Stochastic approximation for shape optimization under uncertainty

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

Joint work with: Kathrin Welker, Estefania Loayza-Romero, Tim Suchan

For further information about the seminar, please visit this webpage.

]]>A posteriori error estimates robust with respect to nonlinearities and final time

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A. Ern, I. Smears, M. Vohralík, Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems, *SIAM J. Numer. Anal.***55** (2017), 2811–2834.

A. Harnist, K. Mitra, A. Rappaport, M. Vohralík, Robust energy a posteriori estimates for nonlinear elliptic problems, HAL Preprint 04033438, 2023.

K. Mitra, M. Vohralík, A posteriori error estimates for the Richards equation, *Math. Comp.* (2024), accepted for publication.

K. Mitra, M. Vohralík, Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on iterative linearization, HAL Preprint 04156711, 2023.

For further information about the seminar, please visit this webpage.

]]>A posteriori error estimates and adaptive error control for electromagnetic coefficient inverse problem in conductive media

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

All reconstruction algorithms are based on optimization approach for finding of stationary point of the Lagrangian. Derivation of a posteriori error estimates for the regularized solution and Tikhonov functional will be presented. Based on these estimates adaptive reconstruction algorithms are developed. Computational tests will show robustness of proposed algorithms in 3D.

For further information about the seminar, please visit this webpage.

]]>PDE-regularized learning

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quantify the accuracy of the PDE model used for regularization and the data coverage.

Furthermore, the discretization of the PDE-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 derive errors bounds for both classes of discretizations and gives first insights into error analysis of variational neural network discretizations of PDEs.

For further information about the seminar, please visit this webpage.

]]>Conforming Space-Time Finite Element Methods

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For further information about the seminar, please visit this webpage.

]]>Harmonic Analysis meets Applications: Wavelet-based Function Spaces and Optimal Approximations of their Embeddings

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

In this lecture, we first review the Fourier analytic approach towards by now well-established (Besov and Triebel-Lizorkin) scales of distribution spaces that measure either isotropic or dominating-mixed regularity. In addition, we introduce new function spaces of hybrid smoothness 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 yield an appropri- ate framework to study break-of-scale embeddings by means of harmonic analysis. We shall present (non-)adaptive wavelet-based multiscale algorithms that approximate such embed- dings at optimal dimension-independent rates of convergence. 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.

The talk is based on a recent paper [1] which represents the first part of a joint work with Glenn Byrenheid (FSU Jena), Markus Hansen (PU Marburg), and Janina Hübner (RU Bochum).

References:

[1] G. Byrenheid, J. Hübner, and M. Weimar. Rate-optimal sparse approximation of compact break-of-scale embeddings. Appl. Comput. Harmon. Anal. 65:40–66, 2023 (arXiv:2203.10011).

For further information about the seminar, please visit this webpage.

]]>A Kernel-based Multilevel Approximation Method for Higher-Dimensional Problems on Sparse Grids

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In machine learning, support vector machines heavily rely on RKHSs. For neural networks Barron spaces are connected to certain RKHSs and offer a possibility for a theoretical analysis of these networks.

Another application of RKHSs is in high(er)-dimensional approximation. For instance in the field of quasi Monte-Carlo methods, kernel-techniques are used to derive an error analysis for high-dimensional quadrature rules. We also developed a novel kernel-based approximation method for higher-dimensional meshfree function reconstruction, based on Smolyak operators.

In this talk I will provide an introduction into the theory 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 used to derive the new tensor product multilevel method for higher- dimensional meshfree approximation, which I will discuss in detail.

For further information about the seminar, please visit this webpage.

]]>Recent Advanced in Interacting particle methods for Bayesian Inference

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In this talk we present enhancement strategies of such ensemble methods based on sample enrichment and homotopy formalism, that ultimately lead to time-dependent drift terms that possible assimilate a larger class of target distributions while providing faster mixing times.

Furthermore, we present an alternative route to construct time-inhomogeneous drift terms based on reverse Diffusion processes that are popular in state-of-the-art Generative Modelling such as Diffusion maps. Here, we propose learning these log-densities by propagation of the target distribution through an Ornstein-Uhlenbeck process. For this, we solve the associated Hamilton-Jabobi-Bellman equation through an adaptive explicit Euler discretization using low-rank compression such as functional Tensor Trains for the spatial discretization.

For further information about the seminar, please visit this webpage.

]]>Convergence of empirical Galerkin FEM for parametric PDEs with sparse TTs

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For further information about the seminar, please visit this webpage.

]]>The cold-plasma model and simulation of wave propagation using B-Spline Finite Elements

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This model has applications in magnetic confinement fusion devices, like the Tokamak. Namely, electromagnetic waves are used to heat up the plasma (Electron cyclotron resonance heating (ECRH)) or for interferometry and reflectometry diagnostics (to measure plasma density and position, etc.).

In the first part of this talk, we introduce the cold-plasma model, together with a qualitative study of the plasma modes which expose the complexity of the problem.

In the second part, we describe the problem and the simplifications we carry out, which yield the indefinite Helmholtz equation. It is solved with B-Spline Finite Elements provided by the Psydac library and some results are shown. Lastly, we discuss the performance and potential ways of preconditioning.

For further information about the seminar, please visit this webpage.

]]>Adaptive Coarse Space for Saddle Point problem

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For further information about the seminar, please visit this webpage.

]]>A least squares Hessian/Gradient recovery method for fully nonlinear PDEs in Hamilton–Jacobi–Bellman form (joint work wit Amireh Mousavi)

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This idea allows the creation of efficient solvers for fully nonlinear elliptic equations, the linearization of which leaves us with an equation in nondivergence form. An important class of fully nonlinear elliptic PDEs can be written in Hamilton--Jacobi--Bellman (Dynamic Programming) form, i.e., as the supremum of a collection of linear operators acting on the unkown.

The least-squares FEM approach, a variant of the nonvariational finite element method, is based on gradient or Hessian recovery and allows the use of FEMs of arbitrary degree. The price to pay for using 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 difference schemes.

Suitable functional spaces and penalties in the least-squares's cost functional must be carefully crafted in order to ensure stability and convergence of the scheme with a good approximation of the gradient (or Hessian) under the Cordes condition on the family of linear operators being optimized.

Furthermore, the nonlinear operator which is not necessarily everywhere differentiable, must be linearized in appropriate functional spaces using semismooth Newton or Howard's policy iteration method. A crucial contribution of our work, is the proof of convergence of the semismooth Newton method at the continuum level, i.e., on infinite dimesional functionals spaces. This allows an easy use of our non-monotone schemes which provides convergence rates as well as a posteriori error estimates.

For further information about the seminar, please visit this webpage.

]]>High-dimensional kernel approximation of parametric PDEs over lattice point sets

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References:

[1] V. Kaarnioja, Y. Kazashi, F. Y. Kuo, F. Nobile, and I. H. Sloan. Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification. Numer. Math. 150:33-77, 2022.

[2] V. Kaarnioja, F. Y. Kuo, and I. H. Sloan. Uncertainty quantification using periodic random variables. SIAM J. Numer. Anal. 58(2):1068-1091, 2020.

[3] V. Kaarnioja, F. Y. Kuo, and I. H. Sloan. Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions. Preprint 2023, arXiv:2303.17755 [math.NA].

For further information about the seminar, please visit this webpage.

]]>Towards robustness in shape optimization

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For further information about the seminar, please visit this webpage.

]]>Non-reflecting boundary conditions and domain decomposition methods for industrial flow acoustics

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[1] Marchner, P., Antoine, X., Geuzaine, C., & Bériot, H. (2022). Construction and numerical assessment of local absorbing boundary conditions for heterogeneous time-harmonic acoustic problems. SIAM 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 method with high-order finite elements for flow acoustics. Computer Methods in Applied Mechanics and Engineering, 369, 113223.

For further information about the seminar, please visit this webpage.

]]>T-coercivity: a practical tool for the study of variational formulations

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In this talk, I will focus on an approach that is completely equivalent to the inf-sup condition for problems set in Hilbert spaces, the **T-coercivity approach**. This approach relies 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 approximation of the exact problem. As a matter of fact, the derivation of the discrete inf-sup condition often becomes elementary, at 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 considered, analysed and solved at once.

In itself, T-coercivity is not a new theory, however it seems that some of its strengths 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 tools and which numerical methods are the most “natural” to analyse 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 classical linear PDEs, and for some generalizations of those models.

For further information about the seminar, please visit this webpage.

]]>Stabilisation — using a multiplier strategy — of spurious high frequencies for high-order discretisations of the wave equation

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For further information about the seminar, please visit this webpage.

]]>On analytic and Gevrey class regularity for parametric elliptic eigenvalue problems (A. Chernov, T. Le)

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For further information about the seminar, please visit this webpage.

]]>Determination of a learnt best basis for two geoscientific inverse problems

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Traditionally, an approximation of such a linear(ized) inverse problem is obtained in one, a-priori chosen basis system: either a global one, e.g. spherical harmonics or polynomials on the ball, or a local one, e.g. radial basis functions and wavelets on the sphere or finite elements on the ball.

In the Geomathematics Group Siegen, we developed methods that enable us to combine different types of trial functions for such an approximation. The idea is to make the most of the benefits of different types of available trial functions. The algorithms are called the (Learning) Inverse Problem Matching Pursuits (LIPMPs). They construct an approximation iteratively from an intentionally overcomplete set of trial functions, the dictionary, such that the Tikhonov functional is reduced. Due to the learning add-on, the dictionary can very well be infinite. Moreover, the computational costs are usually decreased.

In this talk, we give details on the LIPMPs and show some current numerical results.

For further information about the seminar, please visit this webpage.

]]>Riccati-Feedback Stabilization of Optimal Control Problems with PDE Constraints

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On the other hand, the linear-quadratic 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 equations. It’s applicability is limited by the linearity of the dynamical system and the efficient solvability of the quadratic matrix equation.

In this talk, we discuss how certain classes of non-stationary PDEs can be reformulated (after spatial semi-discretization) into structured linear dynamical systems that allow the Riccati feedback to be computed. This allows us to combine both approaches and thus steer solutions of perturbed PDEs back to the optimized trajectories. The key to efficient solvers for the Riccati equations is the usage of the specific structure in the problems and the fact that the Riccati solutions usually feature a strong singular value decay, and thus good low-rank approximability.

For further information about the seminar, please visit this webpage.

]]>Structure-Preserving Model Reduction for Symmetric Second-Order Systems

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Our method guarantees to preserve passivity of the reduced-order model as well as the positive definiteness of the mass and stiffness matrices and admits an a priori gap metric error bound.

Our construction of the second-order reduced model is based on the consideration of an internal symmetry structure and the invariant zeros of the system and their sign-characteristics for which we derive a normal form.

The results are available in [1].

[1] I. Dorschky, T. Reis, and M. Voigt. Balanced truncation model reduction for symmetric second order systems - a passivity-based approach. SIAM J. Matrix Anal. Appl., 42(4):1602--1635, 2021.

For further information about the seminar, please visit this webpage.

]]>Adaptive Approximation of Shapes

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For further information about the seminar, please visit this webpage.

]]>Introduction to finite element error estimates for optimal control problems with PDEs

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For further information about the seminar, please visit this webpage.

]]>Combined iterative methods for solving nonlinear equations with nondifferentiable operators

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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 of systems of nonlinear equations.

For further information about the seminar, please visit this webpage.

]]>Optimality of adaptive stochastic Galerkin methods for affine-parametric elliptic PDEs

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M. Bachmayr and I. Voulis, *An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality*, arXiv:2109:09136

For further information about the seminar, please visit this webpage.

]]>Some results on Perfectly Matched Layers (PML) for non-dispersive and dispersive Maxwell’s equations

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In this talk, we address the questions of well-posedness, stability and convergence of standard and new models of PMLs in the context of electromagnetic waves for non-dispersive and dispersive materials.

For further information about the seminar, please visit this webpage.

]]>Residual and restarting in Krylov subspace evaluation of the φ function

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For further information about the seminar, please visit this webpage.

]]>Adaptive algorithms for solution of an electromagnetic volume integral equation with application to microwave thermometry

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Based on these estimates, different adaptive finite element algorithms are formulated. Numerical examples will show efficiency of the proposed adaptive algorithms to improve quality of 3D reconstruction of target during the process of microwave thermometry which is used in cancer therapies. This is joint work with the group of Biomedical Imaging at the Department of Electrical Engineering at CTH, Chalmers.

For further information about the seminar, please visit this webpage.

]]>Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations

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In this presentation, we will discuss a different approach to ensure that a higher order accurate DG solution satisfies the positivity constraints. Instead of using a limiter, we impose the positivity constraints directly on the algebraic equations resulting from a higher order accurate time-implicit DG discretization using techniques from mathematical optimization theory. 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 equations are then solved using a specially designed semi-smooth Newton method that is well suited to deal with the resulting nonlinear complementarity problem. We will demonstrate the algorithm on several parabolic model problems.

For further information about the seminar, please visit this webpage.

]]>Scales of Knowledge - Multilevel Minimization in Scientific Computing and Machine Learning

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Traditional multilevel decompositions are the basic ingredient of the most efficient class of solution methods for linear systems, i.e. of multigrid methods, which allow to solve certain classes of linear systems with optimal complexity. The transfer of these concepts to 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 how multilevel optimization methods can be constructed and analyzed and we will illustrate the sometimes significant gain in performance, which can be achieved by multilevel minimization techniques. Examples from mechanics, geophysics, and machine learning will illustrate our discussion.

For further information about the seminar, please visit this webpage.

]]>Machine Learning meets Numerical Simulation

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We give examples of different types of combinations using exemplary approaches of simulation-assisted machine learning and machine-learning assisted simulation. We also discuss an advanced pairing where we see particular further potential for hybrid systems.

For further information about the seminar, please visit this webpage.

]]>Waves, learning, and inverse problems

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For further information about the seminar, please visit this webpage.

]]>High order finite element methods for inverse initial value problems and wave propagation in heterogeneous media

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For further information about the seminar, please visit this webpage.

]]>Splines in computer graphics and numerical methods

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For further information about the seminar, please visit this webpage.

]]>Wavenumber-explicit hp-FEM analysis for the Helmholtz equation in heterogeneous media

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Joint work with: Maximilian Bernkopf (TU Wien), Théophile Chaumont-Frelet (Inria).

**References**

[EM11] S. Esterhazy and J.M. Melenk, On stability of discretizations of the Helmholtz equation, in: Numerical Analysis of Multiscale Problems, Graham et al., eds, Springer 2012

[MS10] J.M. Melenk and S. Sauter, Convergence Analysis for Finite Element Discretizations of the Helmholtz equation with 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 discretizations of the Helmholtz equation, SIAM J. Numer. Anal., 49:1210–1243, 2011

[MSP12] J.M. Melenk, S. Sauter, A. Parsania, Generalized DG-methods for highly indefinite Helmholtz problems, J. Sci. Comp. 57:536–581, 2013

For further information about the seminar, please visit this webpage.

]]>A Nonlinear Spectral Analysis of Gradient Flows

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For further information about the seminar, please visit this webpage.

]]>Nonlinear acoustics: time integration, optimization, and open domain problems

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- absorbing boundary conditions for the treatment of open domain problems;
- 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 domain Ω, e.g, for computational purposes, artificial reflections on the boundary ∂Ω have to be avoided. This can be done by imposing so-called absorbing boundary conditions ABC that induce dissipation of outgoing waves. Here it will turn out to be crucial to take into account nonlinearity of the PDE also in these ABC. This is joint work with Igor Shevchenko (Imperial College London).

In the context of applications in HIFU, focusing of nonlinearly propagating waves amounts to optimization problems. The design of ultrasound excitation via piezoelectric transducers leads to a boundary control problem; focusing high intensity ultrasound by a silicone lens requires shape optimization. For both problem classes, we will discuss the derivation of gradient information in order to formulate optimality conditions and drive numerical optimization methods. This is joint work with Christian Clason (University of Duisburg-Essen), Vanja Nikolić (TU München), and Gunther Peichl (University of Graz).

Finally we will provide an outlook on imaging with nonlinearly acoustic waves, which amounts to identifying spatially varying coefficients (sound speed and/or coefficient of nonlinearity) in the Westervelt equation. This is recent joint work with Masahiro Yamamoto (University of Tokyo) and William Rundell (Texas A&M University).

For further information about the seminar, please visit this webpage.

]]>High-order finite element methods for the fractional Laplacian

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For further information about the seminar, please visit this webpage.

]]>Uniqueness and global convergence for a discretized inverse coefficient problem

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We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of determining an unknown inverse Robin transmission coefficient in an elliptic PDE. Using a relation to monotonicity and localized potentials techniques, we show that a piecewise-constant coefficient on an a-priori known partition with a-priori known bounds is uniquely determined by finitely many boundary measurements and that it can be uniquely and stably reconstructed by a globally convergent Newton iteration. We derive a constructive method to identify these boundary measurements, calculate the stability constant and give a numerical example.

For further information about the seminar, please visit this webpage.

Adaptive time-stepping for S(P)DEs with non-Lipschitz drift

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Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with oblique boundary conditions

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

This is joint work with Guillaume Bertoli (Université de Genève) and Christophe Besse (Institut de Mathématiques de Toulouse).

For further information about the seminar, please visit this webpage.

]]>The Liouville equation, its extensions, and related optimal control problems

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This talk provides an introduction to the formulation and solution of optimal control problems 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 these agents are represented in terms of density functions. For this purpose, expected-value cost functionals are considered that include attracting potentials and different costs of the controls, whereas the control mechanism in the governing models is part of the drift or is included in a collision term.

In this talk, theoretical and numerical results concerning ensemble optimal control problems with Liouville, Fokker-Planck and linear Boltzmann equations are presented.

For further information about the seminar, please visit this webpage.

An evolutionary, fast, and oblivious compression approach to Volterra integral operators

Tags: TAG Events Forschung Mathematik, TAG Events DMI]]>

For further information about the seminar, please visit this webpage.

Numerical methods for fractional PDEs with applications to spatial statistics

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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 theoretical outcomes, as well as a statistical application where we use the method for inference, i.e., for parameter estimation given data, and for spatial prediction.

For further information about the seminar, please visit this webpage.

]]>High order finite element discretizations of Helmholtz problems

Tags: TAG Events DMI, TAG Events Forschung Mathematik]]>

It has received much attention since it is widely employed in applications,

but still challenging to numerically simulate 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 to 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 corresponds to the well-studied Poisson equation. We take advantage of this classical setting to recall central concepts of the finite 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 quasi-optimality is lost.

We provide a detailed analysis, as well as numerical examples, which

show that higher order methods are less affected by this phenomena,

and thus more suited to discretize high-frequency problems.

Before drawing our main conclusions, we briefly discuss advanced topics,

such as the use of unfitted meshes in highly heterogeneous media

and mesh refinements around re-entrant corners.

For further information about the seminar, please visit this webpage.

]]>On error estimates for optimal control problems with functions of bounded variation

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We consider a convex optimal control problem governed by a partial differential equation in one space dimension which is controlled by a right-hand-side living in the space of functions with bounded variation. These functions tend to favor optimal controls that are piecewise constant with often finitely many jump poins. We are interested in deriving finite element discretization error estimates for the controls when the state 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, usual techniques for estimating the control error cannot be applied. Instead, these have to be derived from (suboptimal) error estimates for the state, which can later be improved.

For further information about the seminar, please visit this webpage.

]]>Numerical aspects of elliptic diffusion problems on random domains

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For further information about the seminar, please visit this webpage.

]]>Aposteriori error analysis and adaptive schemes for the wave equation

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Notes: This is based on joint work with E.H. Georgoulis, C. Makridakis and J.M. Virtanen.

References:

[1] W. Bangerth and R. Rannacher, J. Comput. Acoust. 9(2):575–591, 2001.

[2] C. Bernardi and E. Süli, Math. Models Methods Appl. Sci. 15(2):199--225, 2005.

[3] E. H. Georgoulis, O. Lakkis, and C. Makridakis. IMA J. Numer. Anal., 33(4):1245–1264, 2013, http://arxiv.org/abs/1003.3641

[4] E. H. Georgoulis, O. Lakkis, C. Makridakis, and J. M. Virtanen. SIAM J. Numer. Anal., 54(1), 2016, http://arxiv.org/abs/1411.7572

For further information about the seminar, please visit this webpage.

]]>Multirate explicit stabilized integrators for stiff differential equations

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The efficiency of these methods deteriorates 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 chosen inversely proportional to the smallest element size in order to achieve stability, thus largely wasting computational resources on locally-refined meshes. We first tackle this issue by replacing the right hand side of the PDE with an averaged force, 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 give rise to multirate RKC schemes, for which the number of expensive 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 stability bottleneck caused by a few of fine mesh elements can be overcome without sacrificing accuracy.

For further information about the seminar, please visit this webpage.

]]>Quantitative seismic imaging using reciprocity waveform inversion with arbitrary probing sources

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We investigate the deployment of new devices developed in the acoustic setting: the dual-sensors, which are able to capture both the pressure field and the vertical velocity of the waves. For solving the inverse problem, we define a new cost function, adapted to these two types of data and based upon the reciprocity. We first note that the stability of the problem can be shown to be Lipschitz, assuming piecewise linear parameters. In addition, reciprocity waveform inversion allows a separation between the observational and numerical acquisitions. In fact, the numerical sources do not have to coincide with the observational ones, offering new possibilities to create adapted computational acquisitions, consequently reducing the numerical cost. We illustrate our approach with three-dimensional medium reconstructions, where we start with minimal information on the target models. We also extend the methodology for elasticity.

Eventually, if time allows, we shall explore the model representation in numerical seismic inversion, where the adaptive eigenspace method appears as a promising approach to have a compromise between number of unknowns and resolution.

References

[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).

[2] E. Beretta, M. V. De Hoop, F. Faucher, and O. Scherzer, Inverse boundary value problem for the Helmholtz equation: quantitative conditional Lipschitz stability estimates. SIAM Journal on Mathematical Analysis, 48(6), pp.3962-3983 (2016).

[3] M. J. Grote, M. Kray, and U. Nahum, Adaptive eigenspace method for inverse scattering problems in the frequency domain. Inverse Problems, 33(2), 025006 (2017).

[4] H. Barucq, F. Faucher, and O. Scherzer, Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation. arXiv preprint arXiv:1903.08991 (2019).

For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage.

Adaptive Iterative Linearization Galerkin Methods for Nonlinear Partial Differential Equations

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For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage. ]]>

Some recent results on the approximation of conservation laws: revisiting the notion of conservation

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However, this is not the end of the story. All these works use a one dimensional way of thinking: the main player is the normal flux across cell interfaces. In addition there are several excellent numerical methods that do not fit the form of the lax Wendroff 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 that an equivalent flux formulation, with a suitable definition of what is a flux, can be explicitly constructed (and computed), so that any reasonable scheme can be put in a finite volume 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 equations-, 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 mathematik seminar-in-numerical-analysis internal link in current>webpage.

]]>

Recent methods for solving the high-frequency Helmholtz equation on a regular mesh

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For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage. ]]>

Multilevel Uncertainty Quantification with Sample-Adaptive Model Hierarchies

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This is joint work with Gianluca Detommaso (Bath), Tim Dodwell (Exeter) and Jens Lang (Darmstadt).

For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage. ]]>

Solution Spaces in Complex Systems Design

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For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage. ]]>

A reduced SQP Method for Shape Optimisation of Flow Control Problems

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For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage. ]]>

Nonlinear Spectral Decomposition

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For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage.

]]>Geometry inpainting with preserving singularities and curvature

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For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis external-link-new-window internal link in current>webpage. ]]>

Probabilistic numerical methods, Bayesian inference and multiscale inverse problems

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In the second part of the talk, we will discuss a Bayesian approach for inverse problems involving elliptic partial differential equations 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. Convergence 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 approach based on numerical homogenization and reduced basis methods 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 for uncertainty quantification in chaotic and geometric numerical integration, Preprint (2018), submitted for publication.

A. Abdulle, A. Di Blasio, Numerical homogenization and model order reduction for multiscale inverse problems, to appear in SIAM MMS.

A. Abdulle, A. Di Blasio, A Bayesian numerical homogenization method for elliptic multiscale inverse problems, Preprint (2018), submitted for publication.

For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage. ]]>

Optimal transport in seismic imaging based on full waveform inversion

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For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis external-link-new-window internal link in current>webpage.

]]>Wavenumber-Explicit Analysis for High-Frequency Maxwell Equations

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This talk comprises joint work with Prof. Markus Melenk, TU Wien.

For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage.

]]>Transparent boundary conditions for sound propagation in human lungs, modelled by fractal trees

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The construction of such transparent conditions relies on the approximation of the Dirichlet-to-Neumann (DtN) operator, whose symbol is a meromorphic function that satisfies a certain non-linear functional equation. We present two approaches to approximate the DtN in the time domain, alternative to the low-order absorbing boundary conditions, which appear inefficient in this case.

The first approach 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 problem. In this case the combination of the explicit leapfrog method for the volumic terms and the implicit trapezoid rule for the boundary terms leads to a second-order scheme stable under the classical CFL condition.

The second approach is motivated by the Engquist-Majda ABCs (cf. [Engquist, Majda 1977]), and consists in approximating the DtN by local operators, obtained from the truncation of the meromorphic series which represents the symbol of the DtN. We show how the respective error can be controlled and provide some complexity estimates.

This is a joint work with Patrick Joly (INRIA, France) and Adrien Semin (TU Darmstadt, Germany).

For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage.

]]>Spline curves in Riemannian Shape Spaces

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On energy-stable discontinuous Galerkin approximations for scattering problems in complex elastic media with adaptive curvilinear meshes

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Optimal transport in seismic imaging based on full waveform inversion

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Machine Learning Driving Innovative Computing Concepts

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An Adaptive Multiscale Approach for Electronic Structure Methods

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Two-level Domain Decomposition preconditioners for Maxwell equations

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[1] I. G. Graham, E. A. Spence, and E. Vainikko. Domain decomposition preconditioning for high-frequency Helmholtz problems with absorption. Mathematics of Computation, 86(307):2089–2127, 2017.

]]>Mesh Refinement for T-splines in any dimension

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As 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.

]]>Optimal adaptivity in finite and boundary element methods

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In this talk, we first consider an adaptive FEM with hierarchical splines of arbitrary 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 generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted-residual a posteriori error estimator. In [1], we proved linear convergence of the error estimator with optimal algebraic rate.

Next, we consider an adaptive BEM with hierarchical splines of arbitrary degree for weakly-singular integral equations of the first kind that arise from the solution of linear elliptic PDE systems of second order with constant coefficients and Dirichlet boundary condition. We assume that the boundary of the geometry is the union of surfaces that can be parametrized over the unit square. Again, we propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions, where adaptivity is driven by some weighted-residual a posteriori error estimator. In [2], we proved linear convergence of the error estimator with optimal algebraic rate. In contrast to prior works, which are restricted to the Laplace model problem, our analysis allows for arbitrary elliptic PDE operators of second order with constant coefficients.

Finally, for one-dimensional boundaries, we investigate an adaptive BEM with standard splines instead of hierarchical splines. We modify the corresponding algorithm so that it additionally uses knot multiplicity increase which results in local smoothness reduction of the ansatz space. In [3], we proved linear convergence of the employed weighted-residual error estimator with optimal algebraic rate.

[1] G. Gantner, D. Haberlik, and Dirk Praetorius, Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines. Math. Mod. Meth. in Appl. S., Vol. 27, 2017.

[2] G. Gantner, Optimal adaptivity for splines in finite and boundary element methods, PhD thesis, TU Wien, 2017.

[3] Michael Feischl, Gregor Gantner, Alexander Haberl, and Dirk Praetorius. Adaptive 2D IGA boundary element methods. Eng. Anal. Bound. Elem., Vol. 62, 2016.

]]>Aspects of Adaptive Galerkin FEM for random PDEs

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A Mixed Finite-Element Method for Gas Transport on Pipeline Networks

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SQP Methods for Shape Optimization Based on Weak Shape Hessians

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The talk introduces shape optimization as a special sub-class of PDE constraint optimization problems. The main focus here will be on generating Newton-like methods for large scale applications. The key for this endeavor is the derivation of the shape Hessian, that is the second directional derivative of a cost functional with respect to geometry changes in a weak form based on material derivatives instead of classical local shape derivatives. To avoid human errors, a computer aided derivation system is also introduced.

The methodologies are tested on problem from fluid dynamics and geometric inverse problems.

]]>Quantum Machine Learning (QML)

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Local time discretisation based on transmission problem for linear wave equations: construction and analysis

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Small volume asymptotics for elliptic equations and their use in impedance tomography

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IGA BEM for Maxwell eigenvalue problems

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To live up to the promises of IGA, namely closing the gap bewteen design and analysis, we suggest an IGA boundary element method (BEM). We will review the state-of-the-art of all relevant building blocks. We will address the B-spline de Rham complex on a boundary manifold, the Galerkin discretization of the electric field integral equation, 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 methods” will also be presented, in particular Adaptive Cross Approximation [5] and Calderón preconditioning [4].

[1] W.-J. Beyn. An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl, 436(10):3839–3863, 2012.

[2] A. Buffa, G. Sangalli, and R. Vázquez. Isogeometric analysis in electromagnetics: B-splines approximation. Comput Method Appl M, 199:1143–1152, 2010.

[3] J. Corno, C. de Falco, H. De Gersem, and S. Schöps. Isogeometric simulation of Lorentz detuning in superconducting accelerator cavities. Comput Phys Commun, 201:1–7, February 2016.

[4] J. Li, D. Dault, B. Liu, Y. Tong, and B. Shanker. Subdivision based isogeometric analysis technique for electric field integral equations for simply connected structures. J Comput Phys, 319:145–162, 2016.

[5] B. Marussig, J. Zechner, G. Beer, and T.-P. Fries. Fast isogeometric boundary element method based on independent field approximation. Comput Method Appl M, 284:458–488, 2015.

The work of Stefan Kurz is supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt.

]]>Optimal integration in reproducing kernel Hilbert spaces

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Reproducing kernel methods

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Furthermore, we will present a deterministic a priori (often exponential) convergence analysis via sampling inequalities which can be employed to analyze a large class of regularized reconstruction schemes.

Such an analysis enables us to derive a priori couplings of various discretization and regularization parameters. Such parameters can range from iteration numbers in numerical linear algebra, numerical evaluation of input parameters to rounding errors.

An important issue is the choice of the reproducing kernel. We will discuss some implications of such choices and address the problem of approximating the solution of a parametric partial differential equation using problem adapted kernels.

This is partly based on joint work with M. Griebel and B. Zwicknagl (both Bonn University).

]]>Numerical homogenization beyond periodicity and scale separation with applications to wave propagation in heterogeneous media

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Quasi-Optimal Schwarz Domain Decomposition Methods for Time Harmonic Waves

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Algorithmic patterns for hierarchical matrices on many-core processors

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We are interested to use these many-core processors for the full H-matrix construction and application process. A motivation for this interest lies in the well-known claim that future 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-matrix algorithms and adaptive cross approximations.

In 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 main part of the talk will be dedicated to the challenges of H-matrix parallelizations on many-core hardware with the specific 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 different parallel formulations of the algorithms.

]]>Verifiable conditions for convergence rates of regularization methods for inverse problems

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Over the last years Variational Source Conditions (VSCs) have become a standard assumption for the analysis of these methods. Compared to spectral source conditions they have a number of advantages: They can be used for general nonquadratic penalty and data 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 tangential cone condition). However, so far only few sufficient conditions for VSCs for specific inverse problems are known.

To 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 solution, and the other one the degree of ill-posedness of the operator. For a number of important linear inverse problems this leads to equivalent characterizations of VSCs in terms of Besov spaces and necessary and sufficient conditions for rates of convergence. We also discuss the application of our strategy to nonlinear parameter identification and inverse medium scattering problems where it provides sufficient conditions for VSCs in terms of standard function spaces.

]]>Hierarchical Tensor Approximation

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We propose an alternative: the Galerkin trial space is constructed using a partition 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 supports of the cut-off functions. The main challenge lies in the construction of the corresponding system matrix, since even determining the sparsity pattern involves interactions between cut-off functions on different levels of the mesh hierarchy.

Our approach leads to a sparse system matrix, the basis functions are convenient tensor products of functions on lower-dimensional domains, and local regularity can be exploited by variable-order interpolation in order to obtain close to optimal complexity.

]]>Towards code modernization and porting on Intel many-core architecture: the story of Gadget code

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Parabolic equations with random coefficients on moving hypersurfaces

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This is a joint work with Charles M. Elliott (University of Warwick, UK), Ralf Kornhuber (Free University Berlin, Germany) and Thomas Ranner (University of Leeds, UK). ]]>

Discontinuous Galerkin Method for seismic imaging in frequency domain

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Robust Coarse Space Construction for Domain Decomposition Methods

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Model-aware Newton-solver for electrical impedance tomography

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Model order reduction with adaptive finite element POD and application to uncertainty quantification

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Standard techniques assume that all snapshots use one and the same spatial mesh. I present a generalization for unsteady adaptive finite elements, where the mesh can change from time step to time step and, in the case of stochastic sampling, from realization to realization. I will answer the following questions: How can the coding effort for creating such a reduced-order model be minimized? How can the union of all snapshot meshes be avoided? What is the main difference between static and adaptive snapshots in the error analysis of Galerkin reduced-order models?

As a numerical test case I consider a two-dimensional viscous Burgers equation with smooth initial data multiplied by a normally distributed random variable. The results illustrate the convergence properties with respect to the number of POD basis functions and indicate possible savings of computation time.

]]>Dimension-adaptive sparse grid simulations of multiscale viscoelastic flows

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Intrinsic Finite Element Methods

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Controllability method for time-harmonic wave equations

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Crystallographic Applications of Uniform Distribution Theory

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Multilevel Monte Carlo methods for multiscale problems

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In this 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 multiscale coefficient and then extended to approximations of statistical solutions to the Navier-Stokes equations.

]]>Conforming Approximations of Bingham Type Fluid Flows

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In the general context of a convex lower semi-continuous functional on a Hilbert space, 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, for which we can apply the regularization method. We consider the time implicit regularized, linearized or not, algorithms, and prove their convergence for general total variation functionals.

]]>Monte Carlo Methods and Mean Field Approximation for Stochastic Particle Systems

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In 2012, my Master thesis developed different versions of Multilevel Monte Carlo (MLMC) for particle systems, both with respect to time steps and number of particles and proposed using particle antithetic estimators for MLMC. In that thesis, I showed moderate savings of MLMC compared 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 proposing the use of our recent Multi-index Monte Carlo method to obtain improved convergence rates.

]]>Schwarz Methods as Preconditioners for Non-linear Problems

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Numerical modeling of light interaction with matter on the nanoscale using DGTD methods

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This is a joint work with Claire Scheid and Jonathan Viquerat.

[1] Fezoui, L., S. Lanteri, S. Lohrengel, and S. Piperno. Convergenceand stability of a discontinuous Galerkin time-domain method for the3D heterogeneous Maxwell equations on unstructured meshes. ESAIM:Math. Model. Numer. Anal., Vol. 39, No. 6, 1149-1176, 2005.

]]>Analytical and Numerical Homogenization of Maxwell's Equations

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This is a joint work with Sonia Fliss and Patrick Ciarlet.

]]>A fully discrete approximation of the one-dimensional stochastic wave equation

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This is a joint work with Lluís Quer-Sardanyons, Universitat Autònoma de Barcelona.

]]>One Shot Inverse Scattering and Source Splitting

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Besides standard regularization methods, which are often iterative, a completely different methodology - so-called qualitative reconstruction methods - has attracted a lot of interest recently. These algorithms recover specific qualitative properties of scattering objects or anomalies 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 or 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 by the unknown objects corresponding to infinitely many primary waves are given - at least theoretically.

We consider the inverse source problem for the Helmholtz equation as a means to provide a qualitative inversion algorithm for inverse scattering problems for acoustic or electromagnetic waves with a single excitation only. Probing an ensemble of obstacles by just one primary wave at a fixed frequency and measuring the far field of the corresponding scattered wave, the inverse scattering problem that we are interested in consists in reconstructing the support of the scatterers. To this end we rewrite the scattering problem as a source problem and apply two recently developed algorithms - the inverse Radon approximation and the convex scattering support - to recover information on the support 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 components. This information is then utilized to split the far field into 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 individual source components as a reconstruction of the individual scatterers. We discuss this algorithm and present numerical results.

]]>BEM++ - Efficient solution of boundary integral equation problems

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Multilevel Monte Carlo methods for multiscale problems

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Performance Engineering and Sparse Matrices: Introduction, applications and supercomputing

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Mathematical Models of Mosquito Population Dynamics and Malaria

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Mathematical models can help to determine more efficient combinations of existing and new interventions in reducing malaria transmission and delaying the spread of resistance. We present difference equation models of mosquito population dynamics and malaria in mosquitoes; and ordinary differential equation models of mosquito movement and population dynamics. We analyse these models to provide threshold conditions for the survival of mosquitoes and show the existence of invariant positive states; and run numerical simulations to provide quantitative comparisons of interventions that target mosquitoes with varying levels of resistance.

]]>Ten good reasons for using kernel reconstructions in adaptive finite volume particle methods

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Minimal energy problems with hypersingular Riesz potentials

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Some examples of numerical resolution of PDEs from the physics with FreeFem++

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Piezoelectric problems

Thermal problems with thermal resistances

Elasticity problems

Problems of fluid mechanics like incompressible Navier-Stokes

Problem of melting and/or solidification of the ice. (Boussinesq with specific heat)

Fourth order energy-preserving locally implicit discretization for linear wave equations

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Preservation of Volumes in Nonholonomic Mechanics

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The study of nonholonomic mechanical systems is challenging 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 identity. One now speaks of an almost Poisson bracket.

