do not require restrictive assumptions such as a t
angential cone condition). However, so far only few sufficient
conditions for VSCs for specific inverse problems are known.\r
\n\r\nTo overcome this drawback, we propose a general strategy
for the verification of VSCs, which consists of two sufficient
conditions: One of them describes the smoothness of the solut
ion, and the other one the degree of ill-posedness of the oper
ator. For a number of important linear inverse problems this l
eads to equivalent characterizations of VSCs in terms of Besov
spaces and necessary and sufficient conditions for rates of c
onvergence. We also discuss the application of our strategy to
nonlinear parameter identification and inverse medium scatter
ing problems where it provides sufficient conditions for VSCs i
n terms of standard function spaces.
END:VEVENT
BEGIN:VEVENT
UID:news-237@dmi.unibas.ch
DTSTAMP:20180716T204932
DTSTART;TZID=Europe/Zurich:20161021T110000
DTEND;TZID=Europe/Zurich:20161021T120000
SUMMARY:Seminar in Numerical Analysis: Steffen Börm (Universität Kiel)
LOCATION:
DESCRIPTION:In the context of stochastic partial differential equation, we
are frequently faced with equations in high-dimensional domain
s. In order to obtain efficient numerical methods for these eq
uations, we have to take local regularity properties of the so
lution into account, e.g., by using locally refined finite ele
ment meshes. Extending standard meshing algorithms to higher d
imensions poses a significant challenge.\r\n\r\nWe propose an a
lternative: the Galerkin trial space is constructed using a pa
rtition of unity. By multiplying local cut-off functions with
polynomials, we can obtain discretizations of arbitrary order,
and local grid refinement can be realized by reducing the sup
ports of the cut-off functions. The main challenge lies in the
construction of the corresponding system matrix, since even de
termining the sparsity pattern involves interactions between c
ut-off functions on different levels of the mesh hierarchy.\r\
n\r\nOur approach leads to a sparse system matrix, the basis fu
nctions are convenient tensor products of functions on lower-d
imensional domains, and local regularity can be exploited by v
ariable-order interpolation in order to obtain close to optima
l complexity.
END:VEVENT
BEGIN:VEVENT
UID:news-238@dmi.unibas.ch
DTSTAMP:20180716T205111
DTSTART;TZID=Europe/Zurich:20161007T110000
DTEND;TZID=Europe/Zurich:20161007T120000
SUMMARY:Seminar in Numerical Analysis: Fabio Baruffa (Leibniz-Rechenzentrum, München)
LOCATION:
DESCRIPTION:In the framework of the Intel Parallel Computing Centre at Leib
niz Supercomputing Centre (LRZ), Fabio Baruffa will present re
cent results on the performance optimization of Gadget-3 on mu
lti and many-core computer architectures, including the new In
tel Xeon Phi processor of second generation, codenamed Knights
Landing (KNL). An overview of results for node-level scalabil
ity, vector efficiency and performance are presented here. Our
work is based on an isolated, representative code kernel, whe
re threading parallelism, data locality and vectorization effi
ciency was improved. The node-level parallel efficiency improv
ed by factors ranging from 5x to 16x on Haswell and KNL nodes,
respectively. Moreover, a vectorization efficiency of 80% (6.6
x) on a prototypical target loop of the code is obtained witho
ut programming using intrinsics instructions.
END:VEVENT
BEGIN:VEVENT
UID:news-239@dmi.unibas.ch
DTSTAMP:20180716T205454
DTSTART;TZID=Europe/Zurich:20160527T110000
DTEND;TZID=Europe/Zurich:20160527T120000
SUMMARY:Seminar in Numerical Analysis: Ana Djurdjevac (FU Berlin)
LOCATION:
DESCRIPTION:Sometimes the partial differential equations with random coeffi
cients can be better formulated on moving domains, especially i
n biological applications. We will introduce and analyse the ad
vection-diffusion equations with random coefficients on moving
hypersurfaces. Under suitable regularity assumptions, using Ban
ach-Necas-Babuska theorem, we will prove existence and uniquene
ss of the weak solution and also we will give some regularity r
esults about the solution. For discretization in space, we will
apply the evolving surface finite element method. In order to
deal with uncertainty, we will use Monte Carlo method. Furtherm
ore, we plan to discuss the case when the velocity of the hyper
suraface is random.This is a joint work with Charles M. Elliott
(University of Warwick, UK), Ralf Kornhuber (Free University B
erlin, Germany) and Thomas Ranner (University of Leeds, UK).
END:VEVENT
BEGIN:VEVENT
UID:news-240@dmi.unibas.ch
DTSTAMP:20180716T205652
DTSTART;TZID=Europe/Zurich:20160520T110000
DTEND;TZID=Europe/Zurich:20160520T120000
SUMMARY:Seminar in Numerical Analysis: Julien Diaz (INRIA)
LOCATION:
DESCRIPTION:Seismic imaging techniques such as Reverse Time Migration (RTM)
or Full Waveform Inversion (FWI) can be applied in time domai
n or in frequency domain. After havind recalled the principle
of RTM and discussed the advantages and drawbacks of both appr
oaches, we focus on frequential domain. We show the usefulness
of Discontinuous for solving acoustic and elastodynamic wav
e equation, et we apply the Interior Penalty Discontinuous Gal
erkin method (IPDG) to the modelling of elasto/acoustic coupli
ng. We then present an alternative method, the Hydridizable Di
scontinuous Galerkin method (HDG), which reduces the number of
unknowns of the global linear system thanks to the introductio
n of a Lagrange multiplier defined only on the faces of the ce
lls of the mesh. We illustrate the efficiency of HDG with resp
ect to IPDG thanks to comparisons on academic and industrial t
est cases.
END:VEVENT
BEGIN:VEVENT
UID:news-241@dmi.unibas.ch
DTSTAMP:20180716T205837
DTSTART;TZID=Europe/Zurich:20160513T110000
DTEND;TZID=Europe/Zurich:20160513T120000
SUMMARY:Seminar in Numerical Analysis: Frédéric Nataf (CNRS Paris 6)
LOCATION:
DESCRIPTION:Optimized Schwarz methods (OSM) are very popular methods which
were introduced by P.L. Lions for elliptic problems and B. Des
pres for propagative wave phenomena. One drawback is the lack
of theoretical results for variable coefficients problems and
overlapping decompositions. We build here a coarse space for w
hich the convergence rate of the two-level method is guarantee
d regardless of the regularity of the coefficients. We do this
by introducing a symmetrized variant of the ORAS (Optimized R
estricted Additive Schwarz) algorithm. Numerical results on ne
arly incompressible elasticity and Stokes system are shown for
systems with hundred of millions of degrees of freedom on high
performance computers.
END:VEVENT
BEGIN:VEVENT
UID:news-242@dmi.unibas.ch
DTSTAMP:20180716T210058
DTSTART;TZID=Europe/Zurich:20160429T110000
DTEND;TZID=Europe/Zurich:20160429T120000
SUMMARY:Seminar in Numerical Analysis: Andreas Rieder (Karlsruhe Institute of Technology)
LOCATION:
DESCRIPTION:Electrical impedance tomography is a non-invasive method for im
aging the electrical conductivity of an object from voltage mea
surements on its surface. This inverse problem suffers threefol
d: it is highly nonlinear, severely ill-posed, and highly under
-determined. To obtain yet reasonable reconstructions, maximal
information needs to be extracted from the data. We will presen
t and analyze a holistic Newton-type method which addresses all
these challenges. Finally, we demonstrate the performance of t
his concept numerically for simulated and measured data.
END:VEVENT
BEGIN:VEVENT
UID:news-246@dmi.unibas.ch
DTSTAMP:20180716T211540
DTSTART;TZID=Europe/Zurich:20151204T110000
DTEND;TZID=Europe/Zurich:20151204T120000
SUMMARY:Seminar in Numerical Analysis: Sebastian Ullmann (TU Darmstadt)
LOCATION:
DESCRIPTION:Surrogate models can be used to decrease the computational cos
t for uncertainty quantification in the context of parabolic P
DEs with stochastic data. Projection based reduced-order model
ing provides surrogates which inherit the spatial structure of
the solution as well as the underlying physics. In my talk I f
ocus on the type of models that is derived by a Galerkin proje
ction onto a proper orthogonal decomposition (POD) of snapshot
s of the solution.\r\n\r\nStandard techniques assume that all
snapshots use one and the same spatial mesh. I present a gener
alization for unsteady adaptive finite elements, where the mes
h can change from time step to time step and, in the case of s
tochastic sampling, from realization to realization. I will ans
wer the following questions: How can the coding effort for cre
ating such a reduced-order model be minimized? How can the uni
on of all snapshot meshes be avoided? What is the main differe
nce between static and adaptive snapshots in the error analysi
s of Galerkin reduced-order models?\r\n\r\nAs a numerical test
case I consider a two-dimensional viscous Burgers equation wi
th smooth initial data multiplied by a normally distributed ra
ndom variable. The results illustrate the convergence properti
es with respect to the number of POD basis functions and indic
ate possible savings of computation time.
END:VEVENT
BEGIN:VEVENT
UID:news-243@dmi.unibas.ch
DTSTAMP:20180716T210827
DTSTART;TZID=Europe/Zurich:20151201T121000
DTEND;TZID=Europe/Zurich:20151201T131500
SUMMARY:Seminar CCCS: Prof. M. Griebel (University of Bonn)
LOCATION:
DESCRIPTION:
END:VEVENT
BEGIN:VEVENT
UID:news-244@dmi.unibas.ch
DTSTAMP:20180716T211049
DTSTART;TZID=Europe/Zurich:20151120T110000
DTEND;TZID=Europe/Zurich:20151120T120000
SUMMARY:Seminar in Numerical Analysis: Stefan Sauter (University of Zürich)
LOCATION:
DESCRIPTION:In this talk we consider an intrinsic approach for the direct c
omputation of the fluxes for problems in potential theory. We p
resent a general method for the derivation of intrinsic conform
ing and non-conforming finite element spaces and appropriate li
fting operators for the evaluation of the right-hand side from
abstract theoretical principles related to the second Strang Le
mma. This intrinsic finite element method is analyzed and conve
rgence with optimal order is proved.
END:VEVENT
BEGIN:VEVENT
UID:news-245@dmi.unibas.ch
DTSTAMP:20180716T211238
DTSTART;TZID=Europe/Zurich:20151106T110000
DTEND;TZID=Europe/Zurich:20151106T120000
SUMMARY:Seminar in Numerical Analysis: Sanna Mönkölä (University of Jyväskylä)
LOCATION:
DESCRIPTION:A wide range of numerical methods have been used for solving t
ime-harmonic wave equations. Typically, the methods are based o
n complex-valued formulations leading to large-scale indefinit
e linear equations. An alternative is to simulate time-depende
nt equations in time, until the time-harmonic solution is reac
hed. However, this approach suffers from poor convergence, par
ticularly in the case of large wavenumbers and complicated dom
ains. We accelerate the convergence rate by employing a contro
llability method. The problem is formulated as a least-squares
optimization problem, which is solved by the conjugate gradie
nt algorithm. The efficiency of the method relies on smart dis
cretizations. For spatial discretization we use the spectral e
lement method or the discrete exterior calculus, and for time e
volution we consider leap-frog style discretization with non-u
niform timesteps or higher-order schemes. For constructing spa
tially isotropic grids for complex geometries, we use non-unif
orm polygonal structures imitating the close packing in crysta
l lattices.
END:VEVENT
BEGIN:VEVENT
UID:news-247@dmi.unibas.ch
DTSTAMP:20180716T211947
DTSTART;TZID=Europe/Zurich:20151016T110000
DTEND;TZID=Europe/Zurich:20151016T120000
SUMMARY:Seminar in Numerical Analysis: Wolfgang Hornfeck (DLR Köln)
LOCATION:
DESCRIPTION:Crystallography, as seen from a mathematician's viewpoint, is m
ainly concerned with "Sphere Packings, Lattices and Groups" (a
s is the title of a famous book of Conway and Sloane). It is c
lear, however, that there are many other connections between c
rystallography and mathematics, ranging from more general appl
ications of graph theory to more special ones such as differen
tial geometry. In my talk I want to present some explorations
into some applications of uniform distribution theory within a
crystallographic context.
END:VEVENT
BEGIN:VEVENT
UID:news-248@dmi.unibas.ch
DTSTAMP:20180716T212124
DTSTART;TZID=Europe/Zurich:20151009T110000
DTEND;TZID=Europe/Zurich:20151009T120000
SUMMARY:Seminar in Numerical Analysis: Andrea Barth (University of Stuttgart)
LOCATION:
DESCRIPTION:Multilevel Monte Carlo methods were introduced to lower the co
mputational complexity for the calculation of, for instance, t
he expectation of a random quantity. More precisely, in compar
ison to standard Monte Carlo methods the computational complex
ity is (asymptotically) equal to the calculation of one sample
of the problem on the finest grid used. The price to pay for
this increase in efficiency is that the problem needs to be sol
ved not only on one (fine) grid, but on a hierarchy of discret
izations. This implies first that the solution has to be repre
sented on all grids and second, that the variance of the detai
l (the difference of approximate solutions on two consecutive
grids) converges with the refinement of the grid.\r\n\r\nIn th
is talk, I will give an introduction to multilevel Monte Carlo
methods in the case when the variance of the detail does not
converge uniformly. The idea is illustrated by the calculation
of the expectation for an elliptic problem with a random multi
scale coefficient and then extended to approximations of stati
stical solutions to the Navier-Stokes equations.
END:VEVENT
BEGIN:VEVENT
UID:news-249@dmi.unibas.ch
DTSTAMP:20180716T212633
DTSTART;TZID=Europe/Zurich:20150925T110000
DTEND;TZID=Europe/Zurich:20150925T120000
SUMMARY:Seminar in Numerical Analysis: Francois Bouchut (Université Paris-Est)
LOCATION:
DESCRIPTION:We study approximations by conforming methods of the solution t
o variational inequalities which arise in the context of invisc
id incompressible Bingham type non-Newtonian fluid flows and of
the total variation flow problem.\r\n\r\nIn the general contex
t of a convex lower semi-continuous functional on a Hilbert spa
ce, we prove the convergence of time implicit space conforming
approximations, without viscosity and for non-smooth data. Then
we introduce a general class of total variation functionals, f
or which we can apply the regularization method. We consider th
e time implicit regularized, linearized or not, algorithms, and
prove their convergence for general total variation functional
s.
END:VEVENT
BEGIN:VEVENT
UID:news-250@dmi.unibas.ch
DTSTAMP:20180716T222212
DTSTART;TZID=Europe/Zurich:20150901T110000
DTEND;TZID=Europe/Zurich:20150901T120000
SUMMARY:Seminar in Numerical Analysis: Abdul-Lateef Haji-Ali (King Abdullah University)
LOCATION:
DESCRIPTION:I discuss using single level and multilevel Monte Carlo method
s to compute quantities of interests of a stochastic particle
system in the mean-field. In this context, the stochastic parti
cles follow a coupled system of Ito stochastic differential eq
uations (SDEs). Moreover, this stochastic particle system conv
erges to a stochastic mean-field limit as the number of partic
les tends to infinity.\r\n\r\nIn 2012, my Master thesis develo
ped different versions of Multilevel Monte Carlo (MLMC) for pa
rticle systems, both with respect to time steps and number of
particles and proposed using particle antithetic estimators fo
r MLMC. In that thesis, I showed moderate savings of MLMC compa
red to Monte Carlo. In this talk, I recall and expand on these
results, emphasizing the importance of antithetic estimators
in stochastic particle systems. I will finally conclude by pro
posing the use of our recent Multi-index Monte Carlo method to
obtain improved convergence rates.
END:VEVENT
BEGIN:VEVENT
UID:news-251@dmi.unibas.ch
DTSTAMP:20180716T213252
DTSTART;TZID=Europe/Zurich:20150529T110000
DTEND;TZID=Europe/Zurich:20150529T120000
SUMMARY:Seminar in Numerical Analysis: Victorita Dolean (University of Nice)
LOCATION:
DESCRIPTION:For linear problems, domain decomposition methods can be used d
irectly as iterative solvers, but also as preconditioners for
Krylov methods. In practice, Krylov acceleration is almost alw
ays used, since the Krylov method finds a much better residual
polynomial than the stationary iteration, and thus converges
much faster. We show in this work that also for non-linear pro
blems, domain decomposition methods can either be used directl
y as iterative solvers, or one can use them as preconditioners
for Newton’s method. For the concrete case of the parallel
Schwarz method, we show that we obtain a preconditioner we call
RASPEN (Restricted Additive Schwarz Preconditioned Exact Newt
on) which is similar to ASPIN (Additive Schwarz Preconditioned
Inexact Newton), but with all components directly defined by
the iterative method. This has the advantage that RASPEN alrea
dy converges when used as an iterative solver, in contrast to
ASPIN, and we thus get a substantially better preconditioner f
or Newton’s method. We illustrate our findings with numerica
l results on the Forchheimer equation and a non-linear diffusi
on problem.
END:VEVENT
BEGIN:VEVENT
UID:news-252@dmi.unibas.ch
DTSTAMP:20180716T213451
DTSTART;TZID=Europe/Zurich:20150522T110000
DTEND;TZID=Europe/Zurich:20150522T120000
SUMMARY:Seminar in Numerical Analysis: Stéphane Lanteri (Inria Sophia Antipolis)
LOCATION:
DESCRIPTION:
END:VEVENT
BEGIN:VEVENT
UID:news-253@dmi.unibas.ch
DTSTAMP:20180716T213628
DTSTART;TZID=Europe/Zurich:20150508T110000
DTEND;TZID=Europe/Zurich:20150508T120000
SUMMARY:Seminar in Numerical Analysis: Christian Stohrer (ENSTA ParisTech)
LOCATION:
DESCRIPTION:Electromagnetic phenomena can be modeled using Maxwell's equati
ons. In particular we are interested in harmonic electromagnet
ic waves propagating through a highly oscillatory material suc
h as e.g. fiber reinforced plastic. The permittivity and the p
ermeability of such materials vary on a microscopic length sca
le. The use of standard edge finite elements is of limited pro
fit, since the microscopic structure requires very refined mes
hes to provide satisfying approximations. This may easily resu
lt in computational costs difficult to manage. However, if one
is only interested in the effective behavior of the solution a
nd not in the microscopic details, homogenization techniques c
an be used to overcome these difficulties. In this talk we rev
iew first the results of analytical homogenization results for
Maxwell's equations. The goal of this theory is to replace th
e oscillatory material with an effective one, such that the ov
erall behavior of the solution remains unchanged. The solution
of the arising equations can be solved with standard numerica
l methods because the effective material depends no longer on
the micro scale. In the second part of the talk we propose a mu
ltiscale scheme following the framework of the finite element
heterogeneous multiscale method (FE-HMM). Contrary to the disc
retization of the analytically homogenized equation, no effect
ive coefficient must be precomputed beforehand. We prove that
the FE-HMM solution converges to the homogenized one for perio
dic materials and show some numerical experiments.\r\n\r\nThis
is a joint work with Sonia Fliss and Patrick Ciarlet.
END:VEVENT
BEGIN:VEVENT
UID:news-254@dmi.unibas.ch
DTSTAMP:20180716T213831
DTSTART;TZID=Europe/Zurich:20150424T110000
DTEND;TZID=Europe/Zurich:20150424T120000
SUMMARY:Seminar in Numerical Analysis: David Cohen (Umeå University)
LOCATION:
DESCRIPTION:A fully discrete approximation of one-dimensional nonlinear sto
chastic wave equations driven by multiplicative noise is presen
ted. A standard finite difference approximation is used in spac
e and a stochastic trigonometric method for the temporal approx
imation. This explicit time integrator allows for error bounds
uniformly in time and space. Moreover, uniform almost sure conv
ergence of the numerical solution is proved.\r\n\r\nThis is a j
oint work with Lluís Quer-Sardanyons, Universitat Autònoma de
Barcelona.
