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Autumn Semester 2020

Date / Time Speaker Title Location
23 September 2020
16:15-17:15 		Prof. Dr. Eduard Feireisl
Czech Academy of Sciences, Prague
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Zurich Colloquium in Applied and Computational Mathematics

Title (S)-convergence and approximation of oscillatory solutions in fluid dynamics
Speaker, Affiliation Prof. Dr. Eduard Feireisl, Czech Academy of Sciences, Prague
Date, Time 23 September 2020, 16:15-17:15
Location Zoom Meeting
Abstract We propose a new concept of (S)-convergence applicable to numerical methods as well as other consistent approximations of the Euler system in gas dynamics. (S)-convergence, based on averaging in the spirit of Strong Law of Large Numbers, reflects the asymptotic properties of a given approximate sequence better than the standard description via Young measures. Similarity with the tools of ergodic theory is discussed.
(S)-convergence and approximation of oscillatory solutions in fluid dynamicsread_more
Zoom Meeting
30 September 2020
16:15-17:15 		Prof. Dr. Richard Nickl
University of Cambridge
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Zurich Colloquium in Applied and Computational Mathematics

Title Bayesian inverse problems, Gaussian processes, and partial differential equations.
Speaker, Affiliation Prof. Dr. Richard Nickl, University of Cambridge
Date, Time 30 September 2020, 16:15-17:15
Location Zoom Meeting
Abstract The Bayesian approach to inverse problems has become very popular in the last decade after seminal work by Andrew Stuart (2010) and collaborators. Particularly in non-linear applications with PDEs and when using Gaussian process priors, this can leverage powerful MCMC methodology to tackle difficult high-dimensional and non-convex inference problems. Little is known in terms of rigorous performance guarantees for such algorithms. After laying out the main ideas behind Bayesian inversion, we will discuss recent progress providing both statistical and computational guarantees for these methods. We will touch upon issues such as how to prove posterior consistency and how to objectively validate posterior uncertainty quantification methods. A main focus will be on very recent results about mixing times of high-dimensional Langevin dynamics that establish the polynomial time computability of posterior measures in some non-linear model examples arising with PDEs.
Bayesian inverse problems, Gaussian processes, and partial differential equations.read_more
Zoom Meeting
14 October 2020
16:15-17:15 		Dr. Kaibo Hu
University of Minnesota
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Zurich Colloquium in Applied and Computational Mathematics

Title Complexes from complexes
Speaker, Affiliation Dr. Kaibo Hu, University of Minnesota
Date, Time 14 October 2020, 16:15-17:15
Location Zoom Meeting
Abstract There is a close connection between the Maxwell equations and the de Rham complex. The perspective of continuous and discrete differential forms has inspired key progress in the mathematical and numerical analysis for electromagnetism. This complex point of view also plays an important role in, e.g., continuum theory of defects, intrinsic theories of elasticity and relativity. In this talk, we derive new differential complexes from the de Rham complexes. The algebraic construction is inspired by the Bernstein-Gelfand-Gelfand (BGG) machinery. The cohomological structures imply various analytic properties. As an example, we construct Sobolev and finite element elasticity complexes (Kröner complex in mechanics and the linearized Calabi complex in geometry) and generalize various results in classical elasticity, e.g., the Korn inequality and the Cesàro-Volterra path integral.
Complexes from complexesread_more
Zoom Meeting
21 October 2020
16:15-17:15 		Prof. Dr. Jakob Zech
Uni Heidelberg, Germany
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Zurich Colloquium in Applied and Computational Mathematics

Title Sparse approximation of triangular transports on bounded domains
Speaker, Affiliation Prof. Dr. Jakob Zech, Uni Heidelberg, Germany
Date, Time 21 October 2020, 16:15-17:15
Location Zoom Meeting
Abstract Transport maps coupling two different measures can be used to sample from arbitrarily complex distributions. One of the main applications of this approach concerns Bayesian inference, where sampling from a posterior distribution facilitates making predictions based on partial and noisy measurments. In this talk we investigate the approximation of triangular transports $T:[-1,1]^d\to [-1,1]^d$ on the $d$-dimensional unit cube by polynomial expansions and ReLU networks. Specifically, given a reference and a target probability measure with positive and analytic Lebesgue densities on $[-1,1]^d$, we show that the unique Knothe-Rosenblatt transport, which pushes forward the reference to the target, can be approximated at an exponential rate in case $d<\infty$. These results are generalized to $d=\infty$, within a setting which incorporates many posterior densities occurring in PDE-driven Bayesian inverse problems. In the infinite dimensional case ($d=\infty$) we verify an algebraic convergence rate, which shows that the curse of dimensionality can be overcome.
Sparse approximation of triangular transports on bounded domainsread_more
Zoom Meeting
11 November 2020
16:15-17:15 		Dr. Cecilia Pagliantini
TU Eindhoven
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Zurich Colloquium in Applied and Computational Mathematics