The failure of the Jacobi identity leads to phenomena that are not shared by usual Hamiltonian systems. Open questions in nonholonomic mechanics that have received attention in recent years include determining general conditions for measure preservation, existence of asymptotic equilibria, relationship between symmetries and con- servation laws, reduction, and integrability.

In the first part of this talk I will present a basic introduction to nonholonomic mechanics. I will then present my recent work with Y. Fedorov and J. C. Marrero in which we study the problem of measure preservation for nonholonomic systems possessing symmetries in a systematic manner. Our method allows us to identify specific parameter values for which there exists a preserved measure for concrete mechanical examples.

]]>Numerical analysis of boundary element methods for impedance transmission conditions

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With impedance transmission conditions (ITCs) we can propose boundary element formulations on the sheet mid-line (or mid-surface) only. In the beginning of the talk we give a motivation of meaningful impedance transmission conditions, which relate jumps and mean values of the Dirichlet and Neumann traces on the mid-line. This relation may

involve surface differential operators, as for boundary conditions of Wentzell's type, and depend on frequency, conductivity, sheet thickness and sheet geometry e.g. curvature). These parameters may take small or large values and may lead to singularly perturbed

boundary integral equations.

We will introduce related boundary element methods in two and three dimensions and analyse well-posedness and discretisation error depending on the model parameters. Numerical experiments confirm the convergence order of the discretisation error of the proposed BEM and that the discretisation error behaves for smooth enough sheets equivalent to the exact solution when varying the model parameters. The results obtained for the eddy current model, for which a Poisson equation has to be solved outside the mid-line, can be transfered to the Helmholtz equation and to transmission conditions

arising from other models.

Analysis of explicit multirate Runge-Kutta schemes for conservation laws

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Sparse grid quadrature and interpolation methods for solving the Schroedinger equation

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Perturbation methods and low rank approximations for the Darcy equation with lognormal permeability

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We consider perturbation methods based on Taylor expansion of the solution of the PDE around the nominal permeability value. Successive higher order corrections to the statistical moments such as pointwise mean and covariance of the solution can be obtained recursively from the computation 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 correlations, we adopt a low-rank format, namely the so called tensor-train (TT) format.

We show that, on the one hand, the Taylor series does not converge globally, so that it only makes sense to compute corrections up to a maximum critical order, beyon which the accuracy of the solution deteriorates insetad of improving. On the other hand, we show on some numerical test cases, the effectiveness of the proposed approach in case of a moderately small variance of the log-normal permeability field.

]]>From asymptotic analysis for simulation to generalized Ventcel's boundary value problem

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I will explain how some mathematical questions about the loss of coercivity arise from the computation of the profiles appearing in the expansion. Ventcel boundary conditions are second order differential conditions that appears when looking for a transparent boundary condition for an exterior boundary value problem in planar linear elasticity. The goal is to bound the infinite domain by a large “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 several works. The wellposedness of such problems was still open. I will present, in the generic case, existence and uniqueness result of the solution for the Ventcel boundary value problem without the sign condition. Then, I will consider perforated geometries and give conditions to remove the genericity restriction.

]]>On the robustification of optimum experimental design problems

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In this talk, we consider robust formulations for optimum experimental design that work under significant uncertainty. Our focus is on problem settings in which the model is described by differential equations of some type that are solved numerically. Our approach is based on a semi-infinite programming formulation in which we exploit additional problem structure, together with sparse grids, to ensure tractability. The talk includes numerical experiments to illustrate and compare the effectiveness of the approaches.

]]>Inside-Outside Duality and Computation of Interior Eigenvalues from Far-Field Data

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Similar connections also exist for inverse exterior scattering problems - for instance, if zero is an eigenvalue of the far-field operator at a fixed wave number, then the squared wave number is an interior eigenvalue. Despite it is in general wrong that interior eigenvalues correspond to zero being an eigenvalue of the far field operator, one can prove a pretty direct characterization of interior eigenvalues via the behavior of the phases of the eigenvalues of the far-field operator.

In this talk, we present this characterization and sketch its proof for Dirichlet, Neumann, and Robin boundary conditions. Then we extend this theory to impenetrable scattering objects and show via a couple of numerical examples that one can indeed use this characterization to compute interior eigenvalues of unknown scattering objects from the spectrum of their far-field operators.

Our motivation to study this so-called inside-outside duality comes from a paper by Eckmann and Pillet (1995). This is joint work with Andreas Kirsch (KIT) and Stefan Peters (University of Bremen).

]]>Efficient multigrid calculation of the far field map of the Helmholtz equation

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The Hessian operator in Full Waveform Inversion: quantitative imaging of complex subsurface structures

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A new approach to solve the inverse scattering problem for the wave equation

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1 - the Time-Reversed Absorbing Condition method: It combines time reversal techniques and absorbing boundary conditions to reconstruct and regularize the signal in a truncated domain that encloses the obstacle. This enables us to reduce the size of the computational domain where we solve the inverse problem, now from virtual internal measurements.

2 - the Adaptive Inversion method: It is an inversion method which looks for the value of the unknown wave propagation speed in a basis composed by eigenvectors of an elliptic operator. Then, it uses an iterative process to adapt the mesh and the basis and improve the reconstruction.

We present several numerical examples in two dimensions to illustrate the efficiency of the combination of both methods. In particular, our strategy allows (a) to reduce the computational cost, (b) to stabilize the inverse problem and (c) to improve the precision of the results.

]]>Adaptive Approximations of Parametric PDE-Constrained Control Problems

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Yet another level of challenge are control problems constrained by evolution PDEs involving stochastic or countably many infinite parametric coefficients: for each instance of the parameters, this requires the solution of the complete control problem.

Our method of attack is based on the following new theoretical paradigm. It is first shown for control problems constrained by evolution PDEs, formulated in full weak space-time form, that state, costate and control are analytic as functions depending on these parameters. Moreover, we establish that these functions allow expansions in terms of sparse tensorized generalized polynomial 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 of an index set, allowing for concurrent approximations of state, co-state and control for which the gpc approximations attain rates of best N-term approximation. These findings serve as the analytical foundation for the development of corresponding sparse realizations in terms of deterministic adaptive Galerkin approximations of state, co-state and control on the entire, possibly infinite-dimensional parameter space.

The results were obtained with Max Gunzburger (Florida State University) and with Christoph Schwab (ETH Zuerich).

]]>Stochastic Shape Optimization by Analogues from Operations Research

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At this point, there is a perfect match with two-stage stochastic programming: after having taken a non-anticipative decision in the first stage, and having observed the random data, a well-defined second-stage problem remains and is solved to optimality. Suitable objective functions complete the formal descriptions of the models, for instance, costs in the stochastic-programming setting and compliance or tracking functionals in shape optimization.

