Zurich Colloquium in Applied and Computational Mathematics

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Archive 2023

Date / Time

Speaker

Title

Location

6 March 2023
13:15-14:15

Prof. Dr. Nana Liu
Shanghai Jiaotong University, China

Event Details

Speaker invited by

Prof. Dr. Siddhartha Mishra

Abstract

What kinds of scientific computing problems are suited to be solved on a quantum device with quantum advantage? It turns out that by transforming a partial differential equation (PDE) into a higher-dimensional space, certain important issues can be resolved while at the same time not incurring a curse of dimensionality, when performed with a quantum algorithm. In this talk, I’ll explore ways in which quantum algorithms can be used to efficiently solve not just linear PDEs but also certain classes of nonlinear PDEs, like nonlinear Hamilton-Jacobi equations and scalar hyperbolic equations, based on the level-set formalism. Using another transformation, PDEs with uncertainty can be tackled. I’ll also introduce a simple new way–called Schrodingerisation– to simulate general linear partial differential equations via quantum simulation. Using a simple new transform and introducing one extra dimension, any linear partial differential equation can be recast into a system of Schrodinger’s equations – in real time — in a straightforward way. This approach is not only applicable to PDEs for classical problems but also those for quantum problems – like the preparation of quantum ground states, Gibbs states and the simulation of quantum states in random media in the semiclassical limit. In this talk, I’ll explore ways in which quantum algorithms can be used to efficiently solve not just linear PDEs but also certain classes of nonlinear PDEs, like nonlinear Hamilton-Jacobi equations and scalar hyperbolic equations, based on the level-set formalism. Using another transformation, PDEs with uncertainty can be tackled. I’ll also introduce a simple new way–called Schrodingerisation– to simulate general linear partial differential equations via quantum simulation. Using a simple new transform and introducing one extra dimension, any linear partial differential equation can be recast into a system of Schrodinger’s equations – in real time — in a straightforward way. This approach is not only applicable to PDEs for classical problems but also those for quantum problems – like the preparation of quantum ground states, Gibbs states and the simulation of quantum states in random media in the semiclassical limit.

Efficient quantum computation for partial differential equations

HG G 19.2

8 March 2023
16:30-17:30

Prof. Dr. Shi Jin
Shanghai Jiaotong University, China

Event Details

Speaker invited by

Prof. Dr. Siddhartha Mishra

Abstract

We first develop random batch methods for classical interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For one of the methods, we give a particle number independent error estimate under some special interactions. This method is also extended to molecular dynamics with Coulomb interactions, in the framework of Ewald summation. We will show its superior performance compared to the current state-of-the-art methods (for example PPPM) for the corresponding problems, in the computational efficiency and parallelizability.

Random Batch Methods for interacting particle systems and molecular dynamics

Y27 H 35/36

15 March 2023
16:30-17:30

Dr. Andrea Manzoni
Politecnico di Milano, Italy

Event Details

Speaker invited by

Prof. Dr. Christoph Schwab

Abstract

Reduced order modeling (ROM) techniques, such as the reduced basis method, provide nowadays an essential toolbox for the efficient approximation of parametrized differential problems, whenever they must be solved either in real-time, or in several different scenarios. These tasks arise in several contexts like, e.g., uncertainty quantification, control and monitoring, as well as data assimilation, ultimately representing key aspects in view of designing predictive digital twins in engineering or medicine. On the other hand, in the last decade deep learning algorithms have witnessed a dramatic blossoming in several fields, ranging from image and signal processing to predictive data-driven models. More recently, deep neural networks have also been exploited for the numerical approximation of differential problems yielding powerful physics-informed surrogate models. In this talk we will explore different contexts in which deep neural networks (DNNs) can enhance the efficiency of ROM techniques, ultimately allowing the real-time simulation of large-scale nonlinear time-dependent problems. We show how to exploit DNNs to build ROMs for parametrized PDEs in a fully non-intrusive way, exploiting deep autoencoders as main building block, ultimately yielding deep learning-based ROMs (DL-ROMs) and their further extension to POD-enhanced DL-ROMs (POD-DL-ROMs). In particular, we will provide some guidelines for the design of deep autoencoders, showing the interplay between their minimal latent dimension and some topological properties of the solution manifold, and illustrating some theoretical results on the approximation errors entailed by the proposed approach, as well as more recent investigations on the use of deep convolutional autoencoders. Other examples of ROM strategies enhanced by deep learning include the use of DNNs for (i) learning nonlinear ROM operators, thus yielding hyper-reduced order models enhanced by deep neural networks (Deep-HyROMnets), or (ii) enhancing the accuracy of low-fidelity ROMs through a multi-fidelity neural network regression technique for the sake of input/output evaluations. Through a set of applications from engineering including, e.g., structural mechanics and fluid dynamics problems, we will highlight the opportunities provided by deep learning in the context of ROMs for parametrized PDEs, as well as those challenges that are still open.

