Zurich Colloquium in Applied and Computational Mathematics

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Archive 2024

Date / Time

Speaker

Title

Location

6 March 2024
16:30-17:30

Prof. Dr. Enrique Zuazua
Friedrich-Alexander-Universität Erlangen-Nürnberg

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

Norbert Wiener defined “Cybernetics” as “the science of control and communication in the animal and the machine”, anticipating some of the goals and the future development of Artificial Intelligence. The traditional Applied Mathematics program, combining modelling, analysis, numerical approximation, and scientific computing, when facing practical applications, must often be complemented by additional efforts to address control issues, to better understand how dynamics changes when varying free parameters. This frequently leads to new complex and fascinating analytical and computational challenges that require significant unexpected further developments. We will lecture on some recent success stories that arise when facing, for instance, source identification problems, and the regulation of collective dynamics. We shall also discuss the issue of the optimal placement of sensors and actuators, which plays a key role when designing efficient control mechanisms. Control techniques also play an unexpected relevant role in other contexts such as the large time asymptotics for partially dissipative systems in fluid mechanics. We will describe the links between these problems and their analytical and numerical treatment, as one further manifestation of the unity and interconnections of all mathematical disciplines. We shall conclude pointing towards some perspective for future research in connection with Machine Learning. We will begin by briefly discussing the origins of mathematical control theory and machine learning, emphasizing their intimate analogies and links. We will then recall some basic results on the control of linear finite-dimensional systems and the Universal Approximation Theorem. Later we will address the problem of supervised learning, formulated as a simultaneous or ensemble control problem for the so-called neural differential equations, driven by Lipschitz nonlinearities, the activation functions in the neural network ansatz for learning. We will present an iterative and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies. The very role that the nonlinear nature of the activation functions plays will be emphasized. Unnecessary technical difficulties will be avoided. Several open problems and perspectives for future research will be formulated.

Dynamics, Control and Numerics

HG E 1.2

13 March 2024
16:30-17:30

Prof. Dr. Sergios Agapiou
University of Cyprus

Event Details

Speaker invited by

Prof. Dr. Christoph Schwab

Abstract

We will consider Bayesian nonparametric settings with functional unknowns and we will be interested in evaluating the asymptotic performance of the posterior in the infinitely informative data limit, in terms of rates of contraction. We will be especially interested in priors which are adaptive to the smoothness of the unknown function. In the last decade, certain hierarchical and empirical Bayes procedures based on Gaussian process priors, have been shown to achieve adaptation to spatially homogenous smoothness. However, we have recently shown that Gaussian priors are suboptimal for spatially inhomogeneous unknowns, that is, functions which are smooth in some areas and rough or even discontinuous in other areas of their domain. In contrast, we have shown that (similar) hierarchical and empirical Bayes procedures based on Laplace (series) priors, achieve adaptation to both homogeneously and inhomogeneously smooth functions. All of these procedures involve the tuning of a hyperparameter of the Gaussian or Laplace prior. After reviewing the above results, we will present a new strategy for adaptation to smoothness based on heavy-tailed priors. We will illustrate it in a variety of nonparametric settings, showing in particular that adaptive rates of contraction in the minimax sense (up to logarithmic factors) are achieved without tuning of any hyperparameters and for both homogeneously and inhomogeneously smooth unknowns. We will also present numerical simulations corroborating the theory. This is joint work with Masoumeh Dashti, Tapio Helin, Aimilia Savva and Sven Wang (Laplace priors) and Ismaël Castillo (heavy-tailed priors)

A new way for achieving Bayesian nonparametric adaptation

HG E 1.2

10 April 2024
16:30-17:30

Dr. Kaibo Hu
University of Edinburgh, UK

Event Details

Speaker invited by

Prof. Dr. Ralf Hiptmair

Abstract

Finite Element Exterior Calculus (FEEC) provides a cohomology framework for structure-preserving discretisation of a large class of PDEs. Differential complexes are important tools in FEEC. The de Rham complex is a basic example, with applications in curl-div related problems such as the Maxwell equations. There is a canonical finite element discretisation of the de Rham complex, which in the lowest order case coincides with discrete differential forms (Whitney forms). Different problems involve different complexes. In this talk, we provide an overview of some efforts towards Finite Element Tensor Calculus, inspired by tensor-valued problems from continuum mechanics and general relativity. On the continuous level, we systematically derive new complexes from the de Rham complexes. On the discrete level, We review the idea of distributional finite elements, and use them to obtain analogies of the Whitney forms for these new complexes. A special case is Christiansen’s finite element interpretation of Regge calculus, a discrete geometric scheme for metric and curvature.

Towards Finite Element Tensor Calculus

HG E 1.2

17 April 2024
16:00-17:00

Prof. Dr. Vincent Perrier
Inria, France

Event Details

Speaker invited by

Prof. Dr. Rémi Abgrall

Abstract

Some hyperbolic systems are known to include implicit preservation of differential constraints: these are for example the time conservation of the vorticity for the first order wave system or divergence preservation for the Maxwell system or the induction equation. In this talk, I will address this problem with the classical discontinuous Galerkin method. Based on discrete de-Rham ideas, I will show that by considering an adapted approximation space (but still discontinuous) for vectors , divergence or curl can be easily preserved under mild assumption on the numerical flux

How to preserve a divergence or a curl constraint in a hyperbolic system with the discontinuous Galerkin method

HG E 1.2

24 April 2024
16:30-18:00

Prof. Dr. Guglielmo Scovazzi
Department of Civil and Environmental Engineering, Duke University

Event Details

Speaker invited by

Prof. Dr. