Zurich Colloquium in Applied and Computational Mathematics

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Archive 2020

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Speaker

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26 February 2020
16:15-17:45

Prof. Dr. Maria Lukacova
Universität Mainz

Event Details

Speaker invited by

Remi Abgrall

Abstract

An iconic example of hyperbolic conservation laws are the Euler equations of gas dynamics expressing the conservation of mass, momentum and energy. Recently, it has been shown that even for smooth initial data the Euler equations may have infinitely many physically admissible, i.e. weak entropy, solutions. Related to the ill-posedness of the Euler equations is the observation that approximate solutions obtained by standard finite volume methods may develop oscillations and cascades of new small scale substructures. Consequently, a fundamental question is: What is the limit of numerical solutions as mesh parameter is refined? In the present talk we will present a concept of K-convergence that can be seen as a new tool in numerical analysis of the ill-posed problems, such as the Euler equations. We will show that the numerical solutions obtained by some standard finite volume methods generate a dissipative measure-valued solution, which is an appropriate probability measure (Young measure). We will also show how to effectively compute its observable quantities, such as the mean and variance and proof their strong convergence in space and time.Theoretical results will be illustrated by a series of numerical simulations. The present research has been done in collaboration with E. Feireisl (Prague/Berlin), H. Mizerova (Bratislava), B. She (Prague) and Y. Wang (Beijing). It has been partially supported by TRR 146 Multiscale simulation methods for soft matter systems and by TRR 165 Waves to weather funded by DFG.

How to compute oscillatory solutions of the Euler equations

Y27 H 46

4 March 2020
16:15-17:15

Prof. Dr. Shi Jin
Shanghai JiaoTong University, Shanghai, China

Event Details

Speaker invited by	Siddhartha Mishra

CANCELLED!

Y27 H 46

1 April 2020
16:15-17:15

Prof. Dr. Didier Lucor
LIMSI-CNRS, Paris-Saclay University

Event Details

Speaker invited by	Siddhartha Mishra

CANCELLED!

Y27 H 46

13 May 2020
16:15-17:15

Prof. Dr. Victor Batista
Yale University

Event Details

Speaker invited by

Vasile Gradinaru

Abstract

We introduce the “tensor-train split-operator Fourier transform” (TT-SOFT) algorithm for simulations of multidimensional nonadiabatic quantum dynamics [J. Chem. Theory Comput. 13: 4034-4042 (2017)]. TT-SOFT is essentially the grid-based SOFT method implemented in dynamically adaptive tensor-train representations. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the grid-based wavepacket tensor components, bypassing the need of actually computing the wave function in its full-rank tensor product grid space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the S1/S2 interconversion dynamics of pyrazine after UV photoexcitation to the S2 state. Our results show that the TT-SOFT method is a powerful computational approach for simulations of quantum dynamics of polyatomic systems since it avoids the exponential scaling problem of full-rank grid-based representations. The development of ultrafast laser technology has also opened the possibility to control ultrafast reaction dynamics in excited electronic states. Thus, we report a Floquet theoretical study of quantum control of the ultrafast cis-trans photoisomerization dynamics of rhodopsin [J. Chem. Theory Comput. 14(3): 1198-1205 (2018)]. The predicted light-induced potentials, including light-induced conical intersections, can open new reaction channels or modify the product yields of existing pathways. The nonadiabatic dynamics is described by a 3-state 2-dimensional wave-packet, coupled to a bath of 23 vibrational modes, evolving according to an empirical model Hamiltonian with frequencies and excited-state gradients parameterized to reproduce the observed resonance Raman excitations of rhodopsin. We analyze the effect of different control pulses on the photoisomerization dynamics, including changes in pulse duration and intensity. We interpret the results in terms of 'dressed states' and we exploit the Floquet description where the effect of control pulses is naturally decoupled along the different channels. Results obtained with 300 fs-long pulses suggest that it should be possible to delay the excited-state isomerization for hundreds of femtoseconds. Our findings are thus particularly relevant to the development of ultrafast optical switches based on visual pigments.

Tensor train decompositions for quantum dynamics simulations

Zoom Meeting

20 May 2020
16:15-17:15

Prof. Dr. George Karniadakis
Brown University, Rhode Island

Event Details

Speaker invited by

Christoph Schwab

Abstract

We will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of physical and biological systems and for discovering hidden physics from noisy data. We will introduce a deep learning approach based on neural networks (NNs) and generative adversarial networks (GANs). We also introduce new NNs that learn functionals and nonlinear operators from functions and corresponding responses for system identification. Unlike other approaches that rely on big data, here we learn from small data by exploiting the information provided by the physical conservation laws, which are used to obtain informative priors or regularize the neural networks. We will also make connections between Gauss Process Regression and NNs and discuss the new powerful concept of meta-learning. We will demonstrate the power of PINNs for several inverse problems in fluid mechanics, solid mechanics and biomedicine including wake flows, shock tube problems, material characterization, brain aneurysms, etc, where traditional methods fail due to lack of boundary and initial conditions or material properties.

Physics-Informed Neural Networks (PINNs): Algorithms & Applications

Zoom Meeting

27 May 2020
16:15-17:15

Prof. Dr. Martin Oberlack
TU Darmstadt, Germany

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

Using the symmetry-based turbulence theory we derive turbulent scaling laws for arbitrarily high moments of the stream-wise velocity $U_1$. In the region of the log-law, the theory predicts an algebraic law with the exponent $t_2 (n-1)$ for moments $n > 1$. The exponent $s_2$ of the $2^{nd}$ moment determines the exponent of all higher moments. Moments here always refer to the instantaneous quantities and not to the fluctuations. For the core region of a Poiseuille flow we find a deficit law for arbitrary moments $n$ of algebraic type with a scaling exponent $n(s_2-s_1)+2s_1-s_2$. Hence, the moments of order one and two with its scaling exponents $s_1$ and $s_2$ determine all higher exponents. To validate the new theoretical results we have conducted a Poiseuille flow DNS at $Re_\tau=10^4$. All of the latter theoretical findings could be verified with high accuracy using DNS data.

High-moment turbulent scaling laws and its roots in statistical symmetries

Zoom Meeting

23 September 2020
16:15-17:15

Prof. Dr. Eduard Feireisl
Czech Academy of Sciences, Prague

Event Details

Speaker invited by

Siddhartha Mishra

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 dynamics

Zoom Meeting

30 September 2020
16:15-17:15

Prof. Dr. Richard Nickl
University of Cambridge

Event Details

Speaker invited by

Christoph Schwab

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.

Zoom Meeting

14 October 2020
16:15-17:15

Dr. Kaibo Hu
University of Minnesota

Event Details

Speaker invited by

Ralf Hiptmair

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 complexes

Zoom Meeting

21 October 2020
16:15-17:15

Prof. Dr. Jakob Zech
Uni Heidelberg, Germany

Event Details

Speaker invited by

Christoph Schwab

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 domains

Zoom Meeting

11 November 2020
16:15-17:15

Dr. Cecilia Pagliantini
TU Eindhoven

Event Details

Speaker invited by

Ralf Hiptmair

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 systems

Zoom Meeting

18 November 2020
16:15-17:15

Prof. Dr. Dirk Pauly
Universität Duisburg-Essen

Event Details

Speaker invited by

Ralf Hiptmair

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 COMPLEXES

Zoom Meeting

9 December 2020
16:15-17:15

Prof. Dr. Helmut Harbrecht
University of Basel

Event Details

Speaker invited by

Christoph Schwab

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 Method

Zoom Meeting

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