Deep learning for reduced order modeling: recent results and open challenges

Y27 H 35/36

22 March 2023
16:00-17:00

Prof. Dr. Anne-Laure Dalibard
Sorbonne University, Paris

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

This talk is devoted to the study of the equation $u u_x - u_{yy}=f$ in the domain $(x_0,x_1)\times (-1,1)$, in the vicinity of the shear flow profile $u(x,y)=y$. This equation serves as a toy model for more complicated fluid equations such as the Prandtl system. The difficulty lies in the fact that we are interested in changing sign solutions. Hence the equation is forward parabolic in the region where $u>0$, and backward parabolic in the region $u<0$. The line $u=0$ is a free boundary and an unknown of the problem. Unexpectedly, we prove that even when the data (i.e. the source term $f$ or the boundary data) are smooth, existence of strong solutions of the equation fails in general. This phenomenon is already present at the linear level, and linked to the existence of singular profiles for the homogeneous linearized equation. In fact, we prove that strong solutions exist (both for the linearized and for the nonlinear system) if and only if the data satisfy a finite number of orthogonality conditions, whose purpose is to avoid the presence of singular profiles in the solution. A key difficulty of our work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. This is a joint work with Frédéric Marbach and Jean Rax.

Nonlinear forward-backward problems

Y27 H 35/36

29 March 2023
16:30-17:30

Prof. Dr. Christian Lubich
Universität Tübingen

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

The scattering of electromagnetic waves from obstacles with wave- material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken here is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. The fields are then evaluated in the exterior domain by a known representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge-Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. The well-posedness analysis of the boundary integral equation as well as the error analysis of the numerical methods relies on frequency-explicit bounds in the Laplace domain. These are then transferred to the time domain and combined with known approximation estimates of the numerical methods. The talk is based on joint work with Balázs Kovács and Jörg Nick.

Time-dependent scattering from thin layers

Y27 H 35/36

10 May 2023
16:30-17:30

Prof. Dr. Silvia Falletta
Politecnico di Torino

Event Details

Speaker invited by

Prof. Dr. Stefan Sauter

Abstract

Soft tissues and other nearly incompressible media pose a challenge for simulating elastic wave propagation, due to the slow propagation of shear waves compared to pressure waves. To overcome this challenge, a classical Helmholtz-Hodge decomposition is used to split the displacement field into scalar pressure (P -) and shear (S-) waves, allowing for separate treatment of the two dynamics and the construction of discretization spaces suited for each type of wave. This presentation focuses on the simulation of 2D soft scattering elastic wave propagation in isotropic homogeneous media, using the scalar potential decomposition in the time-harmonic regime. For problems defined in bounded domains, a Virtual Element Method (VEM) with varying mesh sizes and degrees of accuracy is proposed to approximate the two scalar potentials. For unbounded domains, a boundary element method is coupled with the VEM. The proposed approach performs better than standard methods that directly use the vector formulation, as it allows for tracking the different wave numbers associated with P - and S-speeds of propagation. This makes it possible to use a high-order method for the approximation of waves with higher wave numbers. We establish the stability of our method and present a convergence error estimate in the L2-norm for the displacement field. Notably, our estimate separates the contributions to the error associated with the P - and S- waves. We provide numerical results to demonstrate the effectiveness of the proposed approach. This presentation is the result of collaborative work with M. Ferrari and L. Scuderi from the Polytechnic University of Turin.

Solving 2D linear elastic wave equations via scalar potentials

Y27 H 35/36

17 May 2023
16:30-17:30

Prof. Dr. Stefan Kurz
Bosch Center for Artificial Intelligence and University of Jyväskylä

Event Details

Speaker invited by

Prof. R. Hiptmair

Abstract

The talk will first exemplify Hybrid Modeling, that is combining first-principle based with data-driven models, on a toy example. Next, an approach for formalizing hybrid modeling will be presented, in terms of architectural design patterns. Afterwards, the benefits of Hybrid Modeling will be demonstrated in two applications: (i) data-driven electromagnetic field simulation, where the constitutive law will be directly inferred from data, and (ii) irregular time series, where mathematical structures such as Kálmán filter and stochastic ODEs are integrated within deep neural networks. The talk concludes with some suggestions for research questions.

Hybrid Modeling: Newton + Kepler = Success (joint work with Barbara Rakitsch and Maja Rudolph)

Y27 H 35/36

22 May 2023
16:30-17:30

Prof. Dr. Hai Zhang
Hong Kong Univ. of Science & Technology

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

The developments of topological insulators have provided a new avenue of creating interface modes (or edge modes) in photonic/phononic structures. Such created modes have a distinct property of being topologically protected and are stable with respect to perturbations in certain classes. In this talk, we will report recent results on the existence of in-gap interface modes that are bifurcated from Dirac points in photonic/phononic structures. Both one-dimensional and two-dimensional structures will be discussed.

