Aposteriori error analysis for Galerkin finite element method s have proven very successful tool in developing mathematically rigorous a daptive mesh refinement algorithms for implicit/space-time evolution equat ions. In this work we extend rigorous adaptivity principles to explicit ti me-stepping for the wave equation.
\nI will review in the first part of the talk the state of the art for the wave equation. In the second par t\, I will present recent work in connection to fully practical explicit s chemes such as the Leapfrog method and local time step variants thereof.\n
This talk is mostly based on recent joint work with M Grote\, C San tos and earlier work with EH Georgoulis\, C Makridakis and J Virtanen.
\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20251219T123000 END:VEVENT BEGIN:VEVENT UID:news1962@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20251130T190800 DTSTART;TZID=Europe/Zurich:20251212T110000 SUMMARY:Seminar in Numerical Analysis: Gilles Vilmart (University of Geneva ) DESCRIPTION:Explicit stabilized methods are highly efficient time integrato rs for large and stiff systems of ODEs especially when applied to semi-dis crete parabolic problems. However\, when local spatial mesh refinement is introduced\, their efficiency decreases\, since the stiffness is driven by only the smallest mesh element. A natural approach is to split the system into fast stiff and slower mildly stiff components. In this context\, [A. Abdulle\, M.J. Grote\, and G. Rosilho de Souza 2022] proposed the order o ne multirate explicit stabilized method (mRKC). We extend their approach t o second order and introduce the new multirate ROCK2 method (mROCK2)\, whi ch achieves high precision and allows a step-size strategy with error cont rol.\\r\\nThis talk is based on joint work with Mathieu Benninghoff (Unive rsity of Geneva).\\r\\nFor further information about the seminar\, please visit this webpage [https://dmi.unibas.ch/de/forschung/mathematik/seminar- in-numerical-analysis/]. X-ALT-DESC:Explicit stabilized methods are highly efficient time integra tors for large and stiff systems of ODEs especially when applied to semi-d iscrete parabolic problems. However\, when local spatial mesh refinement i s introduced\, their efficiency decreases\, since the stiffness is driven by only the smallest mesh element. A natural approach is to split the syst em into fast stiff and slower mildly stiff components. In this context\, [ A. Abdulle\, M.J. Grote\, and G. Rosilho de Souza 2022] proposed the order one multirate explicit stabilized method (mRKC). We extend their approach to second order and introduce the new multirate ROCK2 method (mROCK2)\, w hich achieves high precision and allows a step-size strategy with error co ntrol.
\nThis talk is based on joint work with Mathieu Benninghoff ( University of Geneva).
\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20251212T123000 END:VEVENT BEGIN:VEVENT UID:news1961@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20251126T164729 DTSTART;TZID=Europe/Zurich:20251205T110000 SUMMARY:Seminar in Numerical Analysis: Mamadou N'diaye (Université Polytec hnique Hauts-de-France) DESCRIPTION:In this presentation\, we address the numerical solution of the second-order acoustic wave equation using high-order explicit time-integr ation methods on CPU-GPU architectures. After discretizing the spatial dom ain with spectral element method\, we compare several time-integration sch emes for the resulting system of ODEs. We first consider the classical sec ond-order leapfrog method\, followed by higher-order Runge-Kutta-Nyström (RKN) schemes and the modified equation approach. We then focus on the ana lysis of the stability properties of the RKN method. Strategies for incorp orating the damping term\, which involves the first-order time derivative\ , are discussed with particular attention to preserving the order of conve rgence of the schemes. The proposed approaches are compared in terms of co mputational efficiency\, and 3D numerical simulation for the acoustic wave equation are presented.\\r\\nFor further information about the seminar\, please visit this webpage [https://dmi.unibas.ch/de/forschung/mathematik/s eminar-in-numerical-analysis/]. X-ALT-DESC:In this presentation\, we address the numerical solution of t he second-order acoustic wave equation using high-order explicit time-inte gration methods on CPU-GPU architectures. After discretizing the spatial d omain with spectral element method\, we compare several time-integration s chemes for the resulting system of ODEs. We first consider the classical s econd-order leapfrog method\, followed by higher-order Runge-Kutta-Nyströ m (RKN) schemes and the modified equation approach. We then focus on the a nalysis of the stability properties of the RKN method. Strategies for inco rporating the damping term\, which involves the first-order time derivativ e\, are discussed with particular attention to preserving the order of con vergence of the schemes. The proposed approaches are compared in terms of computational efficiency\, and 3D numerical simulation for the acoustic wa ve equation are presented.
\nFor further information about the semin ar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20251205T123000 END:VEVENT BEGIN:VEVENT UID:news1953@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20251109T143006 DTSTART;TZID=Europe/Zurich:20251114T110000 SUMMARY:Seminar in Numerical Analysis: Andrea Moiola (University of Pavia) DESCRIPTION:We propose a new space-time variational formulation for wave eq uation initial-boundary value problems. The key property is that the formu lation is coercive (sign-definite) and continuous in a norm stronger than H1(Q)\, Q being the space-time cylinder. Coercivity holds for constant-coe fficient impedance cavity problems posed in star-shaped domains\, and for a class of impedance-Dirichlet problems. The formulation is defined using simple Morawetz multipliers and its coercivity is proved with elementary a nalytical tools\, following earlier work on the Helmholtz equation. The fo rmulation can be stably discretised with any H2(Q)-conforming discrete spa ce\, leading to quasi-optimal space-time Galerkin schemes. Several numeric al experiments show the excellent properties of the method. We also presen t a continuous-interior-penalty variant\, a posteriori error estimators\, and adaptive discretisations. This is a joint work with Paolo Bignardi and Theophile Chaumont-Frelet.\\r\\n[1] Bignardi\, Moiola\, A space-time cont inuous and coercive formulation for the wave equation\, Numerische Mathema tik 2025. https://doi.org/10.1007/s00211-025-01478-3 [2] Bignardi\, Space- time Morawetz formulations for the wave equation\, PhD thesis\, University of Pavia 2025. https://iris.unipv.it/handle/11571/1519237\\r\\nFor furthe r information about the seminar\, please visit this webpage [https://dmi.u nibas.ch/de/forschung/mathematik/seminar-in-numerical-analysis/]. X-ALT-DESC:We propose a new space-time variational formulation for wave equation initial-boundary value problems. The key property is that the for mulation is coercive (sign-definite) and continuous in a norm stronger tha n H1(Q)\, Q being the space-time cylinder. Coercivity holds for constant-c oefficient impedance cavity problems posed in star-shaped domains\, and fo r a class of impedance-Dirichlet problems. The formulation is defined usin g simple Morawetz multipliers and its coercivity is proved with elementary analytical tools\, following earlier work on the Helmholtz equation. The formulation can be stably discretised with any H2(Q)-conforming discrete s pace\, leading to quasi-optimal space-time Galerkin schemes. Several numer ical experiments show the excellent properties of the method. We also pres ent a continuous-interior-penalty variant\, a posteriori error estimators\ , and adaptive discretisations. This is a joint work with Paolo Bignardi a nd Theophile Chaumont-Frelet.
\n[1] Bignardi\, Moiola\, A space-time continuous and coercive formulation for the wave equation\, Numerische Ma thematik 2025. https://doi.org/10.1007/s00211-025-01478-3 [2] Bignardi\, S pace-time Morawetz formulations for the wave equation\, PhD thesis\, Unive rsity of Pavia 2025. https://iris.unipv.it/handle/11571/1519237
\nFo r further information about the seminar\, please visit this w ebpage.
DTEND;TZID=Europe/Zurich:20251114T123000 END:VEVENT BEGIN:VEVENT UID:news1939@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20251029T175015 DTSTART;TZID=Europe/Zurich:20251107T110000 SUMMARY:Seminar in Numerical Analysis: Rekha Khot (ENPC et Inria\, Paris) DESCRIPTION:We consider the classical wave equation in a Friedrichs-type fo rmulation involving a skew-symmetric spatial differential operator. The fo cus is on the analysis of a fully discrete setting\, employing third- and fourth-order explicit Runge–Kutta schemes (ERK3 and ERK4) for time discr etization\, combined with Hybrid High-Order (HHO) and Weak Galerkin (WG) m ethods for spatial discretization. A first objective is to establish key p roperties that address the static coupling between cell and face unknowns\ , which is intrinsic to hybrid methods.\\r\\nWe investigate two distinct s trategies for error analysis: one based on bounding the operator norm of T aylor polynomials applied to the discrete differential operator\, and anot her that involves testing the error equations with appropriately chosen in crements. The effectiveness of these strategies depends on the specific in terpolation operators involved and the properties that make such analyses viable. Finally\, we outline how the same framework\, using various HHO va riants\, extends to the acoustic-elastic interface problem\, where the cou pled terms contribute no additional error.\\r\\nFor further information ab out the seminar\, please visit this webpage [https://dmi.unibas.ch/de/fors chung/mathematik/seminar-in-numerical-analysis/]. X-ALT-DESC:We consider the classical wave equation in a Friedrichs-type formulation involving a skew-symmetric spatial differential operator. The focus is on the analysis of a fully discrete setting\, employing third- an d fourth-order explicit Runge–Kutta schemes (ERK3 and ERK4) for time dis cretization\, combined with Hybrid High-Order (HHO) and Weak Galerkin (WG) methods for spatial discretization. A first objective is to establish key properties that address the static coupling between cell and face unknown s\, which is intrinsic to hybrid methods.
\nWe investigate two disti nct strategies for error analysis: one based on bounding the operator norm of Taylor polynomials applied to the discrete differential operator\, and another that involves testing the error equations with appropriately chos en increments. The effectiveness of these strategies depends on the specif ic interpolation operators involved and the properties that make such anal yses viable. Finally\, we outline how the same framework\, using various H HO variants\, extends to the acoustic-elastic interface problem\, where th e coupled terms contribute no additional error.
\nFor further inform ation about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20251107T123000 END:VEVENT BEGIN:VEVENT UID:news1919@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20251029T174630 DTSTART;TZID=Europe/Zurich:20251017T110000 SUMMARY:Seminar in Numerical Analysis: Marcella Bonazzoli (Inria\, Institut Polytechnique de Paris) DESCRIPTION:In this talk we are interested in reconstructing the interface between the concrete structure of a hydroelectric gravity dam and the unde rlying rock\, using Full Waveform Inversion. Indeed\, it appears that the roughness of the dam-rock interface has an effect on the sliding stability of gravity dams. We minimize a regularized misfit cost functional by comp uting its shape derivative and iteratively updating the interface shape by the gradient descent method. At each iteration\, we simulate time-harmoni c elasto-acoustic wave propagation models\, coupling linear elasticity in the solid medium with acoustics in the reservoir. Numerical results using realistic noisy synthetic data demonstrate the method ability to accuratel y reconstruct the dam-rock interface\, even with a limited number of measu rements. This is joint work with Mohamed Aziz Boukraa\, Lorenzo Audibert\, Houssem Haddar and Denis Vautrin.\\r\\nFor further information about the seminar\, please visit this webpage [https://dmi.unibas.ch/de/forschung/ma thematik/seminar-in-numerical-analysis/]. X-ALT-DESC:In this talk we are interested in reconstructing the interfac e between the concrete structure of a hydroelectric gravity dam and the un derlying rock\, using Full Waveform Inversion. Indeed\, it appears that th e roughness of the dam-rock interface has an effect on the sliding stabili ty of gravity dams. We minimize a regularized misfit cost functional by co mputing its shape derivative and iteratively updating the interface shape by the gradient descent method. At each iteration\, we simulate time-harmo nic elasto-acoustic wave propagation models\, coupling linear elasticity i n the solid medium with acoustics in the reservoir. Numerical results usin g realistic noisy synthetic data demonstrate the method ability to accurat ely reconstruct the dam-rock interface\, even with a limited number of mea surements. This is joint work with Mohamed Aziz Boukraa\, Lorenzo Audibert \, Houssem Haddar and Denis Vautrin.
\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20251017T123000 END:VEVENT BEGIN:VEVENT UID:news1903@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20250902T144404 DTSTART;TZID=Europe/Zurich:20250919T110000 SUMMARY:Seminar in Numerical Analysis: Niraj Kumar Shukla (Indian Institute of Technology Indore) DESCRIPTION:The generalized translation invariant (GTI) systems unify the d iscrete frame theory of generalized shift-invariant systems with its conti nuous version\, such as wavelets\, shearlets\, Gabor transforms\, and othe rs. This article provides sufficient conditions to construct pairwise orth ogonal Parseval GTI frames in satisfying the local integrability condition (LIC) and having the Calderón sum one\, where G is a second countable lo cally compact abelian group. The pairwise orthogonality plays a crucial ro le in multiple access communications\, hiding data\, synthesizing superfra mes and frames\, etc. Further\, we provide a result for constructing N num bers of GTI Parseval frames\, which are pairwise orthogonal. Consequently\ , we obtain an explicit construction of pairwise orthogonal Parseval frame s in and \, using B-splines as a generating function. In the end\, the res ults are particularly discussed for wavelet systems. This is a joint work with Navneet Redhu and Anupam Gumber.\\r\\nFor further information about t he seminar\, please visit this webpage [https://dmi.unibas.ch/de/forschung /mathematik/seminar-in-numerical-analysis/]. X-ALT-DESC:The generalized translation invariant (GTI) systems unify the discrete frame theory of generalized shift-invariant systems with its con tinuous version\, such as wavelets\, shearlets\, Gabor transforms\, and ot hers. This article provides sufficient conditions to construct pairwise or thogonal Parseval GTI frames in satisfying the local integrability conditi on (LIC) and having the Calderón sum one\, where G is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications\, hiding data\, synthesizing superf rames and frames\, etc. Further\, we provide a result for constructing N n umbers of GTI Parseval frames\, which are pairwise orthogonal. Consequentl y\, we obtain an explicit construction of pairwise orthogonal Parseval fra mes in and \, using B-splines as a generating function. In the end\, the r esults are particularly discussed for wavelet systems. This is a joint wor k with Navneet Redhu and Anupam Gumber.
\nFor further information ab out the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20250919T123000 END:VEVENT BEGIN:VEVENT UID:news1902@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20250825T191230 DTSTART;TZID=Europe/Zurich:20250829T110000 SUMMARY:Seminar in Numerical Analysis: Dinh Dũng (Vietnam National Univers ity) DESCRIPTION:We prove convergence rates of linear sampling recovery of funct ions in abstract Bochner spaces satisfying weighted summability of their g eneralized polynomial chaos expansion coefficients. The underlying algorit hm is a function-valued extension of the least squares method widely used and thoroughly studied in scalar-valued function recovery.\\r\\nWe apply o ur theory to collocation approximation of solutions to parametric ellipt ic or parabolic PDEs with log-normal random inputs and to relevant approxi mation of infinite dimensional holomorphic functions. The application allo ws us to significantly improve known results in Computational Uncertainty Quantification for these problems. Our results are also applicable for par ametric PDEs with affine inputs\, where they match the known rates.\\r\\n\ \r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:We prove convergence rates of linear sampling recovery of fun ctions in abstract Bochner spaces satisfying weighted summability of their generalized polynomial chaos expansion coefficients. The underlying algor ithm is a function-valued extension of the least squares method widely use d and thoroughly studied in scalar-valued function recovery.
\nWe ap ply our theory to \;collocation approximation of solutions to paramet ric elliptic or parabolic PDEs with log-normal random inputs and to releva nt approximation of infinite dimensional holomorphic functions. The applic ation allows us to significantly improve known results in Computational Un certainty Quantification for these problems. Our results are also applicab le for parametric PDEs with affine inputs\, where they match the known rat es.
\n\nFor further information about the seminar\, please visit thi s webpage.
DTEND;TZID=Europe/Zurich:20250829T123000 END:VEVENT BEGIN:VEVENT UID:news1785@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20250417T144636 DTSTART;TZID=Europe/Zurich:20250425T110000 SUMMARY:Seminar in Numerical Analysis: Björn Sprungk (TU Freiberg) DESCRIPTION:In this talk we consider the Bayesian approach to inverse probl ems which allows for uncertainty quantification for the data-driven recons truction of the ground truth. After a brief dicussion of the well-posednes s and local Lipschitz stability of Bayesian inverse problems\, we focus on Markov chain Monte Carlo methods for sampling and integration with respec t to the posterior probability distribution. Here we present our contribut ions to Metropolis-Hastings algorithms in function spaces\, discuss conve rgence in terms of geometric ergodicity and present numerical experiments which show a dimension-indepedent performance which is\, moreover\, robust to the level of observational noise in the data. In the last part of the talk we present recent results of combining Metropolis-Hastings with inter acting particle sampling methods based on Euler-Maruyama discretizations o f stochastic differential equations of McKean-Vlasov type.\\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://pa ge?uid=1115]. X-ALT-DESC:In this talk we consider the Bayesian approach to inverse pro blems which allows for uncertainty quantification for the data-driven reco nstruction of the ground truth. After a brief dicussion of the well-posedn ess and local Lipschitz stability of Bayesian inverse problems\, we focus on Markov chain Monte Carlo methods for sampling and integration with resp ect to the posterior probability distribution. Here we present our contrib utions to Metropolis-Hastings algorithms in function spaces\, discuss \;convergence in terms of geometric ergodicity and present numerical exper iments which show a dimension-indepedent performance which is\, moreover\, robust to the level of observational noise in the data. In the last part of the talk we present recent results of combining Metropolis-Hastings wit h interacting particle sampling methods based on Euler-Maruyama discretiza tions of stochastic differential equations of McKean-Vlasov type.
\n\n< p>For further information about the seminar\, please visit this webpage. DTEND;TZID=Europe/Zurich:20250425T123000 END:VEVENT BEGIN:VEVENT UID:news1826@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20250328T125929 DTSTART;TZID=Europe/Zurich:20250404T110000 SUMMARY:Seminar in Numerical Analysis: Théophile Chaumont-Frelet (Inria L ille) DESCRIPTION:The wave equation is a basic PDE model central to a plethora of physical and engineering applications\, with such applications requiring approximate solutions obtained by numerical schemes. In this talk\, I will focus on the space semi-discretization of the wave equation with a finite element method (and assume that time integration is exactly performed). I n the context of finite element methods\, a posteriori error estimates are a now widely established technique to rigorously control the discretizati on error\, and to drive adaptive processes where the finite element mesh i s iteratively refined. However\, although a posteriori error estimates are widely available for elliptic and parabolic problems\, the literature is much scarcer for hyperbolic problems\, including the time-dependent wave e quation. In this talk\, I will discuss a new a posteriori error estimator that hinges on ideas previously developed for the Helmholtz equation (the time-harmonic version of the wave equation). To the best of my knowledge\, this new error estimator is the first to provide both an upper and a lowe r bound for the error measured in the same norm. I will also briefly quick ly discuss preliminary results concerning time discretization\, and applic ation to adaptive algorithms.\\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:The wave equation is a basic PDE model central to a plethora of physical and engineering applications\, with such applications requirin g approximate solutions obtained by numerical schemes. In this talk\, I wi ll focus on the space semi-discretization of the wave equation with a fini te element method (and assume that time integration is exactly performed). In the context of finite element methods\, a posteriori error estimates a re a now widely established technique to rigorously control the discretiza tion error\, and to drive adaptive processes where the finite element mesh is iteratively refined. However\, although a posteriori error estimates a re widely available for elliptic and parabolic problems\, the literature i s much scarcer for hyperbolic problems\, including the time-dependent wave equation. In this talk\, I will discuss a new a posteriori error estimato r that hinges on ideas previously developed for the Helmholtz equation (th e time-harmonic version of the wave equation). To the best of my knowledge \, this new error estimator is the first to provide both an upper and a lo wer bound for the error measured in the same norm. I will also briefly qui ckly discuss preliminary results concerning time discretization\, and appl ication to adaptive algorithms.
\n\nFor further information about th e seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20250404T123000 END:VEVENT BEGIN:VEVENT UID:news1787@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20250318T095344 DTSTART;TZID=Europe/Zurich:20250328T110000 SUMMARY:Seminar in Numerical Analysis: Daniel Potts (TU Chemnitz) DESCRIPTION:In this talk\, we present algorithms for the approximation of m ultivariate functions. We start with the approximation by trigonometric po lynomials based on sampling of multivariate functions on rank-1 lattices o r on scattered data. To this end\, we study the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. The proposed a lgorithm based mainly on a fast Fourier transforms\, and the arithmetic co mplexity of the algorithm depends only on the cardinality of the support o f the trigonometric polynomial in the frequency domain. After a detailed i ntroduction we will focus on the following questions in more detail.\\r\\n We discuss methods where the support of the trigonometric polynomial i s unknown. We present a method based on the analysis of variance (ANO VA) decomposition that aims to detect the structure of the function\, i.e. \, find out which dimension and dimension interactions are important. This information is then utilized in obtaining an approximation for the functi on. Based on these methods we develop an efficient\, non-intrusive\, adaptive algorithm for the solution of elliptic partial differential equat ions. \\r\\n\\r\\nFor further information about the seminar\, please vis it this webpage [t3://page?uid=1115]. X-ALT-DESC:In this talk\, we present algorithms for the approximation of multivariate functions. We start with the approximation by trigonometric polynomials based on sampling of multivariate functions on rank-1 lattices or on scattered data. To this end\, we study the approximation of functio ns in periodic Sobolev spaces of dominating mixed smoothness. The proposed algorithm based mainly on a fast Fourier transforms\, and the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. After a detailed introduction we will focus on the following questions in more detail.
\nWe discuss methods where the support of the trigonometric polynomial is unknown.
We present a method based on the analysis of variance (ANOVA) decomposition that aims to detect the str ucture of the function\, i.e.\, find out which dimension and dimension int eractions are important. This information is then utilized in obtaining an approximation for the function.
Based on these metho ds we develop an efficient\, non-intrusive\, adaptive algorithm for the so lution of elliptic partial differential equations.
For further information about the seminar\, please visit this webpage .
DTEND;TZID=Europe/Zurich:20250328T123000 END:VEVENT BEGIN:VEVENT UID:news1798@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20250315T000920 DTSTART;TZID=Europe/Zurich:20250321T110000 SUMMARY:Seminar in Numerical Analysis: Zhaonan Dong (Inria Paris) DESCRIPTION:PDE models are often characterized by local features such as so lution singularities\, boundary layers\, domains with complicated boundari es\, and phase transitions. These unique characteristics make designing ac curate numerical solutions challenging or demand substantial computational resources. One effective strategy is to develop novel numerical methods t hat support general meshes composed of polygonal or polyhedral elements\, enabling adaptive refinement that efficiently captures local features.\\r\ \nIn this talk\, we present recent results on a new a posteriori error ana lysis for the discontinuous Galerkin (dG) method applied to general comput ational meshes consisting of polygonal/polyhedral (polytopic) elements wit h an arbitrary number of faces. This analysis\, which first appeared in th e literature\, generalizes known dG methods by allowing an arbitrary numbe r of irregular hanging nodes per element. Moreover\, under practical mesh assumptions\, the new error estimator accommodates nearly any element shap e—even with curved faces. We will also briefly discuss the a posteriori error estimator for the space-time dG method in solving the Allen–Cahn p roblem\, as well as the hp-a posteriori error estimator for the DG method in tackling fourth-order PDEs.\\r\\n\\r\\nFor further information about th e seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:PDE models are often characterized by local features such as solution singularities\, boundary layers\, domains with complicated bounda ries\, and phase transitions. These unique characteristics make designing accurate numerical solutions challenging or demand substantial computation al resources. One effective strategy is to develop novel numerical methods that support general meshes composed of polygonal or polyhedral elements\ , enabling adaptive refinement that efficiently captures local features.\n
In this talk\, we present recent results on a new a posteriori erro r analysis for the discontinuous Galerkin (dG) method applied to general c omputational meshes consisting of polygonal/polyhedral (polytopic) element s with an arbitrary number of faces. This analysis\, which first appeared in the literature\, generalizes known dG methods by allowing an arbitrary number of irregular hanging nodes per element. Moreover\, under practical mesh assumptions\, the new error estimator accommodates nearly any element shape—even with curved faces. We will also briefly discuss the a poster iori error estimator for the space-time dG method in solving the Allen–C ahn problem\, as well as the hp-a posteriori error estimator for the DG me thod in tackling fourth-order PDEs.
\n\nFor further information abou t the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20250321T123000 END:VEVENT BEGIN:VEVENT UID:news1720@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20241213T192501 DTSTART;TZID=Europe/Zurich:20241220T110000 SUMMARY:Seminar in Numerical Analysis: Ana Djurdjevac (FU Berlin) DESCRIPTION:Interacting particle systems provide flexible and powerful mode ls that are useful in many application areas such as sociology (agents)\, molecular dynamics (proteins) etc. However\, particle systems with large n umbers of particles are very complex and difficult to handle\, both analyt ically and computationally. Therefore\, a common strategy is to derive e ffective equations that describe the time evolution of the empirical part icle density. A prototypical example that we will consider is the formal i dentification of a finite system of particles with the singular Dean-Kawas aki equation. We will give a short introduction about the Dean-Kawasaki eq uation and its applications. Our aim is to introduce a well-behaved nonlin ear SPDE that approximates the Dean-Kawasaki equation for a particle syste m with mean-field interaction both in the drift and the noise term. We wan t to study the well-posedness of these nonlinear SPDE models and to contro l the weak error of the SPDE approximation with respect to the particle sy stem using the technique of transport equations on the space of probabilit y measures. This is the joint work with H. Kremp\, N. Perkowski and J. Xia ohao. Furthermore\, we will discuss possible numerical methods for these p roblems. In particular\, we will focus on hybrid methods. This is a joint work with A. Almgren and J. Bell.\\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Interacting particle systems provide flexible and powerful mo dels that are useful in many application areas such as sociology (agents)\ , molecular dynamics (proteins) etc. However\, particle systems with large numbers of particles are very complex and difficult to handle\, both anal ytically and computationally. Therefore\, a common strategy is to derive e ffective equations that describe the time evolution of the empirical part icle density. A prototypical example that we will consider is the formal i dentification of a finite system of particles with the singular Dean-Kawas aki equation. We will give a short introduction about the Dean-Kawasaki eq uation and its applications. Our aim is to introduce a well-behaved nonlin ear SPDE that approximates the Dean-Kawasaki equation for a particle syste m with mean-field interaction both in the drift and the noise term. We wan t to study the well-posedness of these nonlinear SPDE models and to contro l the weak error of the SPDE approximation with respect to the particle sy stem using the technique of transport equations on the space of probabilit y measures. This is the joint work with H. Kremp\, N. Perkowski and J. Xia ohao. Furthermore\, we will discuss possible numerical methods for these p roblems. In particular\, we will focus on hybrid methods. This is a joint work with A. Almgren and J. Bell.
\n\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20241220T120000 END:VEVENT BEGIN:VEVENT UID:news1722@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20241213T192424 DTSTART;TZID=Europe/Zurich:20241213T110000 SUMMARY:Seminar in Numerical Analysis: Robert Scheichl (Heidelberg Universi ty) DESCRIPTION:The fast simulation of Gaussian random fields (GRFs) plays a pi votal role in many research areas such as uncertainty quantification\, dat a science and spatial statistics. In theory\, it is a well understood and solved problem\, but in practice the efficiency and performance of traditi onal sampling procedures degenerates quickly when the random field is disc retised on a grid with spatial resolution going to zero. Most existing alg orithms\, such as Cholesky factorisation samplers\, do not scale well on l arge-scale parallel computers. On the other hand\, stationary\, iterative approaches such as the Gibbs sampler become extremely inefficient at high grid resolution. Already in the late 1980s\, Goodman\, Sokal and their co llaborators wrote a series of papers aimed at accelerating the Gibbs sampl er using multigrid ideas. The key observation is the intricate connection of random samplers\, such as the Gibbs method\, with iterative solvers for linear systems. They proposed the so-called multigrid Monte Carlo (MGMC) method - a random analogue of the multigrid method for solving discretised PDEs. We revisit the MGMC algorithm and provide rigorous theoretical jus tifications for the optimal scalability of the method for large scale prob lems\, with a cost growing linearly with problem size. Most importantly we extend the method and the analysis to the Bayesian setting\, i.e.\, GRFs conditioned on noisy data. By using bespoke\, conditioned variants of the Gibbs sampler on each level of the multigrid hierarchy we are able to samp le directly from the posterior with a fixed\, grid-independent integrated autocorrelation time. Our theoretical analysis is confirmed by numerical experiments. We further generalise the approach by exploiting more flexibl e and robust grid hierarchies that were developed in the context of Algeb raic Multigrid solvers. Finally\, using existing PDE libraries\, such as PETSs\, the sampler is easily parallelised and scales optimally to large p rocessor numbers.\\r\\n\\r\\nFor further information about the seminar\, p lease visit this webpage [t3://page?uid=1115]. X-ALT-DESC:The fast simulation of Gaussian random fields (GRFs) plays a pivotal role in many research areas such as uncertainty quantification\, d ata science and spatial statistics. In theory\, it is a well understood an d solved problem\, but in practice the efficiency and performance of tradi tional sampling procedures degenerates quickly when the random field is di scretised on a grid with spatial resolution going to zero. Most existing a lgorithms\, such as Cholesky factorisation samplers\, do not scale well on large-scale parallel computers. On the other hand\, stationary\, iterativ e approaches such as the Gibbs sampler become extremely inefficient at hig h grid resolution. \;Already in the late 1980s\, Goodman\, Sokal and t heir collaborators wrote a series of papers aimed at accelerating the Gibb s sampler using multigrid ideas. The key observation is the intricate conn ection of random samplers\, such as the Gibbs method\, with iterative solv ers for linear systems. They proposed the so-called multigrid Monte Carlo (MGMC) method - a random analogue of the multigrid method for solving disc retised PDEs. \;We revisit the MGMC algorithm and provide rigorous the oretical justifications for the optimal scalability of the method for larg e scale problems\, with a cost growing linearly with problem size. Most im portantly we extend the method and the analysis to the Bayesian setting\, i.e.\, GRFs conditioned on noisy data. By using bespoke\, conditioned vari ants of the Gibbs sampler on each level of the multigrid hierarchy we are able to sample directly from the posterior with a fixed\, grid-independent integrated autocorrelation time. \;Our theoretical analysis is confir med by numerical experiments. We further generalise the approach by exploi ting more flexible and robust grid hierarchies that were developed in the context of \;Algebraic Multigrid solvers. \;Finally\, using existi ng PDE libraries\, such as PETSs\, the sampler is easily parallelised and scales optimally to large processor numbers.
\n\nFor further informa tion about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20241213T120000 END:VEVENT BEGIN:VEVENT UID:news1728@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20241127T162336 DTSTART;TZID=Europe/Zurich:20241206T110000 SUMMARY:Seminar in Numerical Analysis: Alexandre Imperiale (CEA Paris-Sacla y) DESCRIPTION:We consider the problem of transient wave propagation within tw o domains linked with smooth nonlinear contact conditions at a common inte rface. While standard linear elastodynamics is assumed within each domain\ , at the interface we consider continuity of normal stresses\, and (more i mportantly) a smooth finite compressibility law. We propose an energy pre serving – thus stable – time scheme based upon [1]\, and devise an eff icient time-marching algorithm. We validate our approach with semi-analyti cal results\, and illustrate typical nonlinear waves phenomena (harmonics\ , zero-frequency components) in 2D.\\r\\nReferences\\r\\n[1] O. Gonzalez\, Exact energy and momentum conserving algorithms for general models in non linear elasticity\, Comput. Methods Appl. Mech. Eng.\, 2000\, 190(13-14)\, 1763-1783.\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:We consider the problem of transient wave propagation within two domains linked with smooth nonlinear contact conditions at a common in terface. While standard linear elastodynamics is assumed within each domai n\, at the interface we consider continuity of normal stresses\, and (more importantly) a smooth finite compressibility law. \;We propose an ene rgy preserving – thus stable – time scheme based upon [1]\, and devise an efficient time-marching algorithm. We validate our approach with semi- analytical results\, and illustrate typical nonlinear waves phenomena (har monics\, zero-frequency components) in 2D.
\nReferences
\n[1] O. Gonzalez\, Exact energy and momentum conserving algorithms for general models in nonlinear elasticity\, Comput. Methods Appl. Mech. Eng.\, 2000\, 190(13-14)\, 1763-1783.
\nFor further information about the seminar \, please visit this webpage.
DTEND;TZID=Europe/Zurich:20241206T120000 END:VEVENT BEGIN:VEVENT UID:news1729@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20241106T144114 DTSTART;TZID=Europe/Zurich:20241115T110000 SUMMARY:Seminar in Numerical Analysis: Marco Picasso (EPFL) DESCRIPTION:Anisotropic adaptive meshes\, that is to say adaptive meshes wi th large aspect ratio\, are extremely efficient to approach functions havi ng boundary or internal layers\, some industrial applications will be pres ented. The theory will be justified on elliptic problems\, and on the tran sport equation when using the order two Crank-Nicolson scheme.\\r\\n\\r\\n For further information about the seminar\, please visit this webpage [t3: //page?uid=1115]. X-ALT-DESC:Anisotropic adaptive meshes\, that is to say adaptive meshes with large aspect ratio\, are extremely efficient to approach functions ha ving boundary or internal layers\, some industrial applications will be pr esented. The theory will be justified on elliptic problems\, and on the tr ansport equation when using the order two Crank-Nicolson scheme.
\n\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20241115T120000 END:VEVENT BEGIN:VEVENT UID:news1721@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20241101T133004 DTSTART;TZID=Europe/Zurich:20241108T110000 SUMMARY:Seminar in Numerical Analysis: Jürgen Dölz (University of Bonn) DESCRIPTION:We consider generalized symmetric operator eigenvalue problems with random symmetric perturbations of the operators. This implies that th e eigenpairs of the eigenvalue problem are also random. We investigate sto chastic quantities of interest of eigenpairs of higher but finite multipli city and discuss why for multiplicity larger than one\, only the stochasti c quantities of interest of the eigenspaces are meaningful. To do so\, we characterize the Fréchet derivatives of the eigenpairs with respect to th e perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result\, we prove local analyticity of the eigenspaces. Based on the Fréchet derivatives of the eigenpairs we discu ss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. We present n umerical examples to illustrate the theoretical results.\\r\\n\\r\\nFor fu rther information about the seminar\, please visit this webpage [t3://page ?uid=1115]. X-ALT-DESC:We consider generalized symmetric operator eigenvalue problem s with random symmetric perturbations of the operators. This implies that the eigenpairs of the eigenvalue problem are also random. We investigate s tochastic quantities of interest of eigenpairs of higher but finite multip licity and discuss why for multiplicity larger than one\, only the stochas tic quantities of interest of the eigenspaces are meaningful. To do so\, w e characterize the Fréchet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result\, we prove local analyticity of t he eigenspaces. Based on the Fréchet derivatives of the eigenpairs we dis cuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues a nd develop an uncertainty quantification perturbation approach. We present numerical examples to illustrate the theoretical results.
