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Spring Semester 2023

Date / Time Speaker Title Location
6 March 2023
13:15-14:15 		Prof. Dr. Nana Liu
Shanghai Jiaotong University, China
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Zurich Colloquium in Applied and Computational Mathematics

Title Efficient quantum computation for partial differential equations
Speaker, Affiliation Prof. Dr. Nana Liu, Shanghai Jiaotong University, China
Date, Time 6 March 2023, 13:15-14:15
Location HG G 19.2
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 equationsread_more
HG G 19.2
8 March 2023
16:30-17:30 		Prof. Dr. Shi Jin
Shanghai Jiaotong University, China
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Zurich Colloquium in Applied and Computational Mathematics

Title Random Batch Methods for interacting particle systems and molecular dynamics
Speaker, Affiliation Prof. Dr. Shi Jin, Shanghai Jiaotong University, China
Date, Time 8 March 2023, 16:30-17:30
Location Y27 H 35/36
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 dynamicsread_more
Y27 H 35/36
15 March 2023
16:30-17:30 		Dr. Andrea Manzoni
Politecnico di Milano, Italy
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Zurich Colloquium in Applied and Computational Mathematics

Title Deep learning for reduced order modeling: recent results and open challenges
Speaker, Affiliation Dr. Andrea Manzoni, Politecnico di Milano, Italy
Date, Time 15 March 2023, 16:30-17:30
Location Y27 H 35/36
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 challengesread_more
Y27 H 35/36
22 March 2023
16:00-17:00 		Prof. Dr. Anne-Laure Dalibard
Sorbonne University, Paris
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Zurich Colloquium in Applied and Computational Mathematics

Title Nonlinear forward-backward problems
Speaker, Affiliation Prof. Dr. Anne-Laure Dalibard, Sorbonne University, Paris
Date, Time 22 March 2023, 16:00-17:00
Location Y27 H 35/36
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 problemsread_more
Y27 H 35/36
29 March 2023
16:30-17:30 		Prof. Dr. Christian Lubich
Universität Tübingen
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Zurich Colloquium in Applied and Computational Mathematics

Title Time-dependent scattering from thin layers
Speaker, Affiliation Prof. Dr. Christian Lubich, Universität Tübingen
Date, Time 29 March 2023, 16:30-17:30
Location Y27 H 35/36
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 layersread_more
Y27 H 35/36
10 May 2023
16:30-17:30 		Prof. Dr. Silvia Falletta
Politecnico di Torino
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Zurich Colloquium in Applied and Computational Mathematics

Title Solving 2D linear elastic wave equations via scalar potentials
Speaker, Affiliation Prof. Dr. Silvia Falletta, Politecnico di Torino
Date, Time 10 May 2023, 16:30-17:30
Location Y27 H 35/36
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 read_more
Y27 H 35/36
17 May 2023
16:30-17:30 		Prof. Dr. Stefan Kurz
Bosch Center for Artificial Intelligence and University of Jyväskylä
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Zurich Colloquium in Applied and Computational Mathematics

Title Hybrid Modeling: Newton + Kepler = Success (joint work with Barbara Rakitsch and Maja Rudolph)
Speaker, Affiliation Prof. Dr. Stefan Kurz, Bosch Center for Artificial Intelligence and University of Jyväskylä
Date, Time 17 May 2023, 16:30-17:30
Location Y27 H 35/36
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)read_more
Y27 H 35/36
22 May 2023
16:30-17:30 		Prof. Dr. Hai Zhang
Hong Kong Univ. of Science & Technology
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Zurich Colloquium in Applied and Computational Mathematics

Title A mathematical theory of in-gap interface modes in photonic/phononic structures
Speaker, Affiliation Prof. Dr. Hai Zhang, Hong Kong Univ. of Science & Technology
Date, Time 22 May 2023, 16:30-17:30
Location HG E 23
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 structuresread_more
HG E 23
24 May 2023
16:30-17:30 		Prof. Dr. Patrick Ciarlet
ENSTA Paris | Institut Polytechnique de Paris
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Zurich Colloquium in Applied and Computational Mathematics

Title T-coercivity: a practical tool for the study of variational formulations
Speaker, Affiliation Prof. Dr. Patrick Ciarlet, ENSTA Paris | Institut Polytechnique de Paris
Date, Time 24 May 2023, 16:30-17:30
Location Y27 H 35/36
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 formulationsread_more
Y27 H 35/36

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