END:VEVENT
BEGIN:VEVENT
UID:news-255@dmi.unibas.ch
DTSTAMP:20180716T214015
DTSTART;TZID=Europe/Zurich:20150320T110000
DTEND;TZID=Europe/Zurich:20150320T120000
SUMMARY:Seminar in Numerical Analysis: Roland Griesmaier (Universität Würzburg)
LOCATION:
DESCRIPTION:One of the main themes of inverse scattering theory for time-ha
rmonic acoustic or electromagnetic waves is to determine infor
mation about unknown objects or inhomogeneous media from obser
vations of scattered waves away from these objects or outside
these media. Such inverse problems are typically non-linear an
d severely ill-posed.\r\n\r\nBesides standard regularization me
thods, which are often iterative, a completely different metho
dology - so-called qualitative reconstruction methods - has at
tracted a lot of interest recently. These algorithms recover s
pecific qualitative properties of scattering objects or anomal
ies inside a medium in a reliable and fast way. They avoid the
simulation of forward models and need no a priori information
on physical or topological properties of the unknown objects o
r inhomogeneities to be reconstructed. One of the drawbacks of
currently available qualitative reconstruction methods is the
large amount of data required by most of these algorithms. It
is usually assumed that measurement data of waves scattered b
y the unknown objects corresponding to infinitely many primary
waves are given - at least theoretically.\r\n\r\nWe consider t
he inverse source problem for the Helmholtz equation as a mean
s to provide a qualitative inversion algorithm for inverse sca
ttering problems for acoustic or electromagnetic waves with a s
ingle excitation only. Probing an ensemble of obstacles by jus
t one primary wave at a fixed frequency and measuring the far
field of the corresponding scattered wave, the inverse scatter
ing problem that we are interested in consists in reconstructi
ng the support of the scatterers. To this end we rewrite the s
cattering problem as a source problem and apply two recently d
eveloped algorithms - the inverse Radon approximation and the
convex scattering support - to recover information on the supp
ort of the corresponding source. The first method builds upon
a windowed Fourier transform of the far field data followed by
a filtered backprojection, and although this procedure yields
a rather blurry reconstruction, it can be applied to identify
the number and the positions of well separated source componen
ts. This information is then utilized to split the far field i
nto individual far field patterns radiated by each of the well
separated source components using a Galerkin scheme. Finally
we compute the convex scattering supports associated to the in
dividual source components as a reconstruction of the individu
al scatterers. We discuss this algorithm and present numerical
results.
END:VEVENT
BEGIN:VEVENT
UID:news-256@dmi.unibas.ch
DTSTAMP:20180716T214209
DTSTART;TZID=Europe/Zurich:20150220T110000
DTEND;TZID=Europe/Zurich:20150220T120000
SUMMARY:Seminar in Numerical Analysis: Timo Betcke (University College London)
LOCATION:
DESCRIPTION:The BEM++ boundary element library is a software project that w
as started in 2010 at University College London to provide an
open-source general purpose BEM library for a variety of appli
cation areas. In this talk we introduce the underlying design
concepts of the library and discuss several applications, incl
uding high-frequency preconditioning for ultrasound applicatio
ns, the solution of time-domain problems via convolution quadr
ature, light-scattering from ice crystals, and the solution of
coupled FEM/BEM problems with FEniCS and BEM++.
END:VEVENT
BEGIN:VEVENT
UID:news-257@dmi.unibas.ch
DTSTAMP:20180716T215147
DTSTART;TZID=Europe/Zurich:20141219T110000
DTEND;TZID=Europe/Zurich:20141219T120000
SUMMARY:Seminar in Numerical Analysis: Andrea Barth (Universität Stuttgart)
LOCATION:
DESCRIPTION:Multilevel Monte Carlo methods for multiscale problems
END:VEVENT
BEGIN:VEVENT
UID:news-258@dmi.unibas.ch
DTSTAMP:20180716T215328
DTSTART;TZID=Europe/Zurich:20141212T110000
DTEND;TZID=Europe/Zurich:20141212T120000
SUMMARY:Seminar in Numerical Analysis: Olaf Schenk (Universita della Svizzera italiana)
LOCATION:
DESCRIPTION:We will review the state-of-the art techniques in the parallel
direct solution of linear systems of equations and present sev
eral recent new research directions. This includes (i) fast me
thods for evaluating certain selected elements of a matrix fun
ction that can be used for solving the Kohn-Sham-equation with
out explicit diagonalization and (ii) stochastic optimization
problems under uncertainty from power grid problems from elect
rical power grid systems. Several algorithmic and performance
engineering advances are discussed to sove the underlying spar
se linear algebra problems. The new developments include novel
incomplete augmented multicore sparse factorizations, multicor
e- and GPU-based dense matrix implementations, and communicati
on-avoiding Krylov solvers. We also improve the interprocess c
ommunication on Cray systems to solve e.g. 24-hour horizon pow
er grid problems from electrical power grid systems of realist
ic size with up to 1.95 billion decision variables and 1.94 bi
llion constraints. Full-scale results are reported on Cray X
C30 and BG/Q, where we observe very good parallel efficienci
es and solution times within a operationally defined time inte
rval. To our knowledge, "real-time"-compatible performance on a
broad range of architectures for this class of problems has n
ot been possible prior to present work.
END:VEVENT
BEGIN:VEVENT
UID:news-259@dmi.unibas.ch
DTSTAMP:20180716T215544
DTSTART;TZID=Europe/Zurich:20141205T110000
DTEND;TZID=Europe/Zurich:20141205T120000
SUMMARY:Seminar in Numerical Analysis: Nakul Chitnis (Swiss Tropical Health Institute, Basel)
LOCATION:
DESCRIPTION:Malaria is an infectious disease, spread through mosquito bites
, that is responsible for substantial morbidity and mortality
around the world. In the last decade, through increased fundin
g and a global scale up of control interventions that target m
osquitoes, significant reductions in transmission and disease
burden have been achieved. However, these gains in public heal
th are faced with the twin threat of a decrease in funding for
malaria control and the development of resistance (physiologi
cal and behavioural) in mosquitoes.\r\n\r\nMathematical models
can help to determine more efficient combinations of existing
and new interventions in reducing malaria transmission and del
aying the spread of resistance. We present difference equation
models of mosquito population dynamics and malaria in mosquit
oes; and ordinary differential equation models of mosquito mov
ement and population dynamics. We analyse these models to prov
ide threshold conditions for the survival of mosquitoes and sh
ow the existence of invariant positive states; and run numeric
al simulations to provide quantitative comparisons of interven
tions that target mosquitoes with varying levels of resistance.
END:VEVENT
BEGIN:VEVENT
UID:news-260@dmi.unibas.ch
DTSTAMP:20180716T215716
DTSTART;TZID=Europe/Zurich:20141121T110000
DTEND;TZID=Europe/Zurich:20141121T120000
SUMMARY:Seminar in Numerical Analysis: Armin Iske (Universität Hamburg)
LOCATION:
DESCRIPTION:This talk discusses the utility of meshfree kernel techniques i
n adaptive finite volume particle methods (FVPM). To this end,
we give ten good reasons in favour of using kernel-based reco
nstructions in the recovery step of FVPM, where our discussion
addresses relevant computational aspects concerning numerical
stability and accuracy, as well as more specific points conce
rning efficient implementation. Special emphasis is finally pl
aced on morerecent advances in the construction of adaptive FV
PM, where WENO reconstructions by polyharmonic spline kernelsa
re used in combination with ADER flux evaluations to obtain hi
gh order methods for hyperbolic problems.
END:VEVENT
BEGIN:VEVENT
UID:news-261@dmi.unibas.ch
DTSTAMP:20180716T215846
DTSTART;TZID=Europe/Zurich:20141114T110000
DTEND;TZID=Europe/Zurich:20141114T120000
SUMMARY:Seminar in Numerical Analysis: Wolfgang Wendland (Universität Stuttgart)
LOCATION:
DESCRIPTION:The minimal energy problem for nonnegative charges on a closed
surface Γ in R^3 goes back to C.F. Gauss in 1839. The corresp
onding Riesz kernel is then weakly singular on Γ. If one cons
iders double layer potentials with dipole charges on Γ, the m
inimal energy problem then is based on hypersingular Riesz pot
entials in the form of Hadamard’s partie finie integral oper
ators defining pseudodifferential operators of positive degree
on smooth Γ. Existence and uniqueness results for the minimal
energy problem and a corresponding boundary element method wi
ll be presented.
END:VEVENT
BEGIN:VEVENT
UID:news-262@dmi.unibas.ch
DTSTAMP:20180716T220058
DTSTART;TZID=Europe/Zurich:20141107T110000
DTEND;TZID=Europe/Zurich:20141107T120000
SUMMARY:Seminar in Numerical Analysis: Frédéric Hecht (Université Pierre-et-Marie Curie Paris 6)
LOCATION:
DESCRIPTION:FreeFem++ is a free software for numerical resolution of partia
l differential equations using the finite elements method. Aft
er a short presentation of the possibilities of the software,
we will see through examples how to approach PDEs with mesh ad
aptation and parallel computing. These examples are among Piez
oelectric problemsThermal problems with thermal resistancesElas
ticity problemsProblems of fluid mechanics like incompressible
Navier-StokesProblem of melting and/or solidification of the ic
e. (Boussinesq with specific heat)
END:VEVENT
BEGIN:VEVENT
UID:news-263@dmi.unibas.ch
DTSTAMP:20180716T220342
DTSTART;TZID=Europe/Zurich:20141024T110000
DTEND;TZID=Europe/Zurich:20141024T120000
SUMMARY:Seminar in Numerical Analysis: Juliette Chabassier (INRIA-Bordeaux)
LOCATION:
DESCRIPTION:A family of fourth order coupled implicit-explicit schemes is p
resented as a special case of fourth order implicit-implicit s
chemes for linear wave equations. The domain of interest is de
composed into several regions where different fourth order tim
e discretization are used, chosen among a family of implicit o
r explicit fourth order schemes. The coupling is based on a La
grangian formulation on the boundaries between the several pot
entially non conforming meshes of the regions. A global discre
te energy is shown to be preserved and leads to global fourth
order consistency. Numerical results in 1d and 2d illustrate th
e good behavior of the schemes and their potential for realist
ic highly heterogeneous cases or strongly refined geometries,
for which using everywhere an explicit scheme can be extremely
penalizing because the time step must respect the stability c
ondition adapted to the smallest element or the highest veloci
ties. Accuracy up to fourth order reduces the numerical disper
sion inherent to implicit methods used with a large time step,
and makes this family of schemes attractive compared to secon
d order accurate methods in time. The presented technique could
be an alternative to local time stepping provided that some l
imitations are overcame in the future : treatment of dissipati
ve terms, non trivial boundary conditions, coupling with a PML
region, fluid structure coupling...
END:VEVENT
BEGIN:VEVENT
UID:news-264@dmi.unibas.ch
DTSTAMP:20180716T220655
DTSTART;TZID=Europe/Zurich:20140704T110000
DTEND;TZID=Europe/Zurich:20140704T120000
SUMMARY:Seminar in Numerical Analysis: Luis Garcia Naranjo (UNAM, Mexico City)
LOCATION:
DESCRIPTION:In mechanics, constraints that restrict the possible configurat
ions of the system are termed holonomic. A simple example is th
e fixed length of the rod of a pendulum. Mechanical systems wit
h constraints on the velocities that do not arise as constraint
s on positions are called nonholonomic. These often arise in ro
lling systems, like a sphere rotating without slipping on a tab
le.\r\n\r\nThe study of nonholonomic mechanical systems is chal
lenging because the equations of motion are not Hamil- tonian.
The dynamics of the system can however be described in terms of
a bracket of functions that fails to satisfy the Jacobi identi
ty. One now speaks of an almost Poisson bracket.\r\n\r\nThe fai
lure of the Jacobi identity leads to phenomena that are not sha
red by usual Hamiltonian systems. Open questions in nonholonomi
c mechanics that have received attention in recent years includ
e determining general conditions for measure preservation, exis
tence of asymptotic equilibria, relationship between symmetries
and con- servation laws, reduction, and integrability.\r\n\r\n
In the first part of this talk I will present a basic introduct
ion to nonholonomic mechanics. I will then present my recent wo
rk with Y. Fedorov and J. C. Marrero in which we study the prob
lem of measure preservation for nonholonomic systems possessing
symmetries in a systematic manner. Our method allows us to ide
ntify specific parameter values for which there exists a preser
ved measure for concrete mechanical examples.
END:VEVENT
BEGIN:VEVENT
UID:news-265@dmi.unibas.ch
DTSTAMP:20180716T220851
DTSTART;TZID=Europe/Zurich:20140523T110000
DTEND;TZID=Europe/Zurich:20140523T120000
SUMMARY:Seminar in Numerical Analysis: Kersten Schmidt (TU Berlin)
LOCATION:
DESCRIPTION:Shielding sheets are commonly used in the protection of electro
nic devices. With their large aspect ratios they become a seri
ous issue for the direct application of the boundary element m
ethod (BEM) due to the occuring almost singular integrals.\r\n
\r\nWith impedance transmission conditions (ITCs) we can propo
se boundary element formulations on the sheet mid-line (or mid
-surface) only. In the beginning of the talk we give a motivat
ion of meaningful impedance transmission conditions, which rel
ate jumps and mean values of the Dirichlet and Neumann traces o
n the mid-line. This relation mayinvolve surface differential
operators, as for boundary conditions of Wentzell's type, and
depend on frequency, conductivity, sheet thickness and sheet g
eometry e.g. curvature). These parameters may take small or la
rge values and may lead to singularly perturbedboundary integr
al equations.\r\n\r\nWe will introduce related boundary elemen
t methods in two and three dimensions and analyse well-posedne
ss and discretisation error depending on the model parameters.
Numerical experiments confirm the convergence order of the di
scretisation error of the proposed BEM and that the discretisat
ion error behaves for smooth enough sheets equivalent to the e
xact solution when varying the model parameters. The results o
btained for the eddy current model, for which a Poisson equati
on has to be solved outside the mid-line, can be transfered to
the Helmholtz equation and to transmission conditionsarising
from other models.
END:VEVENT
BEGIN:VEVENT
UID:news-266@dmi.unibas.ch
DTSTAMP:20180716T221031
DTSTART;TZID=Europe/Zurich:20140516T110000
DTEND;TZID=Europe/Zurich:20140516T120000
SUMMARY:Seminar in Numerical Analysis: Valeriu Savcenco (Shell Global Solutions/ TU Eindhoven)
LOCATION:
DESCRIPTION:Multirate methods are highly efficient for large-scale ODE and
PDE problems with widely different time scales. Multirate meth
ods enable one to use large time steps for slowly varying spat
ial regions, and small steps for rapidly varying spatial regio
ns. Multirate schemes for conservation laws seem to come in tw
o flavors: schemes that are locally inconsistent, and schemes
that lack mass-conservation. In this presentation these two de
fects will be discussed for one-dimensional conservation laws.
Particular attention will be given to monotonicity properties
of the multirate schemes, such as maximum principles and the
total variation diminishing (TVD) property. The study of these
properties will be done within the framework of partitioned Ru
nge-Kutta methods. It will also be seen that the incompatibili
ty of consistency and mass-conservation holds for genuine mult
irate schemes, but not for general partitioned methods.
END:VEVENT
BEGIN:VEVENT
UID:news-267@dmi.unibas.ch
DTSTAMP:20180716T221204
DTSTART;TZID=Europe/Zurich:20140509T110000
DTEND;TZID=Europe/Zurich:20140509T120000
SUMMARY:Seminar in Numerical Analysis: Tucker Carrington (Queen's University Kingston)
LOCATION:
DESCRIPTION:To compute the vibrational spectrum of a molecule without negle
cting coupling and anharmonicity one must calculate eigenvalue
s and eigenvectors of a large matrix representing the Hamilton
ian in an appropriate basis. Iterative algorithms (e.g. Lanczo
s, Davidson, Filter Diagonalisation) enable one to compute eig
envalues and eigenvectors. It is easy to efficiently implement
iterative algorithms when a direct product basis is used. How
ever, for a molecule with more than four atoms, a direct produ
ct basis set is large and it is better to reduce the number of
basis functions required to obtain converged eigenvalues by p
runing. This is done without jeopardizing the efficiency of the
matrix-vector products required by all iterative algorithms.
In this talk, I shall present new basis-size reduction ideas t
hat are compatible with efficient matrix-vector products. The
basis is designed to include the product basis functions coupl
ed by the largest terms in the potential and important for com
puting low-lying vibrational levels. To solve the vibrational
Schrödinger equation without approximating the potential, on
e must use quadrature to compute potential matrix elements. Wh
en using iterative methods in conjunction with quadrature, it i
s important to evaluate matrix-vector products by doing sums s
equentially. This is only possible if both the basis and the g
rid have structure. Although it is designed to include only fu
nctions coupled by the largest terms in the potential, the bas
is we use and also the (Smolyak-type) quadrature for doing int
egrals with the basis have enough structure to make efficient
matrix-vector products possible. Using the quadrature methods
of this paper, we evaluate the accuracy of calculations made by
making multimode approximations.
END:VEVENT
BEGIN:VEVENT
UID:news-268@dmi.unibas.ch
DTSTAMP:20180716T221403
DTSTART;TZID=Europe/Zurich:20140307T110000
DTEND;TZID=Europe/Zurich:20140307T120000
SUMMARY:Seminar in Numerical Analysis: Fabio Nobile (EPF Lausanne)
LOCATION:
DESCRIPTION:We consider the Darcy equation to describe the flow in a satura
ted porous medium. The permeability of the medium is described
as a log-normal random field, eventually conditioned to availab
le direct measurements, to account for its relatively large unc
ertainty and heterogeneity.\r\n\r\nWe consider perturbation met
hods based on Taylor expansion of the solution of the PDE aroun
d the nominal permeability value. Successive higher order corre
ctions to the statistical moments such as pointwise mean and co
variance of the solution can be obtained recursively from the c
omputation of high order correlation functions which, on their
turn, solve high dimensional problems. To overcome the curse of
dimensionality in computing and storing such high order correl
ations, we adopt a low-rank format, namely the so called tensor
-train (TT) format.\r\n\r\nWe show that, on the one hand, the T
aylor series does not converge globally, so that it only makes
sense to compute corrections up to a maximum critical order, be
yon which the accuracy of the solution deteriorates insetad of
improving. On the other hand, we show on some numerical test ca
ses, the effectiveness of the proposed approach in case of a mo
derately small variance of the log-normal permeability field.
END:VEVENT
BEGIN:VEVENT
UID:news-269@dmi.unibas.ch
DTSTAMP:20180716T222002
DTSTART;TZID=Europe/Zurich:20140221T110000
DTEND;TZID=Europe/Zurich:20140221T120000
SUMMARY:Seminar in Numerical Analysis: Marc Dambrine (Université de Pau)
LOCATION:
DESCRIPTION:I am interested in the influence of small geometrical perturbat
ions on the solution of elliptic problems. The cases of a sing
le inclusion or several well-separated inclusions have been de
eply studied. I will first recall here the techniques to con
struct an asymptotics expansion in that case. Then I will cons
ider moderately close inclusions, i.e. the distance between th
e inclusions tends to zero more slowly than their characterist
ic size and provide a complete asymptotic description of the s
olution of Laplace equation. I will also present numerical simu
lations based on the multiscale superposition method derived f
rom the first order expansion.\r\n\r\nI will explain how some
mathematical questions about the loss of coercivity arise from
the computation of the profiles appearing in the expansion. V
entcel boundary conditions are second order differential condi
tions that appears when looking for a transparent boundary c
ondition for an exterior boundary value problem in planar line
ar elasticity. The goal is to bound the infinite domain by a la
rge “box” to make numerical approximations possible. Like
Robin boundary conditions, they lead to wellposed variational
problems under a sign condition of a coefficient. Nevertheless
situations where this condition is violated appeared in sever
al works. The wellposedness of such problems was still open.
I will present, in the generic case, existence and uniquenes
s result of the solution for the Ventcel boundary value proble
m without the sign condition. Then, I will consider perforated
geometries and give conditions to remove the genericity restri
ction.
END:VEVENT
BEGIN:VEVENT
UID:news-270@dmi.unibas.ch
DTSTAMP:20180716T230659
DTSTART;TZID=Europe/Zurich:20131206T110000
DTEND;TZID=Europe/Zurich:20131206T120000
SUMMARY:Seminar in Numerical Analysis: Mario S. Mommer (Universität Heidelberg)
LOCATION:
DESCRIPTION:Optimum experimental design (OED) is the problem of finding set
ups for an experiment in such a way that the collected data all
ows for optimally accurate estimation of the parameters of inte
rest - taking into account an experimental budget. In practice,
the parameters are only approximately known as a matter of cou
rse, while at the same time, solving an OED problem is in a way
equivalent to magnifying the dependence of the system response
on these quantities. As a consequence, designs computed on t
he basis of a "good guess" of the parameters may underperform d
ramatically in practice, especially for problems involving nonl
inear models.\r\n\r\nIn this talk, we consider robust formulati
ons for optimum experimental design that work under significant
uncertainty. Our focus is on problem settings in which the mod
el is described by differential equations of some type that are
solved numerically. Our approach is based on a semi-infinite p
rogramming formulation in which we exploit additional problem s
tructure, together with sparse grids, to ensure tractability. T
he talk includes numerical experiments to illustrate and compar
e the effectiveness of the approaches.
END:VEVENT
BEGIN:VEVENT
UID:news-271@dmi.unibas.ch
DTSTAMP:20180716T230846
DTSTART;TZID=Europe/Zurich:20131122T110000
DTEND;TZID=Europe/Zurich:20131122T120000
SUMMARY:Seminar in Numerical Analysis: Armin Lechleiter (Universität Bremen)
LOCATION:
DESCRIPTION:It is well-known that the interior eigenvalues of the Laplacian
in a bounded domain share connections to scattering problems
in the exterior of this domain. For instance, certain boundary
integral equations for exterior scattering problems fail at i
nterior eigenvalues.\r\n\r\nSimilar connections also exist for
inverse exterior scattering problems - for instance, if zero
is an eigenvalue of the far-field operator at a fixed wave num
ber, then the squared wave number is an interior eigenvalue. D
espite it is in general wrong that interior eigenvalues corresp
ond to zero being an eigenvalue of the far field operator, one
can prove a pretty direct characterization of interior eigenv
alues via the behavior of the phases of the eigenvalues of the
far-field operator.\r\n\r\nIn this talk, we present this char
acterization and sketch its proof for Dirichlet, Neumann, and
Robin boundary conditions. Then we extend this theory to impen
etrable scattering objects and show via a couple of numerical
examples that one can indeed use this characterization to comp
ute interior eigenvalues of unknown scattering objects from the
spectrum of their far-field operators.\r\n\r\nOur motivation
to study this so-called inside-outside duality comes from a pa
per by Eckmann and Pillet (1995). This is joint work with Andr
eas Kirsch (KIT) and Stefan Peters (University of Bremen).
END:VEVENT
BEGIN:VEVENT
UID:news-272@dmi.unibas.ch
DTSTAMP:20180716T231105
DTSTART;TZID=Europe/Zurich:20131115T110000
DTEND;TZID=Europe/Zurich:20131115T120000
SUMMARY:Seminar in Numerical Analysis: Wim Vanroose (University of Antwerpen)
LOCATION:
DESCRIPTION:Many imaging systems such phase contrast tomography, internal r
eflection microscopy or reaction microscopes measure the far o
r near field of the scattered wave. We present an efficient an
d scalable method to calculate the far- and near field of a He
lmholtz equation describing a given object. The far and near f
ield solution can be written as an integral of the Greens func
tion multiplied by the solution of the Helmholtz equation with
absorbing boundary conditions. By deforming the contour of th
e integral we only require the numerical solution of the Helmh
oltz equation along a complex valued contour. We show that H
elmholtz equation along this contour is equivalent to a complex
shifted Laplacian that can be solved efficiently by multigrid
. This results in an scalable method to calculated the far and
near field integral. We discuss this numerical method, show i
ts applicability, scalability and discuss its limitations.
END:VEVENT
BEGIN:VEVENT
UID:news-273@dmi.unibas.ch
DTSTAMP:20180716T231305
DTSTART;TZID=Europe/Zurich:20131025T110000
DTEND;TZID=Europe/Zurich:20131025T120000
SUMMARY:Seminar in Numerical Analysis: Ludovic Métivier (Université de Grenoble)
LOCATION:
DESCRIPTION:Full Waveform Inversion is an efficient seismic imaging techniq
ue for quantitative estimation of subsurface parameters such a
s the P-wave and S-wave velocities, density, attenuation and a
nisotropy parameters. The method is based on the iterative min
imization of the misfit between observed and calculated data.
During the past ten years, the method has been successfully ap
plied to real data in 2D acoustic and elastic configuration, a
s well as in 3D acoustic configuration. The inverse Hessian op
erator plays an important role in the reconstruction process.
Particularly, this operator should correct for illumination def
icits, frequency bandlimited effects, and help to restore the
correct amplitude of less illuminated parameters. In this pres
entation, we will focus on the methods we have to approximate
this operator, from preconditioned gradient-based methods, to
quasi-Newton methods (l-BFGS) and truncated Newton methods. We
will present results obtained on 2D synthetic and real data f
or the reconstruction of P-wave velocity which illustrate the
importance of the approximation of this operator. We will also
present a simple illustration of the inverse Hessian operator
effect in a multi-parameter framework. In this context, the op
erator helps to mitigate the trade-off between different class
es of parameters.
END:VEVENT
BEGIN:VEVENT
UID:news-274@dmi.unibas.ch
DTSTAMP:20180716T231504
DTSTART;TZID=Europe/Zurich:20131011T110000
DTEND;TZID=Europe/Zurich:20131011T120000
SUMMARY:Seminar in Numerical Analysis: Maya de Buhan (Université Paris Descartes)
LOCATION:
DESCRIPTION:In this talk, we propose a new method to solve the following in
verse problem: we aim at reconstructing, from boundary measure
ments, the location, the shape and the wave propagation speed
of an unknown obstacle surrounded by a medium whose properties
are known. Our strategy combines two methods recently develop
ed by the authors:\r\n\r\n1 - the Time-Reversed Absorbing Cond
ition method: It combines time reversal techniques and absorbi
ng boundary conditions to reconstruct and regularize the signa
l in a truncated domain that encloses the obstacle. This enabl
es us to reduce the size of the computational domain where we
solve the inverse problem, now from virtual internal measuremen
ts.\r\n\r\n2 - the Adaptive Inversion method: It is an inversi
on method which looks for the value of the unknown wave propag
ation speed in a basis composed by eigenvectors of an elliptic
operator. Then, it uses an iterative process to adapt the mes
h and the basis and improve the reconstruction.\r\n\r\nWe pres
ent several numerical examples in two dimensions to illustrate
the efficiency of the combination of both methods. In particul
ar, our strategy allows (a) to reduce the computational cost,
(b) to stabilize the inverse problem and (c) to improve the pr
ecision of the results.
END:VEVENT
BEGIN:VEVENT
UID:news-275@dmi.unibas.ch
DTSTAMP:20180716T231654
DTSTART;TZID=Europe/Zurich:20130524T110000
DTEND;TZID=Europe/Zurich:20130524T120000
SUMMARY:Seminar in Numerical Analysis: Angela Kunoth (Universität Paderborn)
LOCATION:
DESCRIPTION:Optimization problems constrained by linear parabolic evolution
PDEs are challenging from a computational point of view, as t
hey require to solve a system of PDEs coupled globally in time
and space. For their solution, conventional time-stepping m
ethods quickly reach their limitations due to the enormous dem
and for storage. For such a coupled PDE system, adaptive met
hods which aim at distributing the available degrees of freedo
m in an a-posteriori-fashion to capture singularities in the d
ata or domain, with respect to both space and time, appear to b
e most promising. Employing wavelet schemes for full weak spac
e-time formulations of the parabolic PDEs, we can prove conver
gence and optimal complexity. \r\n\r\nYet another level of cha
llenge are control problems constrained by evolution PDEs invo
lving stochastic or countably many infinite parametric coeffic
ients: for each instance of the parameters, this requires the
solution of the complete control problem. \r\n\r\nOur method o
f attack is based on the following new theoretical paradigm.
It is first shown for control problems constrained by evolutio
n PDEs, formulated in full weak space-time form, that state, c
ostate and control are analytic as functions depending on thes
e parameters. Moreover, we establish that these functions allo
w expansions in terms of sparse tensorized generalized polynom
ial chaos (gpc) bases. Their sparsity is quantified in terms
of p-summability of the coefficient sequences for some 0 < p
<= 1. Resulting a-priori estimates establish the existence o
f an index set, allowing for concurrent approximations of stat
e, co-state and control for which the gpc approximations attain
rates of best N-term approximation. These findings serve as t
he analytical foundation for the development of corresponding
sparse realizations in terms of deterministic adaptive Galerki
n approximations of state, co-state and control on the entire,
possibly infinite-dimensional parameter space.\r\n\r\nThe res
ults were obtained with Max Gunzburger (Florida State Universit
y) and with Christoph Schwab (ETH Zuerich).
END:VEVENT
BEGIN:VEVENT
UID:news-276@dmi.unibas.ch
DTSTAMP:20180716T231855
DTSTART;TZID=Europe/Zurich:20130503T110000
DTEND;TZID=Europe/Zurich:20130503T120000
SUMMARY:Seminar in Numerical Analysis: Rüdiger Schultz (Universität Duisburg-Essen)
LOCATION:
DESCRIPTION:This talk aims at demonstrating how concepts and techniques whi
ch are well-established in operations research may serve as bl
ueprints for approaching shape optimization with linearized
elasticity and stochastic loading. Stochastic shape optimizati
on problems are considered from a two-stage viewpoint: In a fi
rst stage, without anticipation of the random loading, the sha
pe has to be fixed. After realization of the load, the displ
acement obtained from solving the elasticity boundary value pr
oblem then may be seen as a second-stage (or recourse) action,
and the variational problem of the weak formulation as a seco
nd-stage optimization problem.\r\n\r\nAt this point, there is
a perfect match with two-stage stochastic programming: after ha
ving taken a non-anticipative decision in the first stage, a
nd having observed the random data, a well-defined second-stag
e problem remains and is solved to optimality. Suitable object
ive functions complete the formal descriptions of the models,
for instance, costs in the stochastic-programming setting and
compliance or tracking functionals in shape optimization.\r\n\
r\nStochastic programming now offers a wide collection of mode
ls to address shape optimization under uncertainty. This start
s with risk neutral models, is continued by mean-risk optimi
zation involving different risk measures, and will finally lead
to analogues in shape optimization of decision problems with
stochastic-order (or dominance) constraints.\r\n\r\nIn the tal
k we will present these models, discuss solution methods, and r
eport some computational tests.
END:VEVENT
BEGIN:VEVENT
UID:news-277@dmi.unibas.ch
DTSTAMP:20180716T232119
DTSTART;TZID=Europe/Zurich:20130419T110000
DTEND;TZID=Europe/Zurich:20130419T120000
SUMMARY:Seminar in Numerical Analysis: Wolfgang Wendland (Universität Stuttgart)
LOCATION:
DESCRIPTION:As a special case of nonlinear Rieman--Hilbert problems with cl
osed boundary data in multiply connected domains, here a doubl
y connected domain like an annulus is considered.\r\n\r\nThe n
onlinear boundary conditions for the desired holomorphic solut
ions lead to nonlinear singular integral equations on the boun
dary which belong to the class of quasiruled Fredholm maps def
ined on quasicylindrical domains in appropriate separable Bana
ch spaces.\r\n\r\nThe closed boundary data give a priori estim
ates for the modulus of solutions which in turn implies a prio
ri estimates in the Sobolev spaces considered here. For this c
lass of problems, the Shnirelman--Efendiev degree of mappings c
an be defined which allows to investigate the existence of sol
utions if the boundary conditions satisfy some topological ass
umptions.\r\n\r\nThe lifting of the boundary value problem v
ia holomorphic transformation onto the universal covering of t
he unit disc allows to construct a homotopic deformation of th
e lifted nonlinear singular integral equations to a uniquely s
olvable case which implies that the degree of mapping is 1 a
nd existence of (in fact at least two) solutions follows.\r\n\
r\nIf the nonlinear integral equations on the boundary are app
oximated by trigonometric point collocation then the theory als
o implies that approximate solutions exist and converge asympt
otically.
END:VEVENT
BEGIN:VEVENT
UID:news-278@dmi.unibas.ch
DTSTAMP:20180716T232307
DTSTART;TZID=Europe/Zurich:20130412T110000
DTEND;TZID=Europe/Zurich:20130412T120000
SUMMARY:Seminar in Numerical Analysis: Florian Loos (Universität der Bundeswehr München)
LOCATION:
DESCRIPTION:The number of electrical devices in modern cars supplied by hig
h currents grows continuously. In order to avoid hot spot gene
ration and overheating on the one hand, but to save weight and
material on the other hand, electrical connecting structures
have to be dimensioned appropriately. The heat transfer in cur
rent carrying multicables with consideration of the rise of el
ectrical resistivity for higher temperatures is described by a
system of semilinear equations with discontinuous coefficient
s. The effects of convection and radiation are taken into acco
unt by a nonlinear boundary condition.\r\n\r\nSimulation resul
ts and experimental studies show that the positioning of the si
ngle cables has important influence on the maximum temperature
s. In order to find an optimal cable design, i.e. to arrange t
he single cables with fixed cross section and current such tha
t the maximum temperature is minimized, a shape optimization p
roblem is formulated. We derive an adjoint system and the shap
e gradient using the formal Lagrange approach. The effect of t
he discontinuity of some coefficients on the shape gradient is
shown. By application of different (nonlinear) optimizers com
bined with the finite element solver COMSOL Multiphysics, a so
lution is obtained numerically. In this talk, we present the mo
deling of the problem, the derivation of the shape gradient an
d numerical results.\r\n\r\nThis is joint work with Helmut Har
brecht and Thomas Apel.
END:VEVENT
BEGIN:VEVENT
UID:news-279@dmi.unibas.ch
DTSTAMP:20180716T232556
DTSTART;TZID=Europe/Zurich:20121214T110000
DTEND;TZID=Europe/Zurich:20121214T120000
SUMMARY:Seminar in Numerical Analysis: Ulrich Römer (TU Darmstadt)
LOCATION:
DESCRIPTION:Simulation results of magnetic devices with ferromagnetic mater
ials are highly sensitive to the nonlinear material law and th
e geometry. Due to uncertainties inherent to measurements and
the fabrication process, the precise knowledge of model input
data cannot be assumed to be given. Therefore, in recent years
, methods of uncertainty quantification have become more and m
ore important. In this talk a short overview over application
examples (magnets, machines) will be given and the PDEs for th
e magnetic fields will be discussed. Under some simplifications
these are 2D nonlinear elliptic interface equations of monoto
ne type. We will introduce a stochastic collocation method bas
ed on generalized polynomial chaos to quantify the uncertainti
es. Furthermore, we will discuss a worst-case scenario analysi
s to cover cases where the statistics of the inputs is not ava
ilable. Since for the worst-case analysis gradient information
is especially important, sensitivity analysis techniques, e.g
., adjoint equations and shape calculus are required.\r\n\r\nJ
oint work with Sebastian Schöps and Thomas Weiland.
END:VEVENT
BEGIN:VEVENT
UID:news-280@dmi.unibas.ch
DTSTAMP:20180716T232736
DTSTART;TZID=Europe/Zurich:20121207T110000
DTEND;TZID=Europe/Zurich:20121207T120000
SUMMARY:Seminar in Numerical Analysis: Mike Botchev (University of Twente)
LOCATION:
DESCRIPTION:We review some recent advances in Krylov subspace methods to co
mpute an action of the matrix exponential on a given vector.
In particular, we briefly discuss residual-based and shift-an
d-invert Krylov subspace methods in the context of space-discr
etized 3D Maxwell's equations. In our limited experience, a
conventional time stepping, where actions of the matrix expone
ntial have to be repeatedly computed at every time step, are u
sually inefficient. We therefore discuss an alternative appr
oach, based on block Krylov subspaces, where just a couple eva
luations of the matrix exponential suffice to solve the problem
for the whole time interval.
END:VEVENT
BEGIN:VEVENT
UID:news-281@dmi.unibas.ch
DTSTAMP:20180716T232939
DTSTART;TZID=Europe/Zurich:20121109T110000
DTEND;TZID=Europe/Zurich:20121109T120000
SUMMARY:Seminar in Numerical Analysis: Sören Bartels (Universität Freiburg)
LOCATION:
DESCRIPTION:The mathematical description of the elastic deformation of thin
plates can be derived by a dimension reduction from three-dim
ensional elasticity and leads to the minimization of an energy
functional that involves the second fundamental form of the d
eformation and is subject to the constraint that the deformati
on is an isometry. We discuss two approaches to the discretiza
tion of the second order derivatives and the treatment of the
isometry constraint. The first one relaxes the second order de
rivatives via a Reissner-Mindlin approximation and the second
one employs discrete Kirchhoff triangles that define a nonconfo
rming second order derivative. In both cases the deformation i
s decoupled from the deformation gradient and this enables us
to employ techniques developed for the approximation of harmon
ic maps to impose the constraint on the deformation gradient a
t the nodes of a triangulation. The solution of the nonlinear
discrete schemes is done by appropriate gradient flows and we
demonstrate their energy decreasing behaviours under mild cond
itions on step sizes. Numerical experiments show that the prop
osed schemes provide accurate approximations for large vertical
loads as well as compressive boundary conditions.
END:VEVENT
BEGIN:VEVENT
UID:news-282@dmi.unibas.ch
DTSTAMP:20180716T233431
DTSTART;TZID=Europe/Zurich:20121102T110000
DTEND;TZID=Europe/Zurich:20121102T120000
SUMMARY:Seminar in Numerical Analysis: Dominik Schötzau (University of British Columbia)
LOCATION:
DESCRIPTION:We introduce and analyze a new mixed finite element method for
the spatial discretization of an incompressible magnetohydrod
ynamics problem. It is based on divergence-conforming element
s for the fluid velocities and on curl-conforming elements fo
r the magnetic unknowns. The tangential continuity of the velo
cities is enforced by a DG approach. Central features of the
resulting method are that it produces exactly divergence-free
velocity approximations and is provably energy-stable, and t
hat it correctly captures the strongest magnetic singularities
in non-smooth domains. We carry out the error analysis of the
method, and present a comprehensive set of numerical tests in
two and three dimensions. We also discuss some recent ideas re
garding the design of efficient solvers for the matrix systems
.