A mathematical theory of in-gap interface modes in photonic/phononic structures

HG E 23

24 May 2023
16:30-17:30

Prof. Dr. Patrick Ciarlet
ENSTA Paris | Institut Polytechnique de Paris

Event Details

Speaker invited by

Stefan Sauter and Rémi Abgrall

Abstract

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 solve them. Of paramount importance is the inf-sup condition, designed by Ladyzhenskaya, Necas, Babuska and Brezzi in the 1960s and 1970s. As is well-known, it provides sharp conditions to prove well-posedness of the problem, namely existence and uniqueness of the solution, and continuous dependence with respect to the data. Then, to solve the approximate, or discrete, problems, there is the (uniform) discrete inf-sup condition, to ensure existence of the approximate solutions, and convergence of those solutions to the exact solution. Often, the two sides of this problem (exact and approximate) are handled separately, or at least no explicit connection is made between the two. In this talk, I will focus on an approach that is completely equivalent to the inf-sup condition for problems set in Hilbert spaces, the T-coercivity approach. This approach relies on the design of an explicit operator to realize the inf-sup condition. If the operator is carefully chosen, it can provide useful insight for a straightforward definition of the approximation of the exact problem. As a matter of fact, the derivation of the discrete inf-sup condition often becomes elementary, at least when one considers conforming methods, that is when the discrete spaces are subspaces of the exact Hilbert spaces. In this way, both the exact and the approximate problems are considered, analysed and solved at once. In itself, T-coercivity is not a new theory, however it seems that some of its strengths have been overlooked, and that, if used properly, it can be a simple, yet powerful tool to analyse and solve linear PDEs. In particular, it provides guidelines such as, which abstract tools and which numerical methods are the most “natural” to analyse and solve the problem at hand. In other words, it allows one to select simply appropriate tools in the mathematical, or numerical, toolboxes. This claim will be illustrated on classical linear PDEs, and for some generalizations of those models.

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

Y27 H 35/36

27 September 2023
16:30-17:30

Prof. Dr. H. Gimperlein
Leopold-Franzens-Universität Innsbruck

Event Details

Speaker invited by

Prof.Dr. Christoph Schwab

Abstract

Boundary integral formulations are well-known to lead to efficient numerical methods for time-independent scattering and emission problems. In this talk we consider corresponding formulations for the time-dependent acoustic and elastic wave equations. We survey recent work on space-time Galerkin methods for the numerical solution, including higher order approximations by h- and hp-versions, a posteriori error estimates and adaptive mesh refinements, and illustrate them for applications in traffic noise.

Boundary integral equations in space and time: Higher order Galerkin methods and applications

HG E 1.2

4 October 2023
16:30-17:30

Prof. Dr. Wenjia Jing
Tsinghua University

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

We consider elliptic equations with periodic high contrast coefficients and study the asymptotic analysis when the periodicity is sent to zero and/or the contrast parameters are sent to extreme values. Those coefficients model small inclusions that have very different physical properties compared to the surrounding environment. Homogenization captures the macroscopic effects of those inclusions. We report some quantitative results such as the convergence rates of the homogenization (with proper correctors), uniform regularity for the solutions of the heterogeneous equations, and so on. The talk is based on joint works with Mr. Xin Fu.

Quantitative homogenization of elliptic problems in periodic high contrast environments

HG E 1.2

11 October 2023
16:30-17:30

Dr. Théophile Chaumont-Frelet
Inria

Event Details

Speaker invited by

Stefan Sauter

Abstract

Time-harmonic Maxwell's equations model the propagation of electromagnetic waves, and their numerical discretization by finite elements is instrumental in a large array of applications. In the simpler setting of acoustic waves, it is known that (i) the Galerkin Lagrange finite element approximation to a Helmholtz problem becomes asymptotically optimal as the mesh is refined. Similarly, (ii) asymptotically constant-free a posteriori error estimates are available for Helmholtz problems. In this talk, considering Nédélec finite element discretizations of time-harmonic Maxwell's equations, I will show that (i) still holds true and propose an a posteriori error estimator providing (ii). Both results appear to be novel contributions to the existing literature.

Asymptotically optimal a priori and a posteriori error estimates for edge finite element discretizations of time-harmonic Maxwell's equations

HG E 1.2

25 October 2023
16:30-17:30

Prof. Dr. Gianluca Crippa
Departement Mathmatik und Informatik, Universität Basel

Event Details

Speaker invited by

Rémi Abgrall

Abstract

Kolmogorov's K41 theory of fully developed turbulence advances quantitative predictions on anomalous dissipation in incompressible fluids: although smooth solutions of the Euler equations conserve the energy, in a turbulent regime information is transferred to small scales and dissipation can happen even without the effect of viscosity, and it is rather due to the limited regularity of the solutions. In rigorous mathematical terms, however, very little is known. In a recent work in collaboration with M.~Colombo and M.~Sorella we consider the case of passive-scalar advection, where anomalous dissipation is predicted by the Obukhov-Corrsin theory of scalar turbulence. In my talk, I will present the general context and illustrate the main ideas behind our construction of a velocity field and a passive scalar exhibiting anomalous dissipation in the supercritical Obukhov-Corrsin regularity regime. I will also describe how the same techniques provide an example of lack of selection for passive-scalar advection under vanishing diffusivity, and an example of anomalous dissipation for the forced Euler equations in the supercritical Onsager regularity regime (this last result has been obtained in collaboration with E.~Bru\`e, M.~Colombo, C.~De Lellis, and M.~Sorella).

Anomalous dissipation in fluid dynamics

HG E 1.2

1 November 2023
16:30-17:30

Prof. Dr. Daniele Boffi
KAUST

Event Details

Speaker invited by

Prof. Dr. Christoph Schwab

Abstract

In this talk I will discuss the numerical approximation of PDE eigenvalue problems depending on a finite number of deterministic parameters. The parameters can be part of the problem or can be introduced by the discretization. It turns out that eigenvalue problems are influenced by the presence of parameters in a way that doesn't compare to the corresponding source problem. We present several examples and counterexamples, showing the difficulties arising when eigenvalues and eigenfunctions need to be approximated accurately. A crucial aspect of parametric eigenvalue problems is the lack of regularity with respect to the parameter, unless a special sorting is considered, taking into account appropriately possible crossings and clustering. On the other hand, parameters arising from the discretizing scheme can be source of spurious solutions.

On the numerical approximation of parameter dependent PDE eigenvalue problems

HG E 1.2

15 November 2023
16:30-17:30

Prof. Dr. Svitlana Mayboroda
ETH Zurich, Switzerland

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

Waves of all kinds permeate our world. We are surrounded by light (electromagnetic waves), sound (acoustic waves), and mechanical vibrations. Quantum mechanics revealed that, at the atomic level, all matter has a wavelike character. And classical gravitational waves have been very recently detected. At the cutting edge of today’s science, it has become possible to manipulate individual atoms. This provides us with precise measurements of a world that exhibits myriad irregularities — dimensional, structural, orientational, and geometric— simultaneously. For waves, such disorder changes everything. In complex, irregular, or random media, waves frequently exhibit astonishing and mysterious behavior known as ‘localization’. Instead of propagating over extended regions, they remain confined in small portions of the original domain. The Nobel Prize–winning discovery of the Anderson localization in 1958 is only one famous case of this phenomenon. Yet, 60 years later, despite considerable advances in the subject, we still notoriously lack tools to fully understand localization of waves and its consequences. We will discuss modern understanding of the subject, recent results, and the biggest open questions.

Wave Localization

HG E 1.2

29 November 2023
16:30-17:30

Prof. Dr. Daniel Freeman
Saint Louis University, USA

Event Details

Speaker invited by

Prof. Dr. Rima Alaifari

Abstract

A frame (x_j) for a Hilbert space H allows for a linear and stable reconstruction of any vector x in H from the linear measurements (). However, there are many situations where some information of the frame coefficients is lost. In applications such as signal processing, electrical engineering, and digital photography one often uses sensors with an effective range and any measurement above that range is registered as the maximum. Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery. We will discuss a frame theoretic approach to this problem in a similar way to what Balan, Casazza, and Edidin did for phase retrieval. This perspective motivates many interesting open problems. The talk is based on joint work with W. Alharbi, D. Ghoreishi, B. Johnson, and N. Randrianarivony.

Vector recovery from saturated frame coefficients

HG E 1.2

13 December 2023
16:30-17:30

Dr. Dmitry Batenkov
Tel Aviv University

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

The inverse problem of computational super-resolution is to recover fine features of a signal from bandlimited and noisy data. Despite long history of the question and its fundamental importance in science and engineering, relatively little is known regarding optimal accuracy of reconstructing the high resolution signal components, and how to attain it with tractable algorithms. In this talk I will describe recent progress on deriving optimal methods for super-resolving sparse sums of Dirac masses, a popular model in numerous applications such as spectral estimation, direction of arrival, imaging of point sources, and sampling signals below the Nyquist rate. Time permitting, I will also discuss generalizations of the theory and algorithms in several directions.

Super-resolution of sparse measures: recent advances

HG E 1.2

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