Title Structure-preserving dynamical reduced basis methods for parametrized Hamiltonian systems
Speaker, Affiliation Dr. Cecilia Pagliantini, TU Eindhoven
Date, Time 11 November 2020, 16:15-17:15
Location Zoom Meeting
Abstract In real-time and many-query simulations of parametrized differential equations, computational methods need to face high computational costs to provide sufficiently accurate numerical solutions. To address this issue, model order reduction techniques aim at constructing low-complexity high-fidelity surrogate models that allow rapid and accurate solutions under parameter variation. In this talk, we will consider reduced basis methods (RBM) for the model order reduction of parametrized Hamiltonian dynamical systems describing nondissipative phenomena. The development of RBM for Hamiltonian systems is challenged by two main factors: (i) failing to preserve the geometric structure encoding the physical properties of the dynamics might lead to instabilities and unphysical behaviors; (ii) the local low-rank nature of nondissipative dynamics demands large reduced spaces to achieve sufficiently accurate approximations. We will discuss how to address these aspects via nonlinear reduced basis methods based on the characterization of the reduced dynamics on a phase space that evolves in time and is endowed with the geometric structure of the full model.
Structure-preserving dynamical reduced basis methods for parametrized Hamiltonian systemsread_more
Zoom Meeting
18 November 2020
16:15-17:15 		Prof. Dr. Dirk Pauly
Universität Duisburg-Essen
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Zurich Colloquium in Applied and Computational Mathematics

Title FA-TOOLBOX: SOLVING PDES WITH HILBERT COMPLEXES
Speaker, Affiliation Prof. Dr. Dirk Pauly, Universität Duisburg-Essen
Date, Time 18 November 2020, 16:15-17:15
Location Zoom Meeting
Abstract The aim of this talk is to present parts of the so-called Functional Analysis Toolbox (FA- ToolBox), a unified and general approach to solve PDEs. Hilbert Complexes are of particular interest. We shall motivate our concept by discussing the well known and prototypical div-curl-system curlE = F; divE = g; arising, e.g., in electro-magneto statics. Employing techniques from linear functional analysis (FA-ToolBox) we develop a comprehensive (and surprisingly simple) solution theory for static problems of the above type. We will introduce the notion of Hilbert complexes H0 A0! H1 A1! H2; of densely defined and closed linear operators A0 : D(A0) H0 ! H1; A1 : D(A1) H1 ! H2; satisfying the so-called complex property R(A0) N(A1): The latter electro static system is then generalised to A1x = f; A0x = g: The aim is to provide criteria on the complex such that existence and uniqueness of x can be guaranteed. It will turn out that the crucial property is the compactness of the embedding D(A1) \ D(A0) ,! H1;i.e., in classical terms the compactness ofD(curl) \ D(div) ,! L2; the co-called Picard-Weber-Weck selection theorem. Our general theory is not only applicable to the classical de Rham complex involving grad, curl, and div, but also to other important Hilbert complexes, such as the elasticity complex or the biharmonic complex. Moreover, important results can be proved in this general setting, such as general div-curl-type lemmas and informations about generalised Poincare/Friedrichs estimates, e.g., for the Maxwell constants. This talk contains parts of joined work with colleagues from Essen, Linz, and Prag, in particular, with Walter Zulehner (JKU Linz). Some parts are strongly related to the work of Doug Arnold (Minnesota) and Ragnar Winther (Oslo) and their co-authors. Results of this talk can be found in, e.g., [2, 1, 3, 4, 5, 7, 8, 6]. References [1] S. Bauer, D. Pauly, and M. Schomburg, The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions, SIAM Journal on Mathematical Analysis, 48(4), 2912-2943, 2016 [2] D. Pauly, On Maxwell's and Poincare's Constants, Discrete and Continuous Dynamical Systems - Series S,8(3), 607-618, 2015 [3] D. Pauly, A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains and a Cor-responding Generalized A1-A0-Lemma in Hilbert Spaces, Analysis (Munich), 39(2), 33-58, 2019 [4] D. Pauly, On the Maxwell and Friedrichs/Poincare Constants in ND, Mathematische Zeitschrift, 293(3), 957-987, 2019 [5] D. Pauly and J. Valdman, Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div:Theory and Numerical Experiments, Computers and Mathematics with Applications, 79, 3027-3067, 2020 [6] D. Pauly and M. Waurick, The Index of some Mixed Order Dirac-Type Operators and Generalised Dirichlet-Neumann Tensor Fields, arXiv, 2020, https://arxiv.org/abs/2005.07996 [7] D. Pauly and W. Zulehner, The divDiv-Complex and Applications to Biharmonic Equations, Applicable Analysis, 99(9), 1579-1630, 2020 [8] D. Pauly and W. Zulehner, The Elasticity Complex: Compact Embeddings and Regular Decompositions, arXiv, 2020, https://arxiv.org/abs/2001.11007
FA-TOOLBOX: SOLVING PDES WITH HILBERT COMPLEXESread_more
Zoom Meeting
9 December 2020
16:15-17:15 		Prof. Dr. Helmut Harbrecht
University of Basel
Details

Zurich Colloquium in Applied and Computational Mathematics

Title A Fast Isogeometric Boundary Element Method
Speaker, Affiliation Prof. Dr. Helmut Harbrecht, University of Basel
Date, Time 9 December 2020, 16:15-17:15
Location Zoom Meeting
Abstract This talk is concerned with an interpolation-based fast multipole method which is tailored to the context of isogeometric analysis. Hence, the surface is described in terms of a piecewise smooth parameterization by four-sided patches. This surface representation is in contrast to the common approximation of surfaces by flat panels. Nonetheless, parametric surface representations are easily accessible from Computer Aided Design (CAD). Our fast multipole method is based on Galerkin's method with higher-order ansatz functions such as B-splines. Due to an element-based integration scheme, an element-wise clustering is possible. This results in a balanced cluster tree, leading to a superior performance. By performing the interpolation for the fast multipole method directly on the two-dimensional reference domain, we reduce the cost complexity in the polynomial degree by one order. This gain is independent of the application of either $\mathcal{H}$- or $\mathcal{H}^2$-matrices. Numerical examples are provided in order to quantify and qualify the proposed method.
A Fast Isogeometric Boundary Element Methodread_more
Zoom Meeting

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