END:VEVENT
BEGIN:VEVENT
UID:news-283@dmi.unibas.ch
DTSTAMP:20180716T233556
DTSTART;TZID=Europe/Zurich:20121026T110000
DTEND;TZID=Europe/Zurich:20121026T120000
SUMMARY:Seminar Numerical Analysis: Drosos Kourounis (Università della Svizzera italiana)
LOCATION:
DESCRIPTION:Adjoint-based gradients form an important ingredient of fast o
ptimization algorithms for computer-assisted history matching a
nd life-cycle production optimization. Large-scale application
s of adjoint-based reservoir optimization reported so far conc
ern relatively simple physics, in particular two-phase (oil-wa
ter) or three-phase (oil-gas-water) applications. In contrast,
compositional simulation has the added complexity of frequent
flash calculations and high compressibilities which potential
ly complicate both the adjoint computation and gradient-based
optimization, especially in the presence of complex constraint
s. These aspects are investigated using a new adjoint implemen
tation in a research reservoir simulator designed on top of an
automatic differentiation framework coupled to a standard lar
ge-scale nonlinear optimization package. Based on several exa
mples of increasing complexity we conclude that the AD-based a
djoint implementation is capable of accurately and efficiently
computing gradients for multi-component reservoir flow. Howev
er, optimization of strongly compressible flow with constraint
s on well rates or pressures leads to potentially poor perform
ance in conjunction with an external optimization package. We
present a pragmatic but effective strategy to overcome this is
sue.
END:VEVENT
BEGIN:VEVENT
UID:news-284@dmi.unibas.ch
DTSTAMP:20180716T233803
DTSTART;TZID=Europe/Zurich:20121012T110000
DTEND;TZID=Europe/Zurich:20121012T120000
SUMMARY:Seminar in Numerical Analysis: Stefan Volkwein (Universität Konstanz)
LOCATION:
DESCRIPTION:We consider the following problem of error estimation for the o
ptimal control of nonlinear partial differential equations: Le
t an arbitrary admissible control function be given. How far i
s it from the next locally optimal control? Under natural assu
mptions including a second-order sufficient optimality conditi
on for the (unknown) locally optimal control, we estimate the
distance between the two controls. To do this, we need some in
formation on the lowest eigenvalue of the reduced Hessian. We
apply this technique to a model reduced optimal control proble
m obtained by proper orthogonal decomposition (POD). The dista
nce between a local solution of the reduce problem to a local
solution of the original problem is estimated.
END:VEVENT
BEGIN:VEVENT
UID:news-285@dmi.unibas.ch
DTSTAMP:20180716T233949
DTSTART;TZID=Europe/Zurich:20120928T110000
DTEND;TZID=Europe/Zurich:20120928T120000
SUMMARY:Seminar in Numerical Analysis: Frédéric Nataf (Université Pierre et Marie Curie)
LOCATION:
DESCRIPTION:We introduce the time reversed absorbing conditions (TRAC) in t
ime reversal methods. They enable to "recreate the past" with
out knowing the source which has emitted the signals that are
back-propagated. Two applications to inverse problems are gi
ven in both the full and partial aperture case.
END:VEVENT
BEGIN:VEVENT
UID:news-286@dmi.unibas.ch
DTSTAMP:20180716T234139
DTSTART;TZID=Europe/Zurich:20120601T090000
DTEND;TZID=Europe/Zurich:20120601T100000
SUMMARY:Seminar in Numerical Analysis: Walter Gautschi (Purdue University)
LOCATION:
DESCRIPTION:Algorithms are developed for computing the coefficients in the
three-term recurrence relation of repeatedly modified orthogon
al polynomials, the modifications involving division of the or
thogonality measure by a linear function with real or complex
coefficient. The respective Gaussian quadrature rules can be u
sed to account for simple or multiple poles that may be presen
t in the integrand. Several examples are given to illustrate t
his.
END:VEVENT
BEGIN:VEVENT
UID:news-287@dmi.unibas.ch
DTSTAMP:20180716T234452
DTSTART;TZID=Europe/Zurich:20120525T090000
DTEND;TZID=Europe/Zurich:20120525T100000
SUMMARY:Seminar in Numerical Analysis: Wolfgang Bangerth (Texas A&M University)
LOCATION:
DESCRIPTION:In many of the modern biomedical imaging modalities, the measur
able signal can be described as the solution of a partial diff
erential equation that depends nonlinearly on the tissue prope
rties (the "parameters") one would like to image. Consequently
, there are typically no explicit solution formulas for these
so-called "inverse problems" that can recover the parameters f
rom the measurements, and the only way to generate body images
from measurements is through numerical approximation.\r\n\r\n
The resulting parameter estimation schemes have the underlying
partial differential equations as side-constraints, and the s
olution of these optimization problems often requires solving t
he partial differential equation thousands or hundred of thous
ands of times. The development of efficient schemes is therefo
re of great interest for the practical use of such imaging mod
alities in clinical settings. In this talk, the formulation an
d efficient solution strategies for such inverse problems will
be discussed, and we will demonstrate its efficacy using exam
ples from our work on Optical Tomography, a novel way of imagi
ng tumors in humans and animals. The talk will conclude with a
n outlook to even more complex problems that attempt to automa
tically optimize experimental setups to obtain better images.
END:VEVENT
BEGIN:VEVENT
UID:news-288@dmi.unibas.ch
DTSTAMP:20180716T234712
DTSTART;TZID=Europe/Zurich:20120427T090000
DTEND;TZID=Europe/Zurich:20120427T100000
SUMMARY:Seminar in Numerical Analysis: Daniel Kressner (EPFL)
LOCATION:
DESCRIPTION:
END:VEVENT
BEGIN:VEVENT
UID:news-289@dmi.unibas.ch
DTSTAMP:20180716T234846
DTSTART;TZID=Europe/Zurich:20120420T090000
DTEND;TZID=Europe/Zurich:20120420T100000
SUMMARY:Seminar in Numerical Analysis: Ilario Mazzieri (MOX - Politecnico di Milano)
LOCATION:
DESCRIPTION:The study and development of spectral element (SE) methods for
simulating elastic wave propagation in seismic regions has bee
n subjected to a tremendous growth, occurred in the past ten y
ears. SE methods are based on high-order Lagrangian interpolan
ts sampled at the Gauss-Legendre-Lobatto quadrature points, an
d combine the flexibility of finite elements with the accuracy
of spectral techniques. Since they are based on the weak form
ulation of the elastodynamics equations, they handle naturally
both interface continuity and free boundary conditions, allow
ing very accurate resolutions of evanescent interface and surfa
ce waves. Moreover, SE methods retain a high level parallel st
ructure, thus are well suited for massively parallel computati
ons. The main drawback of SE methods is that they usually requ
ire a uniform polynomial order on the whole computational doma
in, and this can lead to an unreasonably large computational e
ffort, in particular in regions where a fine mesh grid is need
ed already to describe accurately the domain geometry.\r\n\r\nH
ere, we consider a Discontinuous Galerkin (DGSE) and a Mortar
(MSE) spectral element methods coupled with the leap-frog time
integration scheme to simulate seismic wave propagations in t
wo and three dimensional heterogeneous media. The main advanta
ge with respect to conforming discretizations, as SE method, i
s that DGSE and MSE discretizations can accommodate discontinu
ities, not only in the parameters, but also in the wavefield,
while preserving the energy. The domain of interest Ω is assu
med to be union of polygonal subdomain Ωi. We allow this subdo
main decomposition to be geometrically non-conforming. Inside e
ach subdomain Ωi, a conforming high order finite element space
associated to a partition Thi(Ωi) is introduced. We consider
different polynomial approximation degrees within different s
ubdomains. To handle non-conforming meshes and non-uniform pol
ynomial degrees across ∂Ωi , a DG or a Mortar discretization
is considered.\r\n\r\nApplications of the DGSE and MSE methods
to simulate realistic seismic wave propagation problems are pr
esented.\r\n\r\nJoint work with: P.F. Antonietti, A. Quarteroni
and F. Rapetti.
END:VEVENT
BEGIN:VEVENT
UID:news-290@dmi.unibas.ch
DTSTAMP:20180716T235028
DTSTART;TZID=Europe/Zurich:20120323T090000
DTEND;TZID=Europe/Zurich:20120323T100000
SUMMARY:Seminar in Numerical Analysis: Reinhold Schneider (TU Berlin)
LOCATION:
DESCRIPTION:The DMRG algorithm (density matrix renormalization group algori
thm) introduced by S. White provides a powerful tool for the
numerical treatment of spin systems. The DMRG version for the
electronic Schrödinger equation, the QDMRG (quantum chemis
try density matrix renormalization group) algorithm is less kn
own. Although it provides an approximaiton of the full CI so
lution within polynomial complexity. Concepts known from spin
systems, e.g. matrix product states, tree tensor networks ha
ve been rediscovered recently in tensor product approximation,
under a different perspective as hierarchical Tucker repres
entation introduced by Hackbusch and coworkers. and on TT-te
nsors (tensor trains) by Oseledets & Tyrtishnikov, offering a
promising approach for the numerical treatment of high dimen
sional differential equation. We have shown that under a ful
l rank condition TT tensors form a manifold and characterize i
ts tangent space, e.g. to apply the Dirac-Frenkel variartion
al principle. We propose an alternating linear scheme (ALS
alternating linear scheme) approach for optimization in the T
T format. A modified alternating linear scheme (MALS) applied
to the electronic Schrödinger equation resembles exactly the
density matrix renormalization group algorithm (QDMRG). Identif
ying the discretized Fock space with the tensor product space
⊗R2 (⊗C2), the formalism of second quantization is direct
ly implemented in the tensor treatment for numerical computatio
n.\r\n\r\nJoint work with Th. Rohwedder and S. Holtz
END:VEVENT
BEGIN:VEVENT
UID:news-291@dmi.unibas.ch
DTSTAMP:20180716T235154
DTSTART;TZID=Europe/Zurich:20120309T090000
DTEND;TZID=Europe/Zurich:20120309T100000
SUMMARY:Seminar in Numerical Analysis: Luca Frediani (University of Tromsø)
LOCATION:
DESCRIPTION:Most modern molecular electronic structure calculations are bas
ed on Density Functional Theory (DFT) due to the very convenie
nt balance between accuracy and required computational resourc
es. The accuracy of the result is then dependent on the qualit
y of the functional and the chosen basis set. By replacing tra
ditional Gaussian Type Orbitals (GTOs) with Multiwavelets, the
basis set can be made practically complete making the lack of
the "exact" functional as the only source of error left. We h
ave implemented a Multiwavelet-based DFT code, which makes use
of the integral formulation of the Kohn-Sham equations of DFT.