\n\nFor f urther information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20241108T120000 END:VEVENT BEGIN:VEVENT UID:news1703@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20241001T122056 DTSTART;TZID=Europe/Zurich:20241025T110000 SUMMARY:Seminar in Numerical Analysis: Andreas Veeser (University of Milan) DESCRIPTION:In the context of finite element methods for boundary value pro blems\, the goal of an a posteriori error analysis is to derive a so-calle d error estimator. Such an estimator should be a quantity that is computab le in terms of the problem data and the finite element solution and equiva lent to its error. Almost all error estimators available in literature\, h owever\, do not meet these requirements. Indeed\, equivalence is typically verified only up to so-called (data) oscillations and their presence has been viewed as the price of computability.\\r\\nThe talk will consist of t wo parts. The first part will argue that the infinite-dimensional nature o f the problem data is an obstruction to computability\, while the typical form of oscillations obstructs equivalence. In other words: the desirable but missing unspoiled equivalence is not only a technical problem but does not hold for the involved oscillations.\\r\\nThe second part will then pr esent the new approach to a posteriori error estimation proposed by [1]. I ts resulting estimators are equivalent to the error and consist of two par ts\, where the first one is of finite-dimensional nature and thus computab le\, while the second one is a new form of data oscillation\, which is alw ays smaller that the old one and whose computability hinges on the knowled ge of the problem data. This splitting of the error estimator is also conv enient in guiding adaptive algorithms\; cf. [2].\\r\\n[1] C. Kreuzer\, A. Veeser\, Oscillation in a posteriori error estimation\, Numer. Math. 148 ( 2021)\, 43-78 [2] A. Bonito\, C. Canuto\, R. H. Nochetto\, A. Veeser\, Ada ptive finite element methods\, Acta Numerica 33 (2024)\, 163-485.\\r\\n\\r \\nFor further information about the seminar\, please visit this webpage [ t3://page?uid=1115]. X-ALT-DESC:In the context of finite element methods for boundary value p roblems\, the goal of an a posteriori error analysis is to derive a so-cal led error estimator. Such an estimator should be a quantity that is comput able in terms of the problem data and the finite element solution and equi valent to its error. Almost all error estimators available in literature\, however\, do not meet these requirements. Indeed\, equivalence is typical ly verified only up to so-called (data) oscillations and their presence ha s been viewed as the price of computability.
\nThe talk will consist of two parts. The first part will argue that the infinite-dimensional nat ure of the problem data is an obstruction to computability\, while the typ ical form of oscillations obstructs equivalence. In other words: the desir able but missing unspoiled equivalence is not only a technical problem but does not hold for the involved oscillations.
\nThe second part will then present the new approach to a posteriori error estimation proposed b y [1]. Its resulting estimators are equivalent to the error and consist of two parts\, where the first one is of finite-dimensional nature and thus computable\, while the second one is a new form of data oscillation\, whic h is always smaller that the old one and whose computability hinges on the knowledge of the problem data. This splitting of the error estimator is a lso convenient in guiding adaptive algorithms\; cf. [2].
\n[1] C. Kr
euzer\, A. Veeser\, Oscillation in a posteriori error estimation\, Numer.
Math. 148 (2021)\, 43-78
[2] A. Bonito\, C. Canuto\, R. H. Nochetto\
, A. Veeser\, Adaptive finite element methods\, Acta Numerica 33 (2024)\,
163-485.
For further information about the seminar\, please visi t this webpage.
DTEND;TZID=Europe/Zurich:20241025T120000 END:VEVENT BEGIN:VEVENT UID:news1684@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240527T101453 DTSTART;TZID=Europe/Zurich:20240531T110000 SUMMARY:Seminar in Numerical Analysis: Philipp Trunschke (University of Nan tes) DESCRIPTION:Many functions of interest exhibit weighted summability of thei r coefficients with respect to some dictionary of basis functions. This s ummability enables efficient sparse and low-rank approximation and ensures that the function can be estimated efficiently from samples. This talk presents some fundamental results on the estimation of sparse and low-rank functions\, like the weighted Stechkin lemma and the restricted isometry property\, and introduces simultaneously sparse and low-rank tensor for mats.\\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Many functions of interest exhibit weighted summability o f their coefficients with respect to some dictionary of basis functions.&n bsp\;This summability enables efficient sparse and low-r ank approximation and ensures that the function can be \;esti mated efficiently from samples. \;This talk presents some fun damental results on the estimation of sparse and low-rank functions\, like the \;weighted Stechkin lemma and the restricted isometry pr operty\, \;and \;introduces simultaneously spars e and low-rank tensor formats.
\n\nFor further information abou t the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20240531T120000 END:VEVENT BEGIN:VEVENT UID:news1665@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240429T095246 DTSTART;TZID=Europe/Zurich:20240503T110000 SUMMARY:Seminar in Numerical Analysis: Andreas Veeser (University of Milano ) DESCRIPTION:TBA\\r\\n\\r\\nFor further information about the seminar\, plea se visit this webpage [t3://page?uid=1115]. X-ALT-DESC:TBA
\n\nFor further information about the seminar\, ple ase visit this webpage.
DTEND;TZID=Europe/Zurich:20240503T120000 END:VEVENT BEGIN:VEVENT UID:news1656@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240415T160416 DTSTART;TZID=Europe/Zurich:20240426T110000 SUMMARY:Seminar in Numerical Analysis: Malte Peter (University of Augsburg) DESCRIPTION:We consider the upscaled linear elasticity problem in the conte xt of periodic homogenisation in the stationary setting as well as in the time-dependent regime where the wavelength is much larger than the microst ructure. Based on measurements of the deformation of the (macroscopic) bou ndary of a body for a given forcing\, the aim is to deduce information on the geometry of the microstructure. After a general introduction to period ic homogenisation in the context of linear elasticity\, we are able to pro ve for a parametrised microstructure that there exists at least one soluti on of the associated minimisation problem based on the L^2-difference of t he measured deformation and the computed deformation for a given parameter vector. To facilitate the use of gradient-based algorithms\, we derive th e Gâteaux derivatives using the Lagrangian method of Céa\, and we presen t numerical experiments showcasing the functioning of the method.\\r\\nThi s is joint work with T. Lochner (University of Augsburg).\\r\\n\\r\\nFor f urther information about the seminar\, please visit this webpage [t3://pag e?uid=1115]. X-ALT-DESC:We consider the upscaled linear elasticity problem in the con text of periodic homogenisation in the stationary setting as well as in th e time-dependent regime where the wavelength is much larger than the micro structure. Based on measurements of the deformation of the (macroscopic) b oundary of a body for a given forcing\, the aim is to deduce information o n the geometry of the microstructure. After a general introduction to peri odic homogenisation in the context of linear elasticity\, we are able to p rove for a parametrised microstructure that there exists at least one solu tion of the associated minimisation problem based on the L^2-difference of the measured deformation and the computed deformation for a given paramet er vector. To facilitate the use of gradient-based algorithms\, we derive the Gâteaux derivatives using the Lagrangian method of Céa\, and we pres ent numerical experiments showcasing the functioning of the method.
\n< p>This is joint work with T. Lochner (University of Augsburg).\n\nF or further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20240426T120000 END:VEVENT BEGIN:VEVENT UID:news1655@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20240402T130832 DTSTART;TZID=Europe/Zurich:20240419T110000 SUMMARY:Seminar in Numerical Analysis: Barbara Verfürth (University of Bon n) DESCRIPTION:In the last years\, there has been an increasing interest in ti me-modulated materials to obtain enhanced properties. As mathematical mode l\, we study the classical wave equation with time-dependent coefficient\, which may also include spatial multiscale features. Based on joint work w ith Bernhard Maier\, we present a numerical multiscale method for spatiall y multiscale\, (slowly) time-evolving coefficients. The method is inspired by the Localized Orthogonal Decomposition (LOD) and entails time-dependen t multiscale spaces. We provide a rigorous a priori error analysis for the considered setting. Numerical examples illustrate the theoretical finding s and investigate an adaptive approach for the computation of the time-dep endent basis functions. On the other hand\, we will also briefly discuss t he setting of spatially homogeneous\, temporal multiscale coefficients. (H igher-order) multiscale expansions may help to interpret effective physica l material properties and are numerically illustrated.\\r\\nFor further in formation about the seminar\, please visit this webpage [t3://page?uid=111 5]. X-ALT-DESC:In the last years\, there has been an increasing interest in
time-modulated materials to obtain enhanced properties. As mathematical mo
del\, we study the classical wave equation with time-dependent coefficient
\, which may also include spatial multiscale features.
Based on join
t work with Bernhard Maier\, we present a numerical multiscale method for
spatially multiscale\, (slowly) time-evolving coefficients. The method is
inspired by the Localized Orthogonal Decomposition (LOD) and entails time-
dependent multiscale spaces. We provide a rigorous a priori error analysis
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 other hand\, we will also bri
efly discuss the setting of spatially homogeneous\, temporal multiscale co
efficients. (Higher-order) multiscale expansions may help to interpret eff
ective physical material properties and are numerically illustrated.
For further information about the seminar\, please visit this webpage< /a>.
DTEND;TZID=Europe/Zurich:20240419T120000 END:VEVENT BEGIN:VEVENT UID:news1550@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231204T091610 DTSTART;TZID=Europe/Zurich:20231215T110000 SUMMARY:Seminar in Numerical Analysis: Caroline Geiersbach (WIAS Berlin) DESCRIPTION:Many problems in shape optimization involve constraints in the form of one or more partial differential equations. In practice\, the mate rial properties of the underlying shape on which a PDE is defined are not known exactly\; it is natural to use a probability distribution based on e mpirical measurements and incorporate this information when designing an o ptimal shape. Additionally\, one might wish to obtain a shape that is robu st in its response to certain external inputs\, such as forces. It is help ful to view shape optimization problems subject to uncertainty through the lens of stochastic optimization\, where a wealth of theory and algorithms already exist for finite-dimensional problems. The focus will be on the a lgorithmic handling of these problems in the case of a high stochastic dim ension. Stochastic approximation\, which dynamically samples from the stoc hastic space over the course of iterations\, is favored in this case\, and we show how these methods can be applied to shape optimization. We study the classical stochastic gradient method\, which was introduced in 1951 by Robbins and Monro and is widely used in machine learning. In particular\, we investigate its application to infinite-dimensional shape manifolds. F urther\, we present numerical examples showing the performance of the meth od\, also in combination with the augmented Lagrangian method for problems with geometric constraints. \\r\\nJoint work with: Kathrin Welker\, Este fania Loayza-Romero\, Tim Suchan\\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Many problems in shape optimization involve constraints in th e form of one or more partial differential equations. In practice\, the ma terial properties of the underlying shape on which a PDE is defined are no t known exactly\; it is natural to use a probability distribution based on empirical measurements and incorporate this information when designing an optimal shape. Additionally\, one might wish to obtain a shape that is ro bust in its response to certain external inputs\, such as forces. It is he lpful to view shape optimization problems subject to uncertainty through t he lens of stochastic optimization\, where a wealth of theory and algorith ms already exist for finite-dimensional problems. The focus will be on the algorithmic handling of these problems in the case of a high stochastic d imension. Stochastic approximation\, which dynamically samples from the st ochastic space over the course of iterations\, is favored in this case\, a nd we show how these methods can be applied to shape optimization. We stud y the classical stochastic gradient method\, which was introduced in 1951 by Robbins and Monro and is widely used in machine learning. In particular \, we investigate its application to infinite-dimensional shape manifolds. Further\, we present numerical examples showing the performance of the me thod\, also in combination with the augmented Lagrangian method for proble ms with geometric constraints. \;
\nJoint work with: Kathrin Wel ker\, Estefania Loayza-Romero\, Tim Suchan
\n\nFor further informati on about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20231215T120000 END:VEVENT BEGIN:VEVENT UID:news1570@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231127T102321 DTSTART;TZID=Europe/Zurich:20231208T110000 SUMMARY:Seminar in Numerical Analysis: Martin Vohralik (Inria Paris) DESCRIPTION:A posteriori estimates enable to certify the error committed in a numerical simulation. In particular\, the equilibrated flux reconstruct ion technique yields a guaranteed error upper bound\, where the flux\, obt ained by a local postprocessing\, is of independent interest since it is a lways locally conservative. In this talk\, we tailor this methodology to m odel nonlinear and time-dependent problems to obtain estimates that are ro bust\, i.e.\, of quality independent of the strength of the nonlinearities and the final time. These estimates include\, and build on\, common itera tive linearization schemes such as Zarantonello\, Picard\, Newton\, or M- and L-ones. We first consider steady problems and conceive two settings: w e either augment the energy difference by the discretization error of the current linearization step\, or we design iteration-dependent norms that f eature weights given by the current iterate. We then turn to unsteady prob lems. Here we first consider the linear heat equation and finally move to the Richards one\, that is doubly nonlinear and exhibits both parabolic– hyperbolic and parabolic–elliptic degeneracies. Robustness with respect to the final time and local efficiency in both time and space are addresse d here. Numerical experiments illustrate the theoretical findings all alon g the presentation. Details can be found in [1-4].\\r\\nA. Ern\, I. Smears \, M. Vohralík\, Guaranteed\, locally space-time efficient\, and polynomi al-degree robust a posteriori error estimates for high-order discretizatio ns of parabolic problems\, SIAM J. Numer. Anal. 55 (2017)\, 2811–2834.\\ r\\nA. Harnist\, K. Mitra\, A. Rappaport\, M. Vohralík\, Robust energy a posteriori estimates for nonlinear elliptic problems\, HAL Preprint 04033 438\, 2023.\\r\\nK. Mitra\, M. Vohralík\, A posteriori error estimates fo r the Richards equation\, Math. Comp. (2024)\, accepted for publication.\\ r\\nK. Mitra\, M. Vohralík\, Guaranteed\, locally efficient\, and robust a posteriori estimates for nonlinear elliptic problems in iteration-depend ent norms. An orthogonal decomposition result based on iterative lineariza tion\, HAL Preprint 04156711\, 2023.\\r\\n\\r\\nFor further information a bout the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:A posteriori estimates enable to certify the error committed in a numerical simulation. In particular\, the equilibrated flux reconstru ction technique yields a guaranteed error upper bound\, where the flux\, o btained by a local postprocessing\, is of independent interest since it is always locally conservative. In this talk\, we tailor this methodology to model nonlinear and time-dependent problems to obtain estimates that are robust\, i.e.\, of quality independent of the strength of the nonlineariti es and the final time. These estimates include\, and build on\, common ite rative linearization schemes such as Zarantonello\, Picard\, Newton\, 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 th e current linearization step\, or we design iteration-dependent norms that feature weights given by the current iterate. We then turn to unsteady pr oblems. Here we first consider the linear heat equation and finally move t o the Richards one\, that is doubly nonlinear and exhibits both parabolic –hyperbolic and parabolic–elliptic degeneracies. Robustness with respe ct to the final time and local efficiency in both time and space are addre ssed here. Numerical experiments illustrate the theoretical findings all a long the presentation. Details can be found in [1-4].
\nA. Ern\, I. Smears\, M. Vohralík\, Guaranteed\, locally space-time efficient\, and po lynomial-degree robust a posteriori error estimates for high-order discret izations of parabolic problems\, SIAM J. Numer. Anal.55< /strong> (2017)\, 2811–2834.
\nA. Harnist\, K. Mitra\, A. Rappapor t\, M. Vohralík\, Robust energy a posteriori estimates for nonlinear elli ptic problems\, HAL Preprint \;04033438\, 2023.
\nK. Mitra\, M. Vohralík\, A posteriori error estimates for the Richards equation\, M ath. Comp. (2024)\, accepted for publication.
\nK. Mitra\, M. V ohralík\, Guaranteed\, locally efficient\, and robust a posteriori estima tes for nonlinear elliptic problems in iteration-dependent norms. An ortho gonal decomposition result based on iterative linearization\, HAL Preprint \;04156711\, 2023.
\n\nFor further information about the semina r\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20231208T120000 END:VEVENT BEGIN:VEVENT UID:news1544@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230922T165627 DTSTART;TZID=Europe/Zurich:20231110T110000 SUMMARY:Seminar in Numerical Analysis: Larisa Beilina (University of Göteb org) DESCRIPTION:An adaptive finite element/finite difference domain decompositi on method for solution of time-dependent Maxwell's equations for electric field in conductive media will be presented. This method is applied for r econstruction of dielectric permittivity and conductivity functions using time-dependent scattered data of electric field at the boundary of the do main.\\r\\nAll reconstruction algorithms are based on optimization approac h for finding of stationary point of the Lagrangian. Derivation of a poste riori error estimates for the regularized solution and Tikhonov functional will be presented. Based on these estimates adaptive reconstruction alg orithms are developed. Computational tests will show robustness of propo sed algorithms in 3D.\\r\\n\\r\\nFor further information about the seminar \, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:An adaptive finite element/finite difference domain decomposi tion \;method for solution of time-dependent Maxwell's equations for e lectric field in conductive media will be presented. This method is applie d for reconstruction of dielectric permittivity and conductivity \;fun ctions using time-dependent scattered data of electric field at the bounda ry of the domain.
\nAll reconstruction algorithms are based on optim ization approach for finding of stationary point of the Lagrangian. Deriva tion of a posteriori error estimates for the regularized solution and Tikh onov functional will be presented. \;Based on these estimates adaptiv e reconstruction algorithms are developed. \;Computational tests will show robustness of proposed algorithms in 3D.
\n\nFor further infor mation about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20231110T120000 END:VEVENT BEGIN:VEVENT UID:news1583@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231024T091747 DTSTART;TZID=Europe/Zurich:20231027T110000 SUMMARY:Seminar in Numerical Analysis: Carsten Gräser (FAU Erlangen-Nürnb erg) 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 error term and a data error term. These error contributions quantify the accuracy of the PDE mod el used for regularization and the data coverage. Furthermore\, the disc retization of the PDE-regularized learning problem by generalized Galerki n methods including finite elements and neural networks approaches is in vestigated. A nonlinear version of Céa's lemma allows to derive errors b ounds for both classes of discretizations and gives first insights into e rror analysis of variational neural network discretizations of PDEs.\\r\ \n\\r\\nFor further information about the seminar\, please visit this webp age [t3://page?uid=1115]. X-ALT-DESC:We consider the regularization of a supervised \;learning
problem by partial differential equations \;(PDEs). For the resulting
regularized problem we \;derive error bounds in terms of a PDE error
term \;and a data error term. These error contributions
quantify
the accuracy of the PDE model used for \;regularization and the data
coverage.
Furthermore\, the discretization of the PDE-regular
ized \;learning problem by generalized Galerkin methods \;includin
g 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 PDE
s.