Stochastic programming now offers a wide collection of models to address shape optimization under uncertainty. This starts with risk neutral models, is continued by mean-risk optimization involving different risk measures, and will finally lead to analogues in shape optimization of decision problems with stochastic-order (or dominance) constraints.

In the talk we will present these models, discuss solution methods, and report some computational tests.

]]>Collocation methods for nonlinear Riemann-Hilbert problems on doubly connected domains

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The nonlinear boundary conditions for the desired holomorphic solutions lead to nonlinear singular integral equations on the boundary which belong to the class of quasiruled Fredholm maps defined on quasicylindrical domains in appropriate separable Banach spaces.

The closed boundary data give a priori estimates for the modulus of solutions which in turn implies a priori estimates in the Sobolev spaces considered here. For this class of problems, the Shnirelman--Efendiev degree of mappings can be defined which allows to investigate the existence of solutions if the boundary conditions satisfy some topological assumptions.

The lifting of the boundary value problem via holomorphic transformation onto the universal covering of the unit disc allows to construct a homotopic deformation of the lifted nonlinear singular integral equations to a uniquely solvable case which implies that the degree of mapping is 1 and existence of (in fact at least two) solutions follows.

If the nonlinear integral equations on the boundary are appoximated by trigonometric point collocation then the theory also implies that approximate solutions exist and converge asymptotically.

]]>On the optimization of current carrying multicables

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Simulation results and experimental studies show that the positioning of the single cables has important influence on the maximum temperatures. In order to find an optimal cable design, i.e. to arrange the single cables with fixed cross section and current such that the maximum temperature is minimized, a shape optimization problem is formulated. We derive an adjoint system and the shape gradient using the formal Lagrange approach. The effect of the discontinuity of some coefficients on the shape gradient is shown. By application of different (nonlinear) optimizers combined with the finite element solver COMSOL Multiphysics, a solution is obtained numerically. In this talk, we present the modeling of the problem, the derivation of the shape gradient and numerical results.

This is joint work with Helmut Harbrecht and Thomas Apel.

]]>On Uncertainties in Low-Frequency Electromagnetics

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Joint work with Sebastian Schöps and Thomas Weiland.

]]>Exponential Krylov subspace time integration for electromagnetic modeling: some recent advances

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Finite element approximation of large bending isometries

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A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics

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Adjoint methods for gradient-based optimization of oil recovery

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A posteriori error analysis for optimal control problems governed by partial differential equations

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Applications of Time Reversed Absorbing Condition

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Repeated modifications of orthogonal polynomials by linear divisors

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Numerics for Inverse Problems in Biomedical Imaging

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The resulting parameter estimation schemes have the underlying partial differential equations as side-constraints, and the solution of these optimization problems often requires solving the partial differential equation thousands or hundred of thousands of times. The development of efficient schemes is therefore of great interest for the practical use of such imaging modalities in clinical settings. In this talk, the formulation and efficient solution strategies for such inverse problems will be discussed, and we will demonstrate its efficacy using examples from our work on Optical Tomography, a novel way of imaging tumors in humans and animals. The talk will conclude with an outlook to even more complex problems that attempt to automatically optimize experimental setups to obtain better images.

]]>Fast methods for computing pseudospectral quantities

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This talk is based on joint work with Nicola Guglielmi, Christian Lubich, and Bart Vandereycken.

]]>Non-conforming high order methods for seismic wave propagations in heterogeneous media

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Here, 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 two and three dimensional heterogeneous media. The main advantage with respect to conforming discretizations, as SE method, is that DGSE and MSE discretizations can accommodate discontinuities, not only in the parameters, but also in the wavefield, while preserving the energy. The domain of interest Ω is assumed to be union of polygonal subdomain Ω_{i}. We allow this subdomain decomposition to be geometrically non-conforming. Inside each subdomain Ω_{i}, a conforming high order finite element space associated to a partition *T*_{hi}(Ω_{i}) is introduced. We consider different polynomial approximation degrees within different subdomains. To handle non-conforming meshes and non-uniform polynomial degrees across ∂Ω_{i} , a DG or a Mortar discretization is considered.

Applications of the DGSE and MSE methods to simulate realistic seismic wave propagation problems are presented.

Joint work with: P.F. Antonietti, A. Quarteroni and F. Rapetti.

]]>The QDMRG algorithm and recent advances in tensor approximation

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Joint work with Th. Rohwedder and S. Holtz

]]>All-electron Density Functional Theory Calculations for atoms and molecules using Multiwavelets and Massively Parallel Architectures

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Reduced Basis Surrogate Models for Parameter Optimization of Evolution Problems

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A Heterogeneous Multiscale Method for highly-oscillatory Hamiltonian systems with solution-dependent frequencies

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We will explain the Heterogeneous Multiscale Method (HMM)[1], which is believed to provide a numerical method for all kind of multiscale systems to overcome the difficulties of numerical integration generated by highly-oscillatory components. We will formulate the HMM for highly-oscillatory Hamiltonian systems with solution-dependent frequencies more precisely for the double spring pendulum with very stiff springs. Finally we will discuss the drawbacks of this method in case of solution-dependent frequencies.

[1] E, W.; Engquist, B.: The heterogeneous multi-scale method, *Comm. Math. Sci.*, **1**, 87--133, 2003.

Time Domain Decomposition based on Multirate Multistep Timestepping Techniques for the Particle-In-Cell Method

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Hierarchical Matrix Approximation with Blockwise Constraints and h-Independent Convergence for Elliptic Problems

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In this talk we present a new technique to locally preserve constraints inside the hierarchical matrix approximation. Numerical experiments indicate that imposing these local constraints leads to constant number of iteration steps when solving elliptic partial differential equations of second order while without preserving these constraints the number of iteration steps grow as *h* → 0. We will further discuss this approach from the theoretical point of view and will sketch why our approximate hierarchical *LU* decomposition leads to a spectral equivalent approximation.

This is joint work with Mario Bebendorf and Michael Bratsch from the University of Bonn.

]]>Explicit continuous and discontinuous Galerkin methods for 3D seismic modeling with the second order wave equation

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Probabilistic UQ for PDEs with Random Data: A Case Study

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We compare two approaches: Gaussian process emulators and stochastic collocation combined with geostatistical techniques for determining the parameters of the input random field's probability law. 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.

]]>Stochastic partial differential equations: examples and recent results

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Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations

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