Different strategies to optimize the density have been attemp
ted and will be presented. The main limitation of the present
approach is the large memory demand of the software compared t
o traditional methods. In order to overcome such a limitation
a massively parallel implementation has been developed for the
kernel of the code: the application of the Green's operator r
epresented in the so called Non-Standard form.
END:VEVENT
BEGIN:VEVENT
UID:news-292@dmi.unibas.ch
DTSTAMP:20180716T235420
DTSTART;TZID=Europe/Zurich:20111216T090000
DTEND;TZID=Europe/Zurich:20111216T100000
SUMMARY:Seminar in Numerical Analysis: Bernard Haasdonk (Universität Stuttgart)
LOCATION:
DESCRIPTION:In this presentation we introduce a method for rapid and certif
ied parameter optimization of problems with PDE constraints gi
ven by evolution equations. We make use of an RB-formulation f
or a general class of evolution problems covering standard ite
rative time-stepping schemes. The reduced spaces for such prob
lems are beneficially constructed by the POD-Greedy procedure,
for which we recently provided theoretical foundation by conv
ergence rate proofs. Extensions of this procedure involve para
meter- and time-partitioning approaches. We will demonstrate,
how these ingredients can be used in iterative direct paramete
r optimization problems. In addition to approximate surrogate
optimization results, we provide rigorous a-posteriori error bo
unds for solutions, outputs, sensitivities and optimal paramet
ers.
END:VEVENT
BEGIN:VEVENT
UID:news-293@dmi.unibas.ch
DTSTAMP:20180716T235602
DTSTART;TZID=Europe/Zurich:20111209T090000
DTEND;TZID=Europe/Zurich:20111209T100000
SUMMARY:Seminar in Numerical Analysis: Daniel Weiss (Universität Tübingen)
LOCATION:
DESCRIPTION:There are many systems which exhibit different scales which act
in the following way: A slow dynamic of interest is driven or
affected by highly-oscillatory components of the systems. A s
imple example is an old-fashioned alarm clock, which moves on
a table driven by the fast movements of the clapper.\r\n\r\nWe
will explain the Heterogeneous Multiscale Method (HMM)[1], wh
ich is believed to provide a numerical method for all kind of
multiscale systems to overcome the difficulties of numerical i
ntegration generated by highly-oscillatory components. We will
formulate the HMM for highly-oscillatory Hamiltonian systems w
ith solution-dependent frequencies more precisely for the doub
le spring pendulum with very stiff springs. Finally we will di
scuss the drawbacks of this method in case of solution-depende
nt frequencies. \r\n\r\n[1] E, W.; Engquist, B.: The heterogene
ous multi-scale method, Comm. Math. Sci., 1, 87--133, 2003.
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BEGIN:VEVENT
UID:news-294@dmi.unibas.ch
DTSTAMP:20180716T235800
DTSTART;TZID=Europe/Zurich:20111202T090000
DTEND;TZID=Europe/Zurich:20111202T100000
SUMMARY:Seminar in Numerical Analysis: Andreas Stock (Universität Stuttgart)
LOCATION:
DESCRIPTION:In the Particle-In-Cell (PIC) method particles and electromagne
tic fields are fully self-consistent coupled to solve the Vlas
ov equation describing the behavior of rarefied plasma flows w
ithout collisions. In the underlying physical model of the PIC
method different time-scales occurs, i.e. the electromagnetic
fields propagating with the speed of light and the particles
moving with speeds much lower than the electromagnetic fields.
Due to the CFL condition for explicit time integration scheme
s the time step is restricted to the largest propagating speed
, yielding an inefficient time discretization for the slower p
articles. We developed a time domain decomposition based on a
multirate multistep technique to treat each component on it's s
pecific time-scale leading to an efficient time-integration al
gorithm.
END:VEVENT
BEGIN:VEVENT
UID:news-295@dmi.unibas.ch
DTSTAMP:20180716T235940
DTSTART;TZID=Europe/Zurich:20111118T090000
DTEND;TZID=Europe/Zurich:20111118T100000
SUMMARY:Seminar in Numerical Analysis: Matthias Bollhöfer (Technische Universität Braunschweig)
LOCATION:
DESCRIPTION:Hierarchical matrix approximations have become an attractive nu
merical approach in solving partial differential equations whe
never the analytic solution can be represented by a kernel fun
ctions that allows for approximate local separable representat
ions. The philosophy of a hierarchical matrix approximation co
nsists of borrowing a matrix partition from an admissibility c
ondition of the underlying analytic model and working with blo
cks that are expected to be of low rank. While the existence o
f hierarchical matrix approximations is relatively well unders
tood, the concrete way of numerically computing a suitable app
roximation still raises some open questions such as the h-indep
endent convergence of the computed approximation.\r\n\r\nIn th
is talk we present a new technique to locally preserve constrai
nts inside the hierarchical matrix approximation. Numerical ex
periments indicate that imposing these local constraints leads
to constant number of iteration steps when solving elliptic p
artial differential equations of second order while without pr
eserving these constraints the number of iteration steps grow
as h → 0. We will further discuss this approach from the theo
retical point of view and will sketch why our approximate hiera
rchical LU decomposition leads to a spectral equivalent approxi
mation.\r\n\r\nThis is joint work with Mario Bebendorf and Mich
ael Bratsch from the University of Bonn.
END:VEVENT
BEGIN:VEVENT
UID:news-296@dmi.unibas.ch
DTSTAMP:20180717T000114
DTSTART;TZID=Europe/Zurich:20111104T090000
DTEND;TZID=Europe/Zurich:20111104T100000
SUMMARY:Seminar in Numerical Analysis: Sara Minisini (Shell Global Solutions International)
LOCATION:
DESCRIPTION:Efficient and accurate modeling of the wave equation is importa
nt for seismic exploration. We compare the performance of expl
icit time stepping with the discontinuous Galerkin method and
with continuous augmented third and fourth-order mass-lumped e
lements for tetrahedra. There are two choices for the last, on
e with a more favorable CFL number. Numerical experiments illu
strate the accuracy, usefulness, and versatility of these meth
ods when solving 3D problems in inhomogeneous media.
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BEGIN:VEVENT
UID:news-297@dmi.unibas.ch
DTSTAMP:20180717T000250
DTSTART;TZID=Europe/Zurich:20111028T090000
DTEND;TZID=Europe/Zurich:20111028T100000
SUMMARY:Seminar in Numerical Analysis: Oliver Ernst (Technische Universität Bergakademie Freiberg)
LOCATION:
DESCRIPTION:We present a case study for probabilistic uncertainty quantific
ation (UQ) applied to groundwater flow in the context of site
assessment for radioactive waste disposal based on data from t
he Waste Isolation Pilot Plant (WIPP) in Carlsbad, New Mexico.
In this context, the primary quantity of interest is the time
it takes for a particle of radioactivity to be transported wi
th the groundwater from the repository to man's environment. T
he mathematical model consists of a stationary diffusion equat
ion for the hydraulic head in which the hydraulic conductivity
coefficient is a random field. Once the (stochastic) hydrauli
c head is computed, contaminant transport can be modelled by p
article tracing in the associated velocity field.\r\n\r\nWe com
pare two approaches: Gaussian process emulators and stochastic
collocation combined with geostatistical techniques for determ
ining the parameters of the input random field's probability l
aw. The second approach involves the numerical solution of the
PDE with random data as a parametrized deterministic system.
The calculation of the statistics of the travel time from the
solution of the stochastic model is formulated for each of the
methods being studied and the results compared.
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BEGIN:VEVENT
UID:news-298@dmi.unibas.ch
DTSTAMP:20180717T000533
DTSTART;TZID=Europe/Zurich:20111021T090000
DTEND;TZID=Europe/Zurich:20111021T100000
SUMMARY:Seminar in Numerical Analysis: Robert C. Dalang (EPFL)
LOCATION:
DESCRIPTION:I will present an introduction to stochastic partial differenti
al equations, with an emphasis on the stochastic heat and wave
equations. These equations describe the motion of a medium subj
ect to random excitation, which is usually taken to be Gaussian
space-time white noise. This probabilistic object will be pres
ented through several examples as well as defined mathematicall
y. Some important questions and results obtained will also be d
iscussed.
END:VEVENT
BEGIN:VEVENT
UID:news-299@dmi.unibas.ch
DTSTAMP:20180717T000852
DTSTART;TZID=Europe/Zurich:20111014T090000
DTEND;TZID=Europe/Zurich:20111014T100000
SUMMARY:Seminar in Numerical Analysis: Annika Lang (ETHZ)
LOCATION:
DESCRIPTION:We analyze the convergence and complexity of Multi-Level Monte
Carlo (MLMC) discretizations of a class of abstract stochastic
, parabolic equations driven by square integrable martingales.
We show, under regularity assumptions on the solution that ar
e minimal under certain criteria, that the combination of pi
ecewise linear, continuous multi-level Finite Element discreti
zations in space and Euler--Maruyama discretizations in time
yields mean square convergence of order one in space and of o
rder 1/2 in time to the expected value of the mild solution. T
he complexity of the multi-level estimator is shown to scale l
og-linearly with respect to the corresponding work to generate
a single solution path on the finest mesh, resp. of the corres
ponding deterministic parabolic problem on the finest mesh. Ex
amples are provided for Levy driven SPDEs as well as equations
for randomly forced surface diffusions.
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