For further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20231027T120000 END:VEVENT BEGIN:VEVENT UID:news1537@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231013T095041 DTSTART;TZID=Europe/Zurich:20231020T110000 SUMMARY:Seminar in Numerical Analysis: Marco Zank (U Wien) 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 appr oach is the usage of space-time methods\, where the space-time domain is d iscretized and the resulting global linear system is solved at once. In th is talk\, some recent developments in space-time finite element methods ar e reviewed. For this purpose\, the heat equation and the wave equation ser ve as model problems. First\, for both model problems\, space-time variati onal formulations and their unique solvability in space-time Sobolev space s are discussed\, where a modified Hilbert transformation is used such tha t ansatz and test spaces are equal. Second\, conforming discretization sch emes\, using piecewise polynomial\, globally continuous functions\, are in troduced. Solvability and stability of these numerical schemes are discuss ed. Next\, we investigate efficient direct solvers for the occurring huge linear systems. The developed solvers are based on the Bartels--Stewart me thod and on the Fast Diagonalization method\, which result in solving a se quence of spatial subproblems. The solver based on the Fast Diagonalizatio n method allows solving these spatial subproblems in parallel\, leading to a full parallelization in time. In the last part of the talk\, numerical examples are shown and discussed.\\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:For the discretization of time-dependent partial differential equations\, the standard approaches are explicit or implicit time-steppin g schemes together with finite element methods in space. An alternative ap proach is the usage of space-time methods\, where the space-time domain is discretized and the resulting global 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 s erve as model problems. First\, for both model problems\, space-time varia tional formulations and their unique solvability in space-time Sobolev spa ces are discussed\, where a modified Hilbert transformation is used such t hat ansatz and test spaces are equal. Second\, conforming discretization s chemes\, using piecewise polynomial\, globally continuous functions\, are introduced. Solvability and stability of these numerical schemes are discu ssed. Next\, we investigate efficient direct solvers for the occurring hug e linear systems. The developed solvers are based on the Bartels--Stewart method and on the Fast Diagonalization method\, which result in solving a sequence of spatial subproblems. The solver based on the Fast Diagonalizat ion method allows solving these spatial subproblems in parallel\, leading to a full parallelization in time. In the last part of the talk\, numerica l examples are shown and discussed.
\n\nFor further information abou t the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20231020T120000 END:VEVENT BEGIN:VEVENT UID:news1562@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230926T153730 DTSTART;TZID=Europe/Zurich:20231013T110000 SUMMARY:Seminar in Numerical Analysis: Markus Weimar (Julius-Maximilians-Un iversität Würzburg) DESCRIPTION:As a rule of thumb in approximation theory\, the asymptotic spe ed of convergence of numerical algorithms is governed by the regularity of the objects we like to approximate. Besides classical isotropic Sobolev s moothness\, in the last decades the notion of so-called dominating- mixed regularity of functions turned out to be an important concept in numerical analysis. Indeed\, it naturally arises in high-dimensional real-world app lications\, e.g.\, related to the electronic Schrödinger equation. Althou gh optimal approximation rates for embeddings within the scales of isotr opic or dominating-mixed Lp-Sobolev spaces are well-understood\, not that much is known for embeddings across those scales (break-of-scale embedd ings).\\r\\nIn this lecture\, we first review the Fourier analytic approac h towards by now well-established (Besov and Triebel-Lizorkin) scales of d istribution spaces that measure either isotropic or dominating-mixed reg ularity. In addition\, we introduce new function spaces of hybrid smoothne ss 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-)ada ptive wavelet-based multiscale algorithms that approximate such embed- din gs at optimal dimension-independent rates of convergence. Important specia l cases cover the approximation of functions having dominating-mixed Sobol ev 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 p art of a joint work with Glenn Byrenheid (FSU Jena)\, Markus Hansen (PU Ma rburg)\, and Janina Hübner (RU Bochum).\\r\\nReferences:\\r\\n[1] G. Byre nheid\, J. Hübner\, and M. Weimar. Rate-optimal sparse approximation of c ompact break-of-scale embeddings. Appl. Comput. Harmon. Anal. 65:40–66\ , 2023 (arXiv:2203.10011).\\r\\nFor further information about the seminar\ , please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:As a rule of thumb in approximation theory\, the asymptotic s peed of convergence of numerical algorithms is governed by the regularity of the objects we like to approximate. Besides classical isotropic Sobolev smoothness\, in the last decades the notion of so-called dominating- mixe d regularity of functions turned out to be an important concept in numeric al analysis. Indeed\, it naturally arises in high-dimensional real-world a pplications\, e.g.\, related to the electronic Schrödinger equation. Alth ough optimal approximation rates for embeddings \;within \;the sca les of isotropic or dominating-mixed \;Lp-Sobolev spaces are well-unde rstood\, not that much is known for embeddings \;across \;those sc ales (break-of-scale embeddings).
\nIn this lecture\, we first revie w 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 introduc e new function spaces of hybrid smoothness which are able to \;simulta neously \;capture both types of regularity at the same time. As a furt her generalization of the aforementioned scales\, they particularly includ e standard Sobolev spaces on domains. On the other hand\, our new spaces y ield an appropri- ate framework to study break-of-scale embeddings by mean s of harmonic analysis. We shall present (non-)adaptive wavelet-based mult iscale algorithms that approximate such embed- dings at optimal dimension- independent rates of convergence. Important special cases cover the approx imation of functions having dominating-mixed Sobolev smoothness w.r.t.&nbs p\;Lp \;in the norm of the (isotropic) energy space \;H1.
\nThe 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).
\nReferences:
\n[1] G. Byre nheid\, J. Hübner\, and M. Weimar. Rate-optimal sparse approximation of c ompact break-of-scale embeddings. Appl. Comput. Harmon. Anal. \;65:40 –66\, 2023 (arXiv:2203.10011).
\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20231013T120000 END:VEVENT BEGIN:VEVENT UID:news1538@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230915T113109 DTSTART;TZID=Europe/Zurich:20230929T110000 SUMMARY:Seminar in Numerical Analysis: Rüdiger Kempf (U Bayreuth) DESCRIPTION:Reproducing kernel Hilbert spaces (RKHSs) and the closely rela ted kernel methods are well-established and well-studied tools in classi cal approximation theory. More recently\, they see many uses in other prob lems in applied and numerical analysis.\\r\\nIn 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 a nalysis of these networks.\\r\\nAnother application of RKHSs is in high(er )-dimensional approximation. For instance in the field of quasi Monte-Carl o methods\, kernel-techniques are used to derive an error analysis for hig h-dimensional quadrature rules. We also developed a novel kernel-based app roximation method for higher-dimensional meshfree function reconstruction\ , based on Smolyak operators.\\r\\nIn this talk I will provide an introduc tion into the theory of RKHSs\, their kernels and associated kernel method s. In particular\, I will focus on a multiscale approximation scheme for r escaled radial basis functions. This method will then be used to derive th e new tensor product multilevel method for higher- dimensional meshfree approximation\, which I will discuss in detail.\\r\\n\\r\\nFor further inf ormation about the seminar\, please visit this webpage [t3://page?uid=1115 ]. X-ALT-DESC:Reproducing kernel Hilbert spaces (RKHSs) \;and the close ly related \;kernel methods \;are well-established and well-studie d tools in classical approximation theory. More recently\, they see many u ses in other problems in applied and numerical analysis.
\nIn machin e learning\, support vector machines heavily rely on RKHSs. For neural net works Barron spaces are connected to certain RKHSs and offer a possibility for a theoretical analysis of these networks.
\nAnother application of RKHSs is in high(er)-dimensional approximation. For instance in the fi eld 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 f unction reconstruction\, based on Smolyak operators.
\nIn this talk I will provide an introduction into the theory of RKHSs\, their kernels an d 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&nbs p\;for higher- dimensional meshfree approximation\, which I will discuss i n detail.
\n\nFor further information about the seminar\, please vis it this webpage.
DTEND;TZID=Europe/Zurich:20230929T120000 END:VEVENT BEGIN:VEVENT UID:news1536@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230904T192551 DTSTART;TZID=Europe/Zurich:20230922T110000 SUMMARY:Seminar in Numerical Analysis: Robert Gruhlke (FU Berlin) DESCRIPTION:Ensemble methods have become ubiquitous for solving Bayesian in ference problems\, in particular the efficient sampling from posterior den sities. State-of-the-art subclasses of Markow-Chain-Monte-Carlo methods r ely on gradient information of the log-density including Langevin samplers such as Ensemble Kalman Sampler (EKS) and Affine Invariant Langevin Dyna mics (ALDI). These dynamics are described by stochastic differential equa tions (SDEs) with time homogeneous drift terms. \\r\\nIn this talk we pre sent enhancement strategies of such ensemble methods based on sample enric hment and homotopy formalism\, that ultimately lead to time-dependent drif t terms that possible assimilate a larger class of target distributions w hile providing faster mixing times. \\r\\nFurthermore\, we present an alt ernative route to construct time-inhomogeneous drift terms based on rever se Diffusion processes that are popular in state-of-the-art Generative Mo delling such as Diffusion maps. Here\, we propose learning these log-dens ities by propagation of the target distribution through an Ornstein-Uhlenb eck process. For this\, we solve the associated Hamilton-Jabobi-Bellman eq uation through an adaptive explicit Euler discretization using low-rank co mpression such as functional Tensor Trains for the spatial discretization. \\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Ensemble methods have become ubiquitous for solving Bayesian inference problems\, in particular the efficient sampling from posterior d ensities. \;State-of-the-art subclasses of Markow-Chain-Monte-Carlo me thods rely on gradient information of the log-density including Langevin s amplers such as \;Ensemble Kalman Sampler (EKS) and Affine Invariant L angevin Dynamics (ALDI). These dynamics are described by \;stochastic differential equations (SDEs) with time homogeneous drift terms. \;
\nIn this talk we present enhancement strategies of such ensemble meth ods based on sample enrichment and homotopy formalism\, that ultimately le ad to time-dependent drift terms that possible \;assimilate a larger c lass of target distributions while providing faster mixing times. \;\n
Furthermore\, we present an alternative route to construct \;ti me-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 ada ptive explicit Euler discretization using low-rank compression such as fun ctional Tensor Trains for the spatial discretization.
\n\nFor furthe r information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20230922T120000 END:VEVENT BEGIN:VEVENT UID:news1464@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230509T105935 DTSTART;TZID=Europe/Zurich:20230512T110000 SUMMARY:Seminar in Numerical Analysis: Martin Eigel (WIAS Berlin) DESCRIPTION:Weighted least squares methods have been examined thouroughly t o obtain quasi-optimal convergence results for a chosen (polynomial) basis of a linear space. A focus in the analysis lies on the construction of op timal sampling measures and the derivation of a sufficient sample complexi ty for stable reconstructions. When considering holomorphic functions such as solutions of common parametric PDEs\, the anisotropic sparsity they ex hibit can be exploited to achieve improved results adapted to the consider ed problem. In particular\, the sparsity of the data transfers to the solu tion sparsity in terms of polynomial chaos coefficients. When using nonlin ear model classes\, it turns out that the known results cannot be used dir ectly. To obtain comparable a priori rates\, we introduce a new weighted v ersion of Stechkin's lemma. This enables to obtain optimal complexity resu lts for a model class of low-rank tensor trains. We also show that the sol ution sparsity results in sparse component tensors and sketch how this can be realised in practical algorithms. A nice application is the reconstruc tion of Galerkin solutions for parametric PDEs. With this\, a provably con verging a posteriori adaptive algorithm can be derived for linear model PD Es with non-affine coefficients.\\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Weighted least squares methods have been examined thouroughly to obtain quasi-optimal convergence results for a chosen (polynomial) bas is of a linear space. A focus in the analysis lies on the construction of optimal sampling measures and the derivation of a sufficient sample comple xity for stable reconstructions. When considering holomorphic functions su ch as solutions of common 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 so lution sparsity in terms of polynomial chaos coefficients. When using nonl inear model classes\, it turns out that the known results cannot be used d irectly. To obtain comparable a priori rates\, we introduce a new weighted version of Stechkin's lemma. This enables to obtain optimal complexity re sults 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 c an be realised in practical algorithms. A nice application is the reconstr uction of Galerkin solutions for parametric PDEs. With this\, a provably c onverging a posteriori adaptive algorithm can be derived for linear model PDEs with non-affine coefficients.
\n\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20230512T120000 END:VEVENT BEGIN:VEVENT UID:news1500@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230427T100910 DTSTART;TZID=Europe/Zurich:20230505T110000 SUMMARY:Seminar in Numerical Analysis: Elena Moral Sánchez (Max-Planck Ins titute for Plasma Physics) DESCRIPTION:The cold-plasma wave equation describes the propagation of an e lectromagnetic wave in a magnetized plasma\, which is an inhomogeneous\, d ispersive and anisotropic medium. The thermal effects are assumed to be ne gligible\, which leads to a linear partial differential equation. Besides\ , we assume that the electromagnetic field of the propagating wave is in t he time-harmonic regime. This model has applications in magnetic confineme nt fusion devices\, like the Tokamak. Namely\, electromagnetic waves are u sed to heat up the plasma (Electron cyclotron resonance heating (ECRH)) or for interferometry and reflectometry diagnostics (to measure plasma densi ty and position\, etc.). In the first part of this talk\, we introduce th e cold-plasma model\, together with a qualitative study of the plasma mode s which expose the complexity of the problem. In the second part\, we desc ribe the problem and the simplifications we carry out\, which yield the in definite Helmholtz equation. It is solved with B-Spline Finite Elements pr ovided by the Psydac library and some results are shown. Lastly\, we discu ss the performance and potential ways of preconditioning.\\r\\n\\r\\nFor f urther information about the seminar\, please visit this webpage [t3://pag e?uid=1115]. X-ALT-DESC:The cold-plasma wave equation describes the propagation of an
electromagnetic wave in a magnetized plasma\, which is an inhomogeneous\,
dispersive and anisotropic medium. The thermal effects are assumed to be
negligible\, which leads to a linear partial differential equation. Beside
s\, we assume that the electromagnetic field of the propagating wave is in
the time-harmonic regime.
This model has applications in magnetic c
onfinement fusion devices\, like the Tokamak. Namely\, electromagnetic wav
es are used to heat up the plasma (Electron cyclotron resonance heating (E
CRH)) or for interferometry and reflectometry diagnostics (to measure plas
ma density and position\, etc.).
In the first part of this ta
lk\, we introduce the cold-plasma model\, together with a qualitative stud
y of the plasma modes which expose the complexity of the problem.
In
the second part\, we describe the problem and the simplifications we carr
y out\, which yield the indefinite Helmholtz equation. It is solved with B
-Spline Finite Elements provided by the Psydac library and some results ar
e shown. Lastly\, we discuss the performance and potential ways of precond
itioning.
For further information about the seminar\, please vis it this webpage.
DTEND;TZID=Europe/Zurich:20230505T120000 END:VEVENT BEGIN:VEVENT UID:news1472@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230417T092542 DTSTART;TZID=Europe/Zurich:20230428T110000 SUMMARY:Seminar in Numerical Analysis: Frédéric Nataf (CNRS — Universit é Pierre et Marie Curie) 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 knowledge of the constra ined space. We assume that all sub matrices are sparse and that the diagon al blocks are spectrally equivalent to a sum of positive semi definite mat rices. The latter assumption enables the design of adaptive coarse space f or DD methods that extends the GenEO theory to saddle point problems. Nume rical results on three dimensional elasticity problems for steel-rubber st ructures discretized by a finite element with continuous pressure are show n for up to one billion degrees of freedom along with comparisons to Algeb raic Multigrid Methods.\\r\\n\\r\\nFor further information about the semin ar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:We introduce a scalable adaptive element-based domain decompo sition (DD) method for solving saddle point problems defined as a block tw o by two matrix. The algorithm does not require any knowledge of the const rained space. We assume that all sub matrices are sparse and that the diag onal blocks are spectrally equivalent to a sum of positive semi definite m atrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory to saddle point problems. Nu merical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are sh own for up to one billion degrees of freedom along with comparisons to Alg ebraic Multigrid Methods.
\n\nFor further information about the semi nar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20230428T120000 END:VEVENT BEGIN:VEVENT UID:news1480@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230418T090618 DTSTART;TZID=Europe/Zurich:20230421T110000 SUMMARY:Seminar in Numerical Analysis: Omar Lakkis (University of Sussex) DESCRIPTION:Least-squares finite element recovery-based methods provide a s imple and practical way to approximate linear elliptic PDEs in nondivergen ce form where standard variational approach either fails or requires techn ically complex modifications.\\r\\nThis idea allows the creation of effici ent 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--Bell man (Dynamic Programming) form\, i.e.\, as the supremum of a collection of linear operators acting on the unkown.\\r\\nThe least-squares FEM approac h\, a variant of the nonvariational finite element method\, is based on gr adient 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-leve l monotonicity (maximum principle)\, which is valid for the PDE and crucia l in proving the convergence of many degree one FEM and finite difference schemes.\\r\\nSuitable functional spaces and penalties in the least-square s'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 b eing optimized.\\r\\nFurthermore\, the nonlinear operator which is not nec essarily everywhere differentiable\, must be linearized in appropriate fun ctional 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 dimes ional functionals spaces. This allows an easy use of our non-monotone sche mes which provides convergence rates as well as a posteriori error estimat es.\\r\\n\\r\\nFor further information about the seminar\, please visit th is webpage [t3://page?uid=1115]. X-ALT-DESC:Least-squares finite element recovery-based methods provide a simple and practical way to approximate linear elliptic PDEs in nondiverg ence form where standard variational approach either fails or requires tec hnically complex modifications.
\nThis idea allows the creation of e fficient solvers for fully nonlinear elliptic equations\, the linearizatio n 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 collecti on of linear operators acting on the unkown.
\nThe least-squares FEM approach\, a variant of the nonvariational finite element method\, is bas ed 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 discr ete-level monotonicity (maximum principle)\, which is valid for the PDE an d crucial in proving the convergence of many degree one FEM and finite dif ference schemes.
\nSuitable functional spaces and penalties in the l east-squares's cost functional must be carefully crafted in order to ensur e 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.
\nFurthermore\, the nonlinear operator wh ich is not necessarily everywhere differentiable\, must be linearized in a ppropriate functional spaces using semismooth Newton or Howard's policy it eration method. A crucial contribution of our work\, is the proof of conve rgence 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.
\n\nFor further information about the seminar\, pl ease visit this webpage.
DTEND;TZID=Europe/Zurich:20230421T120000 END:VEVENT BEGIN:VEVENT UID:news1455@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230403T123724 DTSTART;TZID=Europe/Zurich:20230414T110000 SUMMARY:Seminar in Numerical Analysis: Vesa Kaarnioja (FU Berlin) DESCRIPTION:We describe a fast method for solving elliptic PDEs with uncert ain coefficients using kernel-based interpolation over a rank-1 lattice po int set [1]. By representing the input random field of the system using a model proposed by Kaarnioja\, Kuo\, and Sloan [2]\, in which a countable n umber of independent random variables enter the random field as periodic f unctions\, it is shown that the kernel interpolant can be constructed for the PDE solution (or some quantity of interest thereof) as a function of t he stochastic variables in a highly efficient manner using fast Fourier tr ansform. The method works well even when the stochastic dimension of the p roblem is large\, and we obtain rigorous error bounds which are independen t of the stochastic dimension of the problem. We also outline some techniq ues that can be used to further improve the approximation error and comput ational complexity of the method [3].\\r\\n\\r\\nReferences:\\r\\n[1] V. K aarnioja\, Y. Kazashi\, F. Y. Kuo\, F. Nobile\, and I. H. Sloan. Fast appr oximation by periodic kernel-based lattice-point interpolation with applic ation in uncertainty quantification. Numer. Math. 150:33-77\, 2022.\\r\\n[ 2] V. Kaarnioja\, F. Y. Kuo\, and I. H. Sloan. Uncertainty quantification using periodic random variables. SIAM J. Numer. Anal. 58(2):1068-1091\, 20 20.\\r\\n[3] V. Kaarnioja\, F. Y. Kuo\, and I. H. Sloan. Lattice-based ker nel approximation and serendipitous weights for parametric PDEs in very hi gh dimensions. Preprint 2023\, arXiv:2303.17755 [math.NA].\\r\\n\\r\\nFor further information about the seminar\, please visit this webpage [t3://pa ge?uid=1115]. X-ALT-DESC:We describe a fast method for solving elliptic PDEs with unce rtain 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 Sloan [2]\, in which a countable number of independent random variables enter the random field as periodic functions\, it is shown that the kernel interpolant can be constructed fo r the PDE solution (or some quantity of interest thereof) as a function of the stochastic variables in a highly efficient manner using fast Fourier transform. The method works well even when the stochastic dimension of the problem is large\, and we obtain rigorous error bounds which are independ ent of the stochastic dimension of the problem. We also outline some techn iques that can be used to further improve the approximation error and comp utational complexity of the method [3].
\n\nReferences:
\n[1] V. Kaarnioja\, Y. Kazashi\, F. Y. Kuo\, F. Nobile\, and I. H. Sloan. Fast approximation by periodic kernel-based lattice-point interpolation with ap plication in uncertainty quantification. Numer. Math. 150:33-77\, 2022.
\n[2] V. Kaarnioja\, F. Y. Kuo\, and I. H. Sloan. Uncertainty quantifi cation using periodic random variables. SIAM J. Numer. Anal. 58(2):1068-10 91\, 2020.
\n[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].
\n \nFor further information about the seminar\, please visit this webpag e.
DTEND;TZID=Europe/Zurich:20230414T120000 END:VEVENT BEGIN:VEVENT UID:news1468@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20230403T123812 DTSTART;TZID=Europe/Zurich:20230317T110000 SUMMARY:Seminar in Numerical Analysis: Marc Dambrine (Université de Pau et des Pays de l'Adour) DESCRIPTION:As it is often the case in optimization\, the solution of a sha pe problem is sensitive to the parameters of the problem. For example\, th e loading of a structure to be optimised is known in an imprecise way. In this talk\, I will present the different approaches that have been recentl y proposed to incorporate these uncertainties in the definition of the obj ective. I will present numerical illustrations from structural optimizatio n.\\r\\n\\r\\nFor further information about the seminar\, please visit thi s webpage [t3://page?uid=1115]. X-ALT-DESC:As it is often the case in optimization\, the solution of a s hape problem is sensitive to the parameters of the problem. For example\, the loading of a structure to be optimised is known in an imprecise way. I n this talk\, I will present the different approaches that have been recen tly proposed to incorporate these uncertainties in the definition of the o bjective. I will present numerical illustrations from structural optimizat ion.
\n\nFor further information about the seminar\, please visit th is webpage.
DTEND;TZID=Europe/Zurich:20230317T120000 END:VEVENT BEGIN:VEVENT UID:news1402@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220920T084344 DTSTART;TZID=Europe/Zurich:20221216T110000 SUMMARY:Seminar in Numerical Analysis: Christophe Geuzaine (Université de Liège) DESCRIPTION:This talk is devoted to non-overlapping Schwarz domain decompos ition methods for the resolution of high frequency flow acoustics problems of industrial relevance. First\, we will present recent advances on non-r eflecting boundary techniques that provide local approximations to the Dir ichlet-to-Neumann operator for convected and heterogeneous time-harmonic w ave propagation problems [1]. Then we will show how to adapt a generic dom ain decomposition framework to flow acoustics\, based on these newly desig ned transmission conditions\, and highlight the benefit of the approach on the simulation of three-dimensional noise radiation of a high by-pass rat io turbofan engine intake [2]. [1] Marchner\, P.\, Antoine\, X.\, Geuzain e\, C.\, & Bériot\, H. (2022). Construction and numerical assessment of l ocal absorbing boundary conditions for heterogeneous time-harmonic acousti c problems. SIAM Journal on Applied Mathematics\, 82(2)\, 476-501. [2] Li eu\, A.\, Marchner\, P.\, Gabard\, G.\, Beriot\, H.\, Antoine\, X.\, & Geu zaine\, C. (2020). A non-overlapping Schwarz domain decomposition method w ith high-order finite elements for flow acoustics. Computer Methods in App lied Mechanics and Engineering\, 369\, 113223.\\r\\nFor further informatio n about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:This talk is devoted to non-overlapping Schwarz domain decomp
osition methods for the resolution of high frequency flow acoustics proble
ms of industrial relevance. First\, we will present recent advances on non
-reflecting boundary techniques that provide local approximations to the D
irichlet-to-Neumann operator for convected and heterogeneous time-harmonic
wave propagation problems [1]. Then we will show how to adapt a generic d
omain decomposition framework to flow acoustics\, based on these newly des
igned transmission conditions\, and highlight the benefit of the approach
on the simulation of three-dimensional noise radiation of a high by-pass r
atio turbofan engine intake [2].
[1] Marchner\, P.\, Antoine\
, X.\, Geuzaine\, C.\, &\; Bériot\, H. (2022). Construction and numeri
cal assessment of local absorbing boundary conditions for heterogeneous ti
me-harmonic acoustic problems. SIAM Journal on Applied Mathematics\, 82(2)
\, 476-501.
[2] Lieu\, A.\, Marchner\, P.\, Gabard\, G.\, Ber
iot\, H.\, Antoine\, X.\, &\; Geuzaine\, C. (2020). A non-overlapping S
chwarz domain decomposition method with high-order finite elements for flo
w acoustics. Computer Methods in Applied Mechanics and Engineering\, 369\,
113223.
For further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20221216T120000 END:VEVENT BEGIN:VEVENT UID:news1410@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221130T142441 DTSTART;TZID=Europe/Zurich:20221209T110000 SUMMARY:Seminar in Numerical Analysis: Patrick Ciarlet (ENSTA Paris) DESCRIPTION:Variational formulations are a popular tool to analyse linear P DEs (eg. neutron diffusion\, Maxwell equations\, Stokes equations ...)\, a nd it also provides a convenient basis to design numerical methods to solv e them. Of paramount importance is the inf-sup condition\, designed by Lad yzhenskaya\, Necas\, Babuska and Brezzi in the 1960s and 1970s. As is well -known\, it provides sharp conditions to prove well-posedness of the probl em\, namely existence and uniqueness of the solution\, and continuous depe ndence with respect to the data. Then\, to solve the approximate\, or disc rete\, problems\, there is the (uniform) discrete inf-sup condition\, to e nsure existence of the approximate solutions\, and convergence of those so lutions to the exact solution. Often\, the two sides of this problem (exac t and approximate) are handled separately\, or at least no explicit connec tion is made between the two.\\r\\nIn this talk\, I will focus on an appro ach that is completely equivalent to the inf-sup condition for problems se t in Hilbert spaces\, the T-coercivity approach. This approach relies on t he design of an explicit operator to realize the inf-sup condition. If the operator is carefully chosen\, it can provide useful insight for a straig htforward definition of the approximation of the exact problem. As a matte r of fact\, the derivation of the discrete inf-sup condition often becomes elementary\, at least when one considers conforming methods\, that is whe n the discrete spaces are subspaces of the exact Hilbert spaces. In this w ay\, both the exact and the approximate problems are considered\, analysed and solved at once.\\r\\nIn itself\, T-coercivity is not a new theory\, h owever 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 an d solve linear PDEs. In particular\, it provides guidelines such as\, whic h 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\, toolb oxes. This claim will be illustrated on classical linear PDEs\, and for so me generalizations of those models.\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Variational formulations are a popular tool to analyse linear PDEs (eg. neutron diffusion\, Maxwell equations\, Stokes equations ...)\, and it also provides a convenient basis to design numerical methods to so lve them. Of paramount importance is the inf-sup condition\, designed by Ladyzhenskaya\, Necas\, Babuska and Brezzi in the 1960s an d 1970s. As is well-known\, it provides sharp conditions to prove well-pos edness of the problem\, namely existence and uniqueness of the solution\, and continuous dependence with respect to the data. Then\, to solve the ap proximate\, or discrete\, problems\, there is the (uniform) discre te inf-sup condition\, to ensure existence of the approximate sol utions\, and convergence of those solutions to the exact solution. Often\, the two sides of this problem (exact and approximate) are handled separat ely\, or at least no explicit connection is made between the two.
\n
In this talk\, I will focus on an approach that is completely equivalent t
o the inf-sup condition for problems set in Hilbert spaces\, the T
-coercivity approach. This approach relies on the design of an
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 solv e linear PDEs. In particular\, it provides guidelines such as\, which abst ract tools and which numerical methods are the most “natural” to analy se 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 gen eralizations of those models.
\nFor further information about the se minar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20221209T120000 END:VEVENT BEGIN:VEVENT UID:news1409@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220928T143806 DTSTART;TZID=Europe/Zurich:20221202T110000 SUMMARY:Seminar in Numerical Analysis: Sébastien Imperiale (Inria — LMS\ , Ecole Polytechnique\, CNRS — Université Paris-Saclay\, MΞDISIM) DESCRIPTION:The objective of this work is to propose and analyze numerical schemes to solve transient wave propagation problems that are exponentiall y stable (i.e. the solution decays to zero exponentially fast). Applicatio ns are in data assimilation strategies or the discretisation of absorbing boundary conditions. More precisely the aim of our work is to propose a di scretization process that enables to preserve the exponential stability at the discrete level as well as a high order consistency when using a high- order finite element approximation. The main idea is to add to the wave eq uation a stabilizing term which damps the high-frequency oscillating compo nents of the solutions such as spurious waves. This term is built from a d iscrete multiplier analysis that proves the exponential stability of the s emi-discrete problem at any order without affecting the order of convergen ce.\\r\\nFor further information about the seminar\, please visit this web page [t3://page?uid=1115]. X-ALT-DESC:The objective of this work is to propose and analyze numerica l schemes to solve transient wave propagation problems that are exponentia lly stable (i.e. the solution decays to zero exponentially fast). Applicat ions are in data assimilation strategies or the discretisation of absorbin g boundary conditions. More precisely the aim of our work is to propose a discretization process that enables to preserve the exponential stability at the discrete level as well as a high order consistency when using a hig h-order finite element approximation. The main idea is to add to the wave equation a stabilizing term which damps the high-frequency oscillating com ponents of the solutions such as spurious waves. This term is built from a discrete multiplier analysis that proves the exponential stability of the semi-discrete problem at any order without affecting the order of converg ence.
\nFor further information about the seminar\, please visit thi s webpage.
DTEND;TZID=Europe/Zurich:20221202T120000 END:VEVENT BEGIN:VEVENT UID:news1408@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221110T132504 DTSTART;TZID=Europe/Zurich:20221118T110000 SUMMARY:Seminar in Numerical Analysis: Alexey Chernov (Universität Oldenbu rg) DESCRIPTION:We investigate a class of parametric elliptic eigenvalue proble ms where the coefficients (and hence the solution) may depend on a paramet er y. Understanding the regularity of the solution as a function of y is i mportant for construction of efficient numerical approximation schemes. Se veral approaches are available in the existing literature\, e.g. the comp lex-analytic argument by Andreev and Schwab (2012) and the real-variable a rgument by Gilbert et al. (2019+). The latter proof strategy is more expli cit\, but\, due to the nonlinear nature of the problem\, leads to slightly suboptimal results. In this talk we close this gap and (as a by-product) extend the analysis to the more general class of coefficients.\\r\\nFor fu rther information about the seminar\, please visit this webpage [t3://page ?uid=1115]. X-ALT-DESC:We investigate a class of parametric elliptic eigenvalue prob lems where the coefficients (and hence the solution) may depend on a param eter y. Understanding the regularity of the solution as a function of y is important for construction of efficient numerical approximation schemes. Several approaches are available in the existing literature\, e.g. \;t he complex-analytic argument by Andreev and Schwab (2012) and the real-var iable argument by Gilbert et al. (2019+). The latter proof strategy is mor e explicit\, but\, due to the nonlinear nature of the problem\, leads to s lightly suboptimal results. In this talk we close this gap and (as a by-pr oduct) extend the analysis to the more general class of coefficients.
\ nFor further information about the seminar\, please visit this webpage .
DTEND;TZID=Europe/Zurich:20221118T120000 END:VEVENT BEGIN:VEVENT UID:news1412@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20221021T141230 DTSTART;TZID=Europe/Zurich:20221104T110000 SUMMARY:Seminar in Numerical Analysis: Naomi Schneider (Universität Siegen ) DESCRIPTION:Both the approximation of the gravitational potential via the d ownward continuation of satellite data and of wave velocities via the trav el time tomography using earthquake data are geoscientific ill- posed inve rse problems. To monitor certain aspects of the system Earth\, like the ma ss transport or its geomagnetic field\, it is\, however\, important to tac kle these challenges. Traditionally\, an approximation of such a linear(iz ed) inverse problem is obtained in one\, a-priori chosen basis system: eit her a global one\, e.g. spherical harmonics or polynomials on the ball\, o r 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 devel oped 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 t he (Learning) Inverse Problem Matching Pursuits (LIPMPs). They construct a n approximation iteratively from an intentionally overcomplete set of tria l functions\, the dictionary\, such that the Tikhonov functional is reduce d. Due to the learning add-on\, the dictionary can very well be infinite. Moreover\, the computational costs are usually decreased. In this talk\, w e give details on the LIPMPs and show some current numerical results.\\r\\ nFor further information about the seminar\, please visit this webpage [t3 ://page?uid=1115]. X-ALT-DESC:Both the approximation of the gravitational potential via the
downward continuation of satellite data and of wave velocities via the tr
avel time tomography using earthquake data are geoscientific ill- posed in
verse problems. To monitor certain aspects of the system Earth\, like the
mass transport or its geomagnetic field\, it is\, however\, important to t
ackle these challenges.
Traditionally\, an approximation of such a l
inear(ized) inverse problem is obtained in one\, a-priori chosen basis sys
tem: 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 sp
here or finite elements on the ball.
In the Geomathematics Group Sie
gen\, we developed methods that enable us to combine different types of tr
ial functions for such an approximation. The idea is to make the most of t
he benefits of different types of available trial functions. The algorithm
s are called the (Learning) Inverse Problem Matching Pursuits (LIPMPs). Th
ey construct an approximation iteratively from an intentionally overcomple
te set of trial functions\, the dictionary\, such that the Tikhonov functi
onal 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 num
erical results.
For further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20221104T120000 END:VEVENT BEGIN:VEVENT UID:news1352@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220516T103313 DTSTART;TZID=Europe/Zurich:20220520T110000 SUMMARY:Seminar in Numerical Analysis: Jens Saak (Max Planck Institute for Dynamics of Complex Technical Systems) DESCRIPTION:Optimal control problems subject to constraints given by partia l differential equations are a powerful tool for the improvement of many t asks in science an technology. Classic optimization today is applicable on various problems and tackling nonlinear equations and inclusion of box co nstraints on the solutions is flexible. However\, especially for non-stati onary problems\, small perturbations along the trajectories can easily lea d to large deviations in the desired solutions. Consequently\, optimality may be lost just as easily. On the other hand\, the linear-quadratic regul ator problem in system theory is an approach to make a dynamical system re act to perturbation via feedback controls that can be expressed by the sol utions of matrix Riccati equations. It’s applicability is limited by the linearity of the dynamical system and the efficient solvability of the qu adratic 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 approximabi lity.\\r\\nFor further information about the seminar\, please visit this w ebpage [t3://page?uid=1115]. X-ALT-DESC:Optimal control problems subject to constraints given by part
ial differential equations are a powerful tool for the improvement 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-sta
tionary problems\, small perturbations along the trajectories can easily l
ead to large deviations in the desired solutions. Consequently\, optimalit
y may be lost just as easily.
On the other hand\, the linear-quadrat
ic regulator problem in system theory is an approach to make a dynamical s
ystem react to perturbation via feedback controls that can be expressed by
the solutions of matrix Riccati equations. It’s applicability is limite
d by the linearity of the dynamical system and the efficient solvability o
f the quadratic matrix equation.
In this talk\, we discuss how certa
in classes of non-stationary PDEs can be reformulated (after spatial semi-
discretization) into structured linear dynamical systems that allow the Ri
ccati feedback to be computed. This allows us to combine both approaches a
nd thus steer solutions of perturbed PDEs back to the optimized trajectori
es. 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 solu
tions usually feature a strong singular value decay\, and thus good low-ra
nk approximability.
For further information about the seminar\, pl ease visit this webpage.
DTEND;TZID=Europe/Zurich:20220520T120000 END:VEVENT BEGIN:VEVENT UID:news1351@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220419T163823 DTSTART;TZID=Europe/Zurich:20220422T110000 SUMMARY:Seminar in Numerical Analysis: Matthias Voigt (FernUni Schweiz) DESCRIPTION:We introduce a model reduction approach for linear time-invaria nt second-order systems based on positive real balanced truncation. Our me thod guarantees to preserve passivity of the reduced-order model as well a s the positive definiteness of the mass and stiffness matrices and admits an a priori gap metric error bound. Our construction of the second-order r educed model is based on the consideration of an internal symmetry structu re and the invariant zeros of the system and their sign-characteristics fo r which we derive a normal form. The results are available in [1].\\r\\n[1 ] I. Dorschky\, T. Reis\, and M. Voigt. Balanced truncation model reductio n for symmetric second order systems - a passivity-based approach. SIAM J. Matrix Anal. Appl.\, 42(4):1602--1635\, 2021.\\r\\nFor further informatio n about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:We introduce a model reduction approach for linear time-invar
iant second-order systems based on positive real balanced truncation.
Our method guarantees to preserve passivity of the reduced-order model a
s 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 sy
mmetry structure and the invariant zeros of the system and their sign-char
acteristics for which we derive a normal form.
The results are avail
able in [1].
[1] I. Dorschky\, T. Reis\, and M. Voigt. Balanced tr uncation model reduction for symmetric second order systems - a passivity- based approach. SIAM J. Matrix Anal. Appl.\, 42(4):1602--1635\, 2021.
\ nFor further information about the seminar\, please visit this webpage .
DTEND;TZID=Europe/Zurich:20220422T120000 END:VEVENT BEGIN:VEVENT UID:news1333@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220315T174614 DTSTART;TZID=Europe/Zurich:20220408T110000 SUMMARY:Seminar in Numerical Analysis: Ralf Hiptmair (ETH Zürich) DESCRIPTION:We consider scalar-valued shape functionals on sets of shapes w hich are small perturbations of a reference shape. The shapes are describe d by parameterizations and their closeness is induced by a Hilbert space s tructure on the parameter domain. We justify a heuristic for finding the b est low-dimensional parameter subspace with respect to uniformly approxima ting a given shape functional. We also propose an adaptive algorithm for a chieving a prescribed accuracy when representing the shape functional with a small number of shape parameters.\\r\\nFor further information about th e seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:We consider scalar-valued shape functionals on sets of shapes which are small perturbations of a reference shape. The shapes are descri bed by parameterizations and their closeness is induced by a Hilbert space structure on the parameter domain. We justify a heuristic for finding the best low-dimensional parameter subspace with respect to uniformly approxi mating a given shape functional. We also propose an adaptive algorithm for achieving a prescribed accuracy when representing the shape functional wi th a small number of shape parameters.
\nFor further information abo ut the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20220408T120000 END:VEVENT BEGIN:VEVENT UID:news1334@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220325T150305 DTSTART;TZID=Europe/Zurich:20220401T110000 SUMMARY:Seminar in Numerical Analysis: Johannes Pfefferer (Technische Unive rsität München) DESCRIPTION:Many areas of science and engineering involve optimal control o f processes that are modeled through partial differential equations. This talk will introduce the theoretical foundation and numerical methods based on finite elements for solving PDE constrained optimal control problems. We will discuss different discretization concepts and corresponding discre tization error estimates. The discussion will include the consideration of optimal control problems with control constraints as well as with state c onstraints.\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Many areas of science and engineering involve optimal control of processes that are modeled through partial differential equations. Thi s talk will introduce the theoretical foundation and numerical methods bas ed on finite elements for solving PDE constrained optimal control problems . We will discuss different discretization concepts and corresponding disc retization error estimates. The discussion will include the consideration of optimal control problems with control constraints as well as with state constraints.
\nFor further information about the seminar\, please v isit this webpage.
DTEND;TZID=Europe/Zurich:20220401T120000 END:VEVENT BEGIN:VEVENT UID:news1340@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220321T111220 DTSTART;TZID=Europe/Zurich:20220325T110000 SUMMARY:Seminar in Numerical Analysis: Stepan Shakhno (Ivan Franko National University of Lviv) DESCRIPTION:In this talk\, one- and two-step methods for solving nonlinear equations with nondifferentiable operators are proposed. These methods are based on two methods: using derivatives and using divided differences. Th e 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.\\r\\nFor further inf ormation about the seminar\, please visit this webpage [t3://page?uid=1115 ]. X-ALT-DESC:In this talk\, one- and two-step methods for solving nonlinea r equations with nondifferentiable operators are proposed. These methods a re based on two methods: using derivatives and using divided differences.< br /> The local and semi-local convergence of the proposed methods is stud ied and the order of their convergence is established. We apply our result s to the numerical solving of systems of nonlinear equations.
\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20220325T120000 END:VEVENT BEGIN:VEVENT UID:news1335@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220310T115442 DTSTART;TZID=Europe/Zurich:20220318T110000 SUMMARY:Seminar in Numerical Analysis: Markus Bachmayr (Universität Mainz) 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 spatial discretisation limits the achievable overall convergence rate)\, previous approaches for a posteriori selection of polynomial terms and corresponding spatial di scretizations do not guarantee optimal complexity in the sense of computat ional costs scaling linearly in the number of degrees of freedom. We show that one can achieve optimality of an adaptive Galerkin scheme for discret izations by spline wavelets in the spatial variable when a multiscale rep resentation of the affinely parameterized random coefficients is used. \\ r\\nM. Bachmayr and I. Voulis\, An adaptive stochastic Galerkin method ba sed on multilevel expansions of random fields: Convergence and optimality\ , arXiv:2109:09136 [https://arxiv.org/abs/2109.09136]\\r\\nFor further in formation about the seminar\, please visit this webpage [t3://page?uid=111 5]. X-ALT-DESC:We consider the computational complexity of approximating ell iptic PDEs with random coefficients by sparse product polynomial expansion s. Except for special cases (for instance\, when the spatial discretisatio n limits the achievable overall convergence rate)\, previous approaches fo r \;a posteriori \;selection of polynomial terms and corr esponding spatial discretizations do not guarantee optimal complexity in t he sense of computational costs scaling linearly in the number of degrees of freedom. We show that one can achieve optimality of an adaptive Galerki n scheme for discretizations by spline wavelets \;in the spatial varia ble when a multiscale representation of the affinely parameterized random coefficients is used. \;
\nM. Bachmayr and I. Voulis\, \;An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality\, \;arXiv:2109:09136
\nFor further informati on about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20220318T120000 END:VEVENT BEGIN:VEVENT UID:news1277@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211204T191942 DTSTART;TZID=Europe/Zurich:20211217T110000 SUMMARY:Seminar in Numerical Analysis: Eliane Bécache (POEMS\, CNRS\, INRI A\, ENSTA Paris\, Institut Polytechnique de Paris) DESCRIPTION:The PML method is one of the most widely used for the numerical simulation of wave propagation problems set in unbounded domains. However difficulties arise when the exterior domain has some complexity which p revents from using classical approaches. For instance\, it is well-known t hat PML may be unstable for time-domain eslastodynamic waves in some aniso tropic materials. More recently\, is has also been noticed that standard P ML cannot work in presence of some dispersive materials. In some cases\, new stable PMLs have been designed.\\r\\nIn this talk\, we address the qu estions 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.\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:The PML method is one of the most widely used for the numeric al simulation of wave propagation problems set in unbounded domains. Howev er \;difficulties arise when the exterior domain has some complexity which prevents from using classical approaches. For instance\, it is well- known that PML may be unstable for time-domain eslastodynamic waves in som e anisotropic materials. More recently\, is has also been noticed that sta ndard PML cannot work in presence of some dispersive materials. \;In some cases\, new stable PMLs have been designed.
\nIn this talk\, we address the questions of well-posedness\, stability and convergence of st andard and new models of PMLs in the context of electromagnetic waves for non-dispersive and dispersive materials.
\nFor further information a bout the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20211217T120000 END:VEVENT BEGIN:VEVENT UID:news1276@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211204T143630 DTSTART;TZID=Europe/Zurich:20211210T110000 SUMMARY:Seminar in Numerical Analysis: Mike Botchev (Keldysh Institute of A pplied Mathematics) DESCRIPTION:An efficient Krylov subspace algorithm for computing actions of the phi matrix function for large matrices is proposed. This matrix funct ion is widely used in exponential time integration\, Markov chains\, and n etwork 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 n umerical range in the stable complex half-plane\, that the restarted metho d is guaranteed to converge for any Krylov subspace dimension. Numerical t ests demonstrate efficiency of our approach for solving large scale evolut ion problems resulting from discretized in space time-dependent PDEs\, in particular\, diffusion and convection-diffusion problems.\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid= 1115]. X-ALT-DESC:An efficient Krylov subspace algorithm for computing actions of the phi matrix function for large matrices is proposed. This matrix fun ction is widely used in exponential time integration\, Markov chains\, and network analysis and many other applications. Our algorithm is based on&n bsp\;a reliable residual based stopping criterion and a new efficient rest arting procedure. We analyze residual convergence and prove\, for matrices with numerical range in the stable complex half-plane\, that the restarte d method is guaranteed to converge for any Krylov subspace dimension. Nume rical tests demonstrate efficiency of our approach for solving large scale evolution problems resulting from discretized in space time-dependent PDE s\, in particular\, diffusion and convection-diffusion problems.
\nF or further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20211210T120000 END:VEVENT BEGIN:VEVENT UID:news1258@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211025T123428 DTSTART;TZID=Europe/Zurich:20211203T110000 SUMMARY:Seminar in Numerical Analysis: Larisa Beilina (Chalmers tekniska h ögskola) DESCRIPTION:We will discuss how to apply an adaptive finite element metho d (AFEM) for numerical solution of an electromagnetic volume integral equ ation. The problem of solution of this equation is formulated as an optim al control problem for minimizing of the Tikhonov's regularization funct ional. A posteriori error estimates for the error in the obtained finite element reconstruction and error in the Tikhonov's functional will be pr esented.\\r\\nBased on these estimates\, different adaptive finite element algorithms are formulated. Numerical examples will show efficiency of t he proposed adaptive algorithms to improve quality of 3D reconstruction of target during the process of microwave thermometry which is used in ca ncer therapies. This is joint work with the group of Biomedical Imaging a t the Department of Electrical Engineering at CTH\, Chalmers.\\r\\nFor fu rther information about the seminar\, please visit this webpage [t3://page ?uid=1115]. X-ALT-DESC:We will \; discuss how to apply an adaptive finite elemen t method (AFEM) for numerical solution of \;an electromagnetic volume integral equation. \;The problem of solution of this equation is formu lated as an optimal \;control problem for minimizing of the Tikhonov's regularization \;functional. \;A posteriori error estimates for t he error in the obtained finite \;element reconstruction and error in the Tikhonov's functional will be presented.
\nBased on these estima tes\, different adaptive finite element algorithms \;are formulated.&n bsp\;Numerical examples will show efficiency of the \;proposed adaptiv e algorithms to improve quality of 3D reconstruction \;of target durin g the process of microwave thermometry which is used in \;cancer thera pies. This is joint work with the group of \;Biomedical Imaging at the Department of Electrical \;Engineering at CTH\, Chalmers.
\nFor further information about the seminar\, please visit this webpage. DTEND;TZID=Europe/Zurich:20211203T120000 END:VEVENT BEGIN:VEVENT UID:news1260@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211109T170252 DTSTART;TZID=Europe/Zurich:20211119T110000 SUMMARY:Seminar in Numerical Analysis: Jaap van der Vegt (Universiteit Twen te) DESCRIPTION:In the numerical solution of partial differential equations\, i t is frequently necessary to ensure that certain variables\, e.g.\, densit y\, pressure\, or probability density distribution\, remain within strict bounds. Strict observation of these bounds is crucial\, otherwise unphysic al solutions will be obtained that might result in the failure of the nume rical algorithm. Bounds on certain variables are generally ensured in disc ontinuous Galerkin (DG) discretizations using positivity preserving limite rs\, which locally modify the solution to ensure that the constraints are satisfied\, while preserving higher order accuracy. In practice this appro ach is mostly limited to DG discretizations combined with explicit time in tegration methods. The combination of (positivity preserving) limiters in DG discretizations and implicit time integration methods results\, however \, in serious problems. Many positivity preserving limiters are not easy t o apply in time-implicit DG discretizations and have a non-smooth formulat ion\, which hampers the use of standard Newton methods to solve the nonlin ear algebraic equations resulting from the time-implicit DG discretization . This often results in poor convergence.\\r\\nIn this presentation\, we w ill 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 equation s resulting from a higher order accurate time-implicit DG discretization u sing techniques from mathematical optimization theory. This approach ens ures that the positivity constraints are satisfied and does not affect the higher order accuracy of the time-implicit DG discretization. The resulti ng algebraic equations are then solved using a specially designed semi-smo oth Newton method that is well suited to deal with the resulting nonlinear complementarity problem. We will demonstrate the algorithm on several par abolic model problems.\\r\\nFor further information about the seminar\, pl ease visit this webpage [t3://page?uid=1115]. X-ALT-DESC:
In the numerical solution of partial differential equations\, it is frequently necessary to ensure that certain variables\, e.g.\, dens ity\, pressure\, or probability density distribution\, remain within stric t bounds. Strict observation of these bounds is crucial\, otherwise unphys ical solutions will be obtained that might result in the failure of the nu merical algorithm. Bounds on certain variables are generally ensured in di scontinuous Galerkin (DG) discretizations using positivity preserving limi ters\, which locally modify the solution to ensure that the constraints ar e satisfied\, while preserving higher order accuracy. In practice this app roach is mostly limited to DG discretizations combined with explicit time integration methods. The combination of (positivity preserving) limiters i n DG discretizations and implicit time integration methods results\, howev er\, in serious problems. Many positivity preserving limiters are not easy to apply in time-implicit DG discretizations and have a non-smooth formul ation\, which hampers the use of standard Newton methods to solve the nonl inear algebraic equations resulting from the time-implicit DG discretizati on. This often results in poor convergence.
\nIn this presentation\, we will discuss a different approach to ensure that a higher order accura te DG solution satisfies the positivity constraints. Instead of using a li miter\, we impose the positivity constraints directly on the algebraic equ ations resulting from a higher order accurate time-implicit DG discretizat ion using techniques from mathematical optimization theory. \;This ap proach ensures that the positivity constraints are satisfied and does not affect the higher order accuracy of the time-implicit DG discretization. T he resulting algebraic equations are then solved using a specially designe d semi-smooth Newton method that is well suited to deal with the resulting nonlinear complementarity problem. We will demonstrate the algorithm on s everal parabolic model problems.
\nFor further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20211119T120000 END:VEVENT BEGIN:VEVENT UID:news1257@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211025T121842 DTSTART;TZID=Europe/Zurich:20211112T110000 SUMMARY:Seminar in Numerical Analysis: Rolf Krause (Università della Svizz era italiana) DESCRIPTION:Non-convex minimization problems show up in manifold applicatio ns: non-linear elasticity\, phase field models\, fracture propagation\, or the training of neural networks. Traditional multilevel decompositions a re the basic ingredient of the most efficient class of solution methods for linear systems\, i.e. of multigrid methods\, which allow to solve cert ain classes of linear systems with optimal complexity. The transfer of the se concepts to non-linear problems\, however\, is not straightforward\, ne ither in terms of the design of the multilevel decomposition nor in terms of convergence properties. In this talk\, we will discuss multilevel dec ompositions for convex\, non-convex and possibly non-smooth minimization p roblems. We will discuss in detail how multilevel optimization methods c an be constructed and analyzed and we will illustrate the sometimes sign ificant gain in performance\, which can be achieved by multilevel minimiza tion techniques. Examples from mechanics\, geophysics\, and machine learni ng will illustrate our discussion.\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Non-convex minimization problems show up in manifold applicat
ions: non-linear elasticity\, phase field models\, fracture propagation\,
or the training of neural networks.
Traditional multilevel de
compositions are the basic ingredient \;of the most efficient class o
f 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 s
traightforward\, neither in terms of the design of the multilevel decompos
ition nor in terms of convergence properties. In this talk\, we will discu
ss \;multilevel decompositions for convex\, non-convex and possibly n
on-smooth minimization problems. We will discuss in detail how \;mult
ilevel optimization methods can be constructed and analyzed and we will il
lustrate \;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< /a>.
DTEND;TZID=Europe/Zurich:20211112T120000 END:VEVENT BEGIN:VEVENT UID:news1256@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211025T121402 DTSTART;TZID=Europe/Zurich:20211029T110000 SUMMARY:Seminar in Numerical Analysis: Jochen Garke (Rheinische Friedrich-W ilhelms-Universität Bonn) DESCRIPTION:We present a conceptual framework that helps to bridge the know ledge gap between the two individual communities from machine learning an d numerical simulation to identify potential combined approaches and to promote the development of hybrid systems. We give examples of different types of combinations using exemplary approaches of simulation-assisted m achine learning and machine-learning assisted simulation. We also discuss an advanced pairing where we see particular further potential for hybrid systems.\\r\\nFor further information about the seminar\, please visit th is webpage [t3://page?uid=1115]. X-ALT-DESC:We present a conceptual framework that helps to bridge the kn
owledge gap \;between the two individual communities from machine lear
ning and \;numerical simulation to identify potential combined approac
hes and to \;promote the development of hybrid systems.
W
e give examples of different types of combinations using exemplary \;a
pproaches of simulation-assisted machine learning and machine-learning&nbs
p\;assisted simulation. We also discuss an advanced pairing where we see&n
bsp\;particular further potential for hybrid systems.
For further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20211029T120000 END:VEVENT BEGIN:VEVENT UID:news1157@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210627T202913 DTSTART;TZID=Europe/Zurich:20210604T110000 SUMMARY:Seminar in Numerical Analysis: Ivan Dokmanić (Universität Basel) DESCRIPTION:This talk will be an overview of my group's research between de ep learning and inverse problems. I will first describe the current (?) st ate of the field and then present a medley of our results\, including 1) a neural network architecture for wave-based inverse problems derived from Fourier integral operators\; 2) an approach to nonlinear traveltime tomogr aphy based on neural priors\; and 3) provably injective neural networks th at are universal approximators of probability measures supported on low-di mensional manifolds. My secret hope is to spark discussions that could evo lve to collaborations.\\r\\nFor further information about the seminar\, pl ease visit this webpage [t3://page?uid=1115]. X-ALT-DESC:This talk will be an overview of my group's research between deep learning and inverse problems. I will first describe the current (?) state of the field and then present a medley of our results\, including 1) a neural network architecture for wave-based inverse problems derived fro m Fourier integral operators\; 2) an approach to nonlinear traveltime tomo graphy based on neural priors\; and 3) provably injective neural networks that are universal approximators of probability measures supported on low- dimensional manifolds. My secret hope is to spark discussions that could e volve to collaborations.
\nFor further information about the seminar \, please visit this webpage.
DTEND;TZID=Europe/Zurich:20210604T120000 END:VEVENT BEGIN:VEVENT UID:news1158@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210627T202830 DTSTART;TZID=Europe/Zurich:20210507T110000 SUMMARY:Seminar in Numerical Analysis: Erik Burman (University College Lond on) DESCRIPTION:In many applications both in medical science and in the geoscie nces the accurate approximation of solutions to wave equations is an impor tant component for optimisation or inverse identification. Examples inclu de thermoacoustic imaging or high frequency ultrasound treatments in medic ine (HIFU) or fault slip analysis in seismology. These problems have in c ommon the need for computational solution of an inverse problem where the forward problem is set in a heterogeneous domain. Indeed typically the sou nd speed in the bulk domain jumps over material interfaces. Sometimes ther e 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 these topics\, motivated by two such applications: HIFU and the propagation of seismic waves. After a brief introduction of the applications we will first discuss the analysis of some approximation methods for inverse initial value problems subject to the wave equation. W e will then consider a hybrid high order method for the approximation of w ave propagation in heterogeneous media\, using cut element techniques to a void meshing of interfaces. Finally we will discuss some open problems tha t remain in order to understand the approximation of the inverse initial v alue problem in heterogeneous media using high order methods.\\r\\nFor fur ther information about the seminar\, please visit this webpage [t3://page? uid=1115]. X-ALT-DESC:In many applications both in medical science and in the geosc iences the accurate approximation of solutions to wave equations is an imp ortant \;component for optimisation or inverse identification. Example s include thermoacoustic imaging or high frequency ultrasound treatments i n medicine (HIFU) \;or fault slip analysis in seismology. These proble ms have in common the need for computational solution of an inverse proble m where the forward problem is set in a heterogeneous domain. Indeed typic ally the sound speed in the bulk domain jumps over material interfaces. So metimes there is even a need for coupling of the acoustic and elastodynami c equations in the presence of liquid inclusions. In this talk we will giv e a snapshot of our ongoing work in these topics\, motivated by two such a pplications: HIFU and the propagation of seismic waves. After a brief intr oduction of the applications we will first discuss the analysis of some ap proximation methods for inverse initial value problems subject to the wave equation. We will then consider a hybrid high order method for the approx imation of wave propagation in heterogeneous media\, using cut element tec hniques to avoid meshing of interfaces. Finally we will discuss some open problems that remain in order to understand the approximation of the inver se initial value problem in heterogeneous media using high order methods.< /p>\n
For further information about the seminar\, please visit this web page.
DTEND;TZID=Europe/Zurich:20210507T120000 END:VEVENT BEGIN:VEVENT UID:news1184@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210627T202737 DTSTART;TZID=Europe/Zurich:20210430T110000 SUMMARY:Seminar in Numerical Analysis: Pieter Barendrecht (KAUST) DESCRIPTION:In this talk\, we're going to take a closer look at the basics of both univariate and bivariate splines\, including Bézier- and B-spline curves\, box splines and subdivision surfaces. Next\, we'll shift our foc us to applications of smooth spline surfaces of arbitrary manifold topolog y within the realm of computer graphics. Finally\, a couple of aspects and applications of splines in the context of numerical methods will be discu ssed. Expect more illustrations than equations\, and in addition a couple of (interactive) live software demos!\\r\\nFor further information about t he seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:In this talk\, we're going to take a closer look at the basic s of both univariate and bivariate splines\, including Bézier- and B-spli ne curves\, box splines and subdivision surfaces. Next\, we'll shift our f ocus to applications of smooth spline surfaces of arbitrary manifold topol ogy within the realm of computer graphics. Finally\, a couple of aspects a nd applications of splines in the context of numerical methods will be dis cussed. Expect more illustrations than equations\, and in addition a coupl e of (interactive) live software demos!
\nFor further information ab out the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20210430T120000 END:VEVENT BEGIN:VEVENT UID:news1167@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210627T202614 DTSTART;TZID=Europe/Zurich:20210423T110000 SUMMARY:Seminar in Numerical Analysis: Markus Melenk (TU Wien) DESCRIPTION:We consider the Helmholtz equation with piecewise analytic coef ficients at large wavenumber k > 0. The interface where the coefficients j ump is assumed to be analytic. We develop a k-explicit regularity theory f or the solution that takes the form of a decomposition into two components : the first component is a piecewise analytic\, but highly oscillatory fun ction and the second one has finite regularity but features wavenumber-ind ependent bounds. This decomposition generalizes earlier decompositions of [MS10\, MS11\, EM11\, MSP12]\, which considered the Helmholtz equation wit h constant coefficients\, to the case of (piecewise) analytic coefficients . This regularity theory allows to show for high order Galerkin discretiza tions (hp-FEM) of the Helmholtz equation that quasi-optimality is reached if (a) the approximation order p is selected as p = O(log k) and (b) the m esh 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-optimality of hp-FEM for the Helmholtz equation to the case of the heterogeneous Helmhol tz equation.\\r\\nJoint work with: Maximilian Bernkopf (TU Wien)\, Théoph ile Chaumont-Frelet (Inria).\\r\\nReferences [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\, Sp ringer 2012 [MS10] J.M. Melenk and S. Sauter\, Convergence Analysis f or 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. Nu mer. Anal.\, 49:1210–1243\, 2011 [MSP12] J.M. Melenk\, S. Sauter\, A. Pa rsania\, Generalized 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 [t3://page?uid=1115]. X-ALT-DESC:We consider the Helmholtz equation with piecewise analytic co efficients at large wavenumber k >\; 0. The interface where the coeffici ents jump is assumed to be analytic. We develop a k-explicit regularity th eory for the solution that takes the form of a decomposition into two comp onents: the first component is a piecewise analytic\, but highly oscillato ry function and the second one has finite regularity but features wavenumb er-independent bounds. This decomposition generalizes earlier decompositio ns of [MS10\, MS11\, EM11\, MSP12]\, which considered the Helmholtz equati on with constant coefficients\, to the case of (piecewise) analytic coeffi cients. This regularity theory allows to show for high order Galerkin disc retizations (hp-FEM) of the Helmholtz equation that quasi-optimality is re ached if (a) the approximation 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 results of [MS10\, MS11\, EM11\, MSP12] about the onset of quasi-optimali ty of hp-FEM for the Helmholtz equation to the case of the heterogeneous H elmholtz equation.
\nJoint work with: Maximilian Bernkopf (TU Wien)\ , Théophile Chaumont-Frelet (Inria).
\nReferences<
br /> [EM11] \; \; \;S. Esterhazy and J.M. Melenk\, On stabil
ity of discretizations of the Helmholtz equation\, in: Numerical \;Ana
lysis 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
[MS
11] \; \;J.M. Melenk and S. Sauter\, Wavenumber explicit converg
ence 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 H
elmholtz problems\, J. Sci. Comp. 57:536–581\, 2013
For further information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20210423T120000 END:VEVENT BEGIN:VEVENT UID:news1154@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210627T202454 DTSTART;TZID=Europe/Zurich:20210416T110000 SUMMARY:Seminar in Numerical Analysis: Guy Gilboa (Technion - Israel Instit ute of Technology) 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\, based on the TV gradien t flow\, and its application in image processing. We then present new resu lts on analyzing gradient flows of homogeneous nonlinear operators (such a s the p-Laplacian). Our framework allows a thorough investigation of Dynam ic-Mode-Decomposition (DMD)\, a central dimensionality reduction method fo r time series data. We present analytic solutions of simple nonlinear case s\, reveal shortcomings of DMD and propose improved decomposition methods. \\r\\nFor further information about the seminar\, please visit this webpag e [t3://page?uid=1115]. X-ALT-DESC:Recent studies on nonlinear eigenvalue problems show surprisi ng analogies to harmonic analysis (e.g. Fourier or wavelets). \;In thi s talk we first show the total-variation (TV) transform\, based on the TV gradient flow\, and its application in image processing. We then present n ew results on analyzing gradient flows of homogeneous nonlinear operators (such as the p-Laplacian). Our framework allows a thorough investigation o f Dynamic-Mode-Decomposition (DMD)\, a central dimensionality reduction me thod for time series data. We present analytic solutions of simple nonline ar cases\, reveal shortcomings of DMD and propose improved decomposition m ethods.
\nFor further information about the seminar\, please visit t his webpage.
DTEND;TZID=Europe/Zurich:20210416T120000 END:VEVENT BEGIN:VEVENT UID:news1161@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210627T202314 DTSTART;TZID=Europe/Zurich:20210409T110000 SUMMARY:Seminar in Numerical Analysis: Barbara Kaltenbacher (Universität K lagenfurt) DESCRIPTION:High intensity (focused) ultrasound HIFU is used in numerous me dical and industrial applications ranging from litotripsy and thermotherap y via ultrasound cleaning and welding to sonochemistry. In this talk\, we will highlight two computational aspects related to the relevant nonlinea r acoustic phenomena\, namely\\r\\n absorbing boundary conditions for the treatment of open domain problems\; optimization tasks for ultrasound fo cusing. \\r\\nStrictly speaking\, acoustic sound propagation takes place i n full space or at least in a domain that is typically much larger than th e 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 absorbin g boundary conditions ABC that induce dissipation of outgoing waves. Here it will turn out to be crucial to take into account nonlinearity of the PD E also in these ABC. This is joint work with Igor Shevchenko (Imperial Co llege London). In the context of applications in HIFU\, focusing of nonli nearly propagating waves amounts to optimization problems. The design of u ltrasound excitation via piezoelectric transducers leads to a boundary con trol problem\; focusing high intensity ultrasound by a silicone lens requi res shape optimization. For both problem classes\, we will discuss the der ivation of gradient information in order to formulate optimality condition s and drive numerical optimization methods. This is joint work with Chris tian Clason (University of Duisburg-Essen)\, Vanja Nikolić (TU München) \, and Gunther Peichl (University of Graz). Finally we will provide an ou tlook on imaging with nonlinearly acoustic waves\, which amounts to identi fying 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 Uni versity).\\r\\nFor further information about the seminar\, please visit th is webpage [t3://page?uid=1115]. X-ALT-DESC:High intensity (focused) ultrasound HIFU is used in numerous medical and industrial applications ranging from litotripsy and thermother apy via ultrasound cleaning and welding to sonochemistry. \;In this ta lk\, we will highlight two computational aspects related to the relevant n onlinear acoustic phenomena\, namely
\nStrictly speaking\, acousti
c 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 atte
ntion to a bounded domain Ω\, e.g\, for computational purposes\, artifici
al 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 joi
nt work with Igor Shevchenko (Imperial College London).
In th
e context of applications in HIFU\, focusing of nonlinearly propagating wa
ves amounts to optimization problems. The design of ultrasound excitation
via piezoelectric transducers leads to a boundary control problem\; focusi
ng high intensity ultrasound by a silicone lens requires shape optimizatio
n. For both problem classes\, we will discuss the derivation of gradient i
nformation in order to formulate optimality conditions and drive numerical
optimization methods. \;This is joint work with Christian Clason (Uni
versity of Duisburg-Essen)\, Vanja Nikolić \;(TU München)\, and Gunt
her Peichl (University of Graz).
Finally we will provide an o
utlook on imaging with nonlinearly acoustic waves\, which amounts to ident
ifying \; spatially varying coefficients (sound speed and/or coefficie
nt of nonlinearity) in the Westervelt equation. \;This is recent joint
work with Masahiro Yamamoto (University of Tokyo) and William Rundell (Te
xas A&\;M University).
For further information about the semina r\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20210409T120000 END:VEVENT BEGIN:VEVENT UID:news1127@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210623T193706 DTSTART;TZID=Europe/Zurich:20201211T110000 SUMMARY:Seminar in Numerical Analysis: Heiko Gimperlein (Heriot-Watt Univer sity) DESCRIPTION:Diffusion processes beyond Brownian motion have recently attrac ted significant interest from different communities in mathematics\, the p hysical and biological sciences. They are described by partial differentia l equations involving nonlocal operators with singular non-integrable kern els\, such as fractional Laplacians. This talk discusses the challenges of their approximation by finite elements and discusses our recent results o n the a priori analysis of h\, p and hp-versions for the integral fraction al Laplacian\, as well as their preconditioning. \\r\\nFor further inform ation about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Diffusion processes beyond Brownian motion have recently attr acted significant interest from different communities in mathematics\, the physical and biological sciences. They are described by partial different ial equations involving nonlocal operators with singular non-integrable ke rnels\, such as fractional Laplacians. This talk discusses the challenges of their approximation by finite elements and discusses our recent results on the a priori analysis of h\, p and hp-versions for the integral fracti onal Laplacian\, as well as their preconditioning. \;
\nFor furt her information about the seminar\, please visit this webpage.
DTEND;TZID=Europe/Zurich:20201211T120000 END:VEVENT BEGIN:VEVENT UID:news1098@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210623T193612 DTSTART;TZID=Europe/Zurich:20201204T110000 SUMMARY:Seminar in Numerical Analysis: Bastian von Harrach (Goethe-Universi tät Frankfurt) DESCRIPTION:We derive a simple criterion that ensures uniqueness\, Lipschit z stability and global convergence of Newton's method for the finite dim ensional zero-finding problem of a continuously differentiable\, pointwis e convex and monotonic function. Our criterion merely requires to evaluat e the directional derivative of the forward function at finitely many eva luation points and for finitely many directions.\\r\\nWe then demonstrate that this result can be used to prove uniqueness\, stability and global c onvergence for an inverse coefficient problem with finitely many measurem ents. We consider the problem of determining an unknown inverse Robin tra nsmission coefficient in an elliptic PDE. Using a relation to monotonicit y 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 c an be uniquely and stably reconstructed by a globally convergent Newton i teration. We derive a constructive method to identify these boundary meas urements\, calculate the stability constant and give a numerical example. \\r\\n For further information about the seminar\, please visit this webpa ge [t3://page?uid=1115]. X-ALT-DESC:We derive a simple criterion that ensures uniqueness\, Lipsch itz \;stability and global convergence of Newton's method for the fini te \;dimensional zero-finding problem of a continuously differentiable \, \;pointwise convex and monotonic function. Our criterion merely req uires \;to evaluate the directional derivative of the forward function at \;finitely many evaluation points and for finitely many directions .
\nWe then demonstrate that this result can be used to prove unique ness\, \;stability and global convergence for an inverse coefficient p roblem with \;finitely many measurements. We consider the problem of d etermining an \;unknown inverse Robin transmission coefficient in an e lliptic PDE. Using \;a relation to monotonicity and localized potentia ls techniques\, we show \;that a piecewise-constant coefficient on an a-priori known partition \;with a-priori known bounds is uniquely dete rmined by finitely many \;boundary measurements and that it can be uni quely and stably \;reconstructed by a globally convergent Newton itera tion. We derive a \;constructive method to identify these boundary mea surements\, calculate \;the stability constant and give a numerical ex ample.
\n
For further information about the seminar\, please v
isit this webpage.
We examine how time step adaptivity can be used to control&nb sp\;potential \;instability arising from \;non-Lipschitz \;ter ms \;for \;stochastic \;partial differential \;equations ( SPDEs). I will give a brief introduction to SPDEs and illustrate the stabi lity issue with the standard uniform step Euler method \;to \;moti vate the adaptive method. I \;will present a \;strong convergence& nbsp\;result and outline the \;steps of the proof. To illustrate the&n bsp\;method \;we \;examine the stochastic Allen-Cahn\, Swift-Hohen berg\, \; Kuramoto-Sivashinsky equations and finally will discuss a po tential use of the adaptivity for the deterministic system. This is joint work with Stuart Campbell.
DTEND;TZID=Europe/Zurich:20201120T120000 END:VEVENT BEGIN:VEVENT UID:news1102@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210623T193410 DTSTART;TZID=Europe/Zurich:20201113T110000 SUMMARY:Seminar in Numerical Analysis: Gilles Vilmart (Université de Genè ve) DESCRIPTION:We show that the Strang splitting method applied to a diffusion -reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nic olson method\, while order reduction occurs in general if using other Ru nge-Kutta schemes or even the exact flow itself for the diffusion part. We also show that this method recovers stationary states in contrast with sp litting methods in general.We prove these results when the source term onl y depends on the space variable. Numerical experiments suggest that the s econd order of convergence persists with general nonlinearities.\\r\\nThi s is joint work with Guillaume Bertoli (Université de Genève) and Chris tophe Besse (Institut de Mathématiques de Toulouse).\\r\\nFor further inf ormation about the seminar\, please visit this webpage [t3://page?uid=111 5]. X-ALT-DESC:We show that the Strang splitting method applied to a diffusi on-reaction \;equation with inhomogeneous general oblique boundary con ditions is of \;order two when the diffusion equation is solved with t he Crank-Nicolson \;method\, while order reduction occurs in general i f using other \;Runge-Kutta schemes or even the exact flow itself for the diffusion part. We also show that this method recovers stationary stat es in contrast with splitting methods in general.We prove these results wh en the source term only depends on the space \;variable. Numerical exp eriments suggest that the second order of \;convergence persists with general nonlinearities.
\nThis is joint work with Guillaume Bertoli (Université de Genève) and \;Christophe Besse (Institut de Mathémat iques de Toulouse).
\nFor further information about the seminar\, pl ease visit this \;webpage.
DTEND;TZID=Europe/Zurich:20201113T120000 END:VEVENT BEGIN:VEVENT UID:news1101@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20210623T193336 DTSTART;TZID=Europe/Zurich:20201030T110000 SUMMARY:Seminar in Numerical Analysis: Alfio Borzi (Universität Würzburg) DESCRIPTION:The Liouville equation is the fundamental building block of mod els that govern the evolution of density functions of multi-particle syst ems. These models include different Fokker-Planck and Boltzmann equations that arise in many application fields ranging from gas dynamics to pedes trians' motion where the need arises to control these systems.\\r\\nThis talk provides an introduction to the formulation and solution of optimal control problems governed by the Liouville equation and related models. T he purpose of this framework is the design of robust controls to steer th e motion of particles\, pedestrians\, etc.\, where these agents are repre sented in terms of density functions. For this purpose\, expected-value c ost functionals are considered that include attracting potentials and dif ferent costs of the controls\, whereas the control mechanism in the gover ning models is part of the drift or is included in a collision term.\\r\\ nIn this talk\, theoretical and numerical results concerning ensemble opt imal control problems with Liouville\, Fokker-Planck and linear Boltzmann equations are presented.\\r\\n For further information about the seminar\ , please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:The Liouville equation is the fundamental building block of m odels that \;govern the evolution of density functions of multi-partic le systems. These \;models include different Fokker-Planck and Boltzma nn equations that arise \;in many application fields ranging from gas dynamics to pedestrians' motion \;where the need arises to control the se systems.
\nThis talk provides an introduction to the formulation and solution \;of optimal control problems governed by the Liouville e quation \;and related models. The purpose of this framework \;is t he design of robust controls to steer the motion of particles\, pedestrian s\, etc.\, \;where these agents are represented in terms of density fu nctions. \;For this purpose\, expected-value cost functionals are cons idered \;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.
\nIn th is talk\, theoretical and numerical results concerning ensemble \;opti mal control problems with Liouville\, Fokker-Planck and linear \;Boltz mann equations are presented.
\n
For further information about
the seminar\, please visit this webpage.
We propose an efficient algorithm for the treatment of Volter ra integral operators based on H2-matrix compression techniques. The algor ithm is built in an evolutionary manner\, and therefore\, is well suited f or the problems\, where the right hand side depends on the solution itself and is not known for all time steps a priori. The resulting algorithm is of linear complexity O(N) w.r.t. to the number of time steps\, and require s O(N) active memory. The memory consumption can be reduced to O(log N) fo r the kernels of convolution type using the Laplace inversion techniques i ntroduced by Lubich et al\; the connection to the FOCQ algorithm is drawn. We demonstrate the effectiveness of our algorithm on a series of numerica l examples.
\n
For further information about the seminar\, ple
ase visit this \;webpage.
Many models in spatial statistics are based on Gaussian Maté
rn fields. Motivated by the relation between this class of Gaussian 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 show
s different notions of convergence at explicit and sharp rates. We further
more 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 fo
r inference\, i.e.\, for parameter estimation given data\, and for spatial
prediction.
For further information about the seminar\, please vi sit this \;webpage.
DTEND;TZID=Europe/Zurich:20191220T120000 END:VEVENT BEGIN:VEVENT UID:news931@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191210T105146 DTSTART;TZID=Europe/Zurich:20191213T110000 SUMMARY:Seminar in Numerical Analysis: Théophile Chaumont-Frelet (INRIA\, Nice/Sophia Antipolis) DESCRIPTION:The Helmholtz equation models the propagation of a time-harmoni c wave. It has received much attention since it is widely employed in appl ications\, but still challenging to numerically simulate in the high-frequ ency 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 me thods (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 se tting to recall central concepts of the finite element theory such as quas i-optimality and interpolation error. The second part of the seminar is d evoted to the high-frequency case. We show that without restrictive assump tions 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.\\r\\nFor further information about the seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:The Helmholtz equation models the propagation of a time-harmo
nic wave.
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 acou
stic waves for the sake of simplicity
and consider finite element di
scretizations. The main goal of the
presentation is to highlight the
improved performance of high order
methods (as compared to linear f
inite elements) when the frequency is large.
We will very bri
efly cover the zero-frequency case\, that corresponds to the well-studied
Poisson equation. We take advantage of this classical setting to recall ce
ntral concepts of the finite element theory such as quasi-optimality and i
nterpolation error.
The second part of the seminar is devoted
to the high-frequency case.
We show that without restrictive assump
tions 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 les
s affected by this phenomena\,
and thus more suited to discretize hi
gh-frequency problems.
Before drawing our main conclusions\,
we briefly discuss advanced topics\,
such as the use of unfitted mes
hes in highly heterogeneous media
and mesh refinements around re-ent
rant corners.
For further information about the seminar\, please v isit this \;webpage.
DTEND;TZID=Europe/Zurich:20191213T120000 END:VEVENT BEGIN:VEVENT UID:news926@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191130T081127 DTSTART;TZID=Europe/Zurich:20191206T110000 SUMMARY:Seminar in Numerical Analysis: Ira Neitzel (Universität Bonn) DESCRIPTION:joint work with Dominik Hafemeyer\, Florian Mannel and Boris Ve xler\\r\\n We consider a convex optimal control problem governed by a par tial 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 const ant with often finitely many jump poins. We are interested in deriving fi nite element discretization error estimates for the controls when the sta te ist discretized with usual piecewise linear finite elements\, and the c ontrols is either variationally discrete or piecwise constant. Due to the structure of the objective function\, usual techniques for estimating th e control error cannot be applied. Instead\, these have to be derived fro m (suboptimal) error estimates for the state\, which can later be improved . \\r\\nFor further information about the seminar\, please visit this web page [t3://page?uid=current]. X-ALT-DESC:joint work with Dominik Hafemeyer\, Florian Mannel and Boris Vexler
\n
We consider a convex optimal control problem govern
ed by a partial differential equation in one space dimension which is con
trolled by a right-hand-side living in the space of functions with bounde
d variation. These functions tend to favor optimal controls that are piece
wise constant with often finitely many jump poins. We are interested in
deriving finite element discretization error estimates for the controls w
hen 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 es
timating 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 v isit this \;webpage.
DTEND;TZID=Europe/Zurich:20191206T120000 END:VEVENT BEGIN:VEVENT UID:news930@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191107T104152 DTSTART;TZID=Europe/Zurich:20191115T110000 SUMMARY:Seminar in Numerical Analysis: Michael Multerer (USI Lugano) DESCRIPTION:The numerical simulation of physical phenomena is very well und erstood given that the input data are given exactly. However\, in practic e\, the collection of these data is usually subjected to measurement erro rs. The goal of uncertainty quantification is to assess those errors and their possible impact on simulation results.In this talk\, we address diff erent numerical aspects of uncertainty quantification in elliptic partial differential equations on random domains. Starting from the modelling of random domains via random vector fields\, wediscuss how the corresponding Karhunen-Loève expansion can efficiently becomputed. For the discretisat ion of the partial differential equation\, we apply an adaptive Galerkin framework. An a posteriori error estimator is presented\, which allows fo r the problem-dependent iterative refinement of all discretisation paramet ers and the assessment of the achieved error reduction. The proposed appr oach is demonstrated in numerical benchmark problems.\\r\\nFor further in formation about the seminar\, please visit this webpage [t3://page?uid=11 15]. X-ALT-DESC:The numerical simulation of physical phenomena is very well u nderstood given that \;the input data are given exactly. However\, in practice\, the collection of these \;data is usually subjected to meas urement errors. The goal of uncertainty quantification \;is to assess those errors and their possible impact on simulation results.In this talk\ , we address different numerical aspects of uncertainty quantification&nbs p\;in elliptic partial differential equations on random domains. \;Sta rting from the modelling of random domains via random vector fields\, wedi scuss how the corresponding Karhunen-Loève expansion can efficiently beco mputed. For the discretisation of the partial differential equation\, \;we apply an adaptive Galerkin framework. An a posteriori error estimator is presented\, \;which allows for the problem-dependent iterative ref inement of all discretisation parameters \;and the assessment of the a chieved error reduction. The proposed approach is demonstrated \;in nu merical benchmark problems.
\nFor further information about the semi nar\, please visit this \;webpage.
DTEND;TZID=Europe/Zurich:20191115T120000 END:VEVENT BEGIN:VEVENT UID:news920@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191104T081039 DTSTART;TZID=Europe/Zurich:20191108T110000 SUMMARY:Seminar in Numerical Analysis: Omar Lakkis (University of Sussex) DESCRIPTION:Aposteriori error estimates provide a rigorous foundation for t he derivation of efficient adaptive algorithms for the approximation of so lutions of partial differential equations (PDEs). While the literature i s rich with results for the approximation of elliptic and parabolic PDEs\, it is much less developed for the hyperbolic equations such as the acoust ic or elastic wave equations. 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 further dev elopments to lower order Sobolev norms based on Baker’s Trick [3] for ba ckward Euler schemes. Subsequent focus will be given to practically rele vant methods such as Verlet\, Cosine\, or Newmark methods\, a popular exam ple of which is the Leap-frog method [4].\\r\\nNotes: This is based on joi nt work with E.H. Georgoulis\, C. Makridakis and J.M. Virtanen.\\r\\nRefer ences:\\r\\n[1] W. Bangerth and R. Rannacher\, J. Comput. Acoust. 9(2):575 –591\, 2001.[2] C. Bernardi and E. Süli\, Math. Models Methods Appl. Sc i. 15(2):199--225\, 2005.[3] E. H. Georgoulis\, O. Lakkis\, and C. Makrida kis. 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. Vir tanen. SIAM J. Numer. Anal.\, 54(1)\, 2016\, http://arxiv.org/abs/1411.757 2 \\r\\nFor further information about the seminar\, please visit this web page [t3://page?uid=1115]. X-ALT-DESC:Aposteriori error estimates provide a rigorous foundation for the derivation of efficient adaptive algorithms for the approximation of solutions of partial differential equations (PDEs). \;While the liter ature 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 equations. \;In this talk\, I will review s ome of the "\;standard"\; aposteriori results for the scalar linea r wave equation\, including those of [1] and [2]\, and present recent impr ovements and further developments to lower order Sobolev norms based on Ba ker’s Trick [3] for backward Euler schemes. \;Subsequent focus will be given to practically relevant methods such as Verlet\, Cosine\, or New mark methods\, a popular example of which is the Leap-frog method [4].
\n
Notes: This is based on joint work with E.H. Georgoulis\, C. Ma
kridakis and J.M. Virtanen.
References:
\n[1] W. Bange
rth 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 \;< a href="t3://page?uid=1115" title="Opens internal link in current window" class="internal-link">webpage
. DTEND;TZID=Europe/Zurich:20191108T120000 END:VEVENT BEGIN:VEVENT UID:news932@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20191023T141150 DTSTART;TZID=Europe/Zurich:20191101T110000 SUMMARY:Seminar in Numerical Analysis: Giacomo De Souza (EPFL) DESCRIPTION:Traditional explicit Runge--Kutta schemes\, though computationa lly inexpensive\, are inefficient for the integration of stiff ordinary di fferential equations due to stability issues. Conversely\, implicit scheme s are stable but can be overly expensive due to the solution of possibly l arge nonlinear systems with Newton-like methods\, whose convergence is nei ther guaranteed for large time steps. Explicit stabilized schemes such as the Runge--Kutta--Chebyshev method (RKC) represent a viable compromise\, as the width of their stability domain grows quadratically with respect t o the number of function evaluations\, thus presenting enhanced stability properties with a reasonable computational cost. These methods are particu larly efficient for systems arising from the space discretization of parab olic partial differential equations (PDEs).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 prop ortional to the smallest element size in order to achieve stability\, thus largely wasting computational resources on locally-refined meshes. We fir st tackle this issue by replacing the right hand side of the PDE with an a veraged 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 R KC schemes\, for which the number of expensive function evaluations is ind ependent of the small elements' size.The stability properties of our metho d are demonstrated on a model problem and numerical experiments confirm th at the stability bottleneck caused by a few of fine mesh elements can be o vercome without sacrificing accuracy.\\r\\nFor further information about t he seminar\, please visit this webpage [t3://page?uid=1115]. X-ALT-DESC:Traditional explicit Runge--Kutta schemes\, though computatio
nally inexpensive\, are inefficient for the integration of stiff ordinary
differential equations due to stability issues. Conversely\, implicit sche
mes are stable but can be overly expensive due to the solution of possibly
large nonlinear systems with Newton-like methods\, whose convergence is n
either guaranteed for large time steps. Explicit stabilized schemes such a
s the Runge--Kutta--Chebyshev method (RKC) \; represent a viable compr
omise\, as the width of their stability domain grows quadratically with re
spect to the number of function evaluations\, thus presenting enhanced sta
bility properties with a reasonable computational cost. These methods are
particularly efficient for systems arising from the space discretization o
f parabolic partial differential equations (PDEs).
The efficiency of
these methods deteriorates as the system becomes stiffer\, even if stiffne
ss is induced by only few degrees of freedom. In the framework of discreti
zed parabolic PDEs\, the number of function evaluations has to be chosen i
nversely proportional to the smallest element size in order to achieve sta
bility\, thus largely wasting computational resources on locally-refined m
eshes. 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 d
own using the dissipative effect of the equation itself and which is cheap
to evaluate. Combining RKC methods with the averaged force we give rise t
o multirate RKC schemes\, for which the number of expensive function evalu
ations is independent of the small elements' size.
The stability prop
erties of our method are demonstrated on a model problem and numerical exp
eriments confirm that the stability bottleneck caused by a few of fine mes
h elements can be overcome without sacrificing accuracy.
For furth er information about the seminar\, please visit this \;webpage.
DTEND;TZID=Europe/Zurich:20191101T120000 END:VEVENT BEGIN:VEVENT UID:news887@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20190514T111137 DTSTART;TZID=Europe/Zurich:20190525T110000 SUMMARY:Seminar in Numerical Analysis: Florian Faucher (Université de Pau) DESCRIPTION:We study the inverse problem associated with the propagation of time-harmonic waves. In the seismic context\, the available measurements correspond with partial reflection data\, obtained from one side illuminat ion (only from the Earth surface). The inverse problem aims at recovering the subsurface Earth medium parameters and we employ the Full Waveform Inv ersion (FWI) method\, which relies on an iterative minimization algorithm of the difference between the measurement and simulation. \\r\\nWe investi gate the deployment of new devices developed in the acoustic setting: the dual-sensors\, which are able to capture both the pressure field and the v ertical 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 t he reciprocity. We first note that the stability of the problem can be sho wn to be Lipschitz\, assuming piecewise linear parameters. In addition\, r eciprocity waveform inversion allows a separation between the observationa l and numerical acquisitions. In fact\, the numerical sources do not have to coincide with the observational ones\, offering new possibilities to cr eate adapted computational acquisitions\, consequently reducing the numeri cal cost. We illustrate our approach with three-dimensional medium reconst ructions\, where we start with minimal information on the target models. W e also extend the methodology for elasticity. \\r\\nEventually\, if time a llows\, we shall explore the model representation in numerical seismic inv ersion\, where the adaptive eigenspace method appears as a promising appro ach to have a compromise between number of unknowns and resolution. \\r\\nReferences \\r\\n[1] G. Alessandrini\, M. V. de Hoop\, F. Faucher\, R. Gaburro and E. Sincich\, Inverse problem for the Helmholtz e quation with Cauchy data: reconstruction with conditional well-posedness d riven iterative regularization\, ESAIM: M2AN (2019). \\r\\n[2] E. Berett a\, M. V. De Hoop\, F. Faucher\, and O. Scherzer\, Inverse boundary value problem for the Helmholtz equation: quantitative conditional Lipschitz sta bility estimates. SIAM Journal on Mathematical Analysis\, 48(6)\, pp.3962- 3983 (2016).\\r\\n[3] M. J. Grote\, M. Kray\, and U. Nahum\, Adaptive ei genspace method for inverse scattering problems in the frequency domain. I nverse Problems\, 33(2)\, 025006 (2017). \\r\\n[4] H. Barucq\, F. Fauche r\, 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 webpage. X-ALT-DESC:We study the inverse problem associated with the propagation of time-harmonic waves. In the seismic context\, the available measurements c orrespond with partial reflection data\, obtained from one side illuminati on (only from the Earth surface). The inverse problem aims at recovering t he subsurface Earth medium parameters and we employ the Full Waveform Inve rsion (FWI) method\, which relies on an iterative minimization algorithm o f the difference between the measurement and simulation. \nWe 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 vertic al velocity of the waves. For solving the inverse problem\, we define a ne w cost function\, adapted to these two types of data and based upon the re ciprocity. We first note that the stability of the problem can be shown to be Lipschitz\, assuming piecewise linear parameters. In addition\, recipr ocity waveform inversion allows a separation between the observational and numerical acquisitions. In fact\, the numerical sources do not have to co incide with the observational ones\, offering new possibilities to create adapted computational acquisitions\, consequently reducing the numerical c ost. We illustrate our approach with three-dimensional medium reconstructi ons\, where we start with minimal information on the target models. We als o extend the methodology for elasticity. \nEventually\, if time allows\, w e shall explore the model representation in numerical seismic inversion\, where the adaptive eigenspace method appears as a promising approach to ha ve a compromise between number of unknowns and resolution.Piezoelectric problems
Thermal problems with thermal resistances
Elasticity pro blems
Problems of fluid mechanics like incompressible Navie r-Stokes
Problem of melting and/or solidification of the ic e. (Boussinesq with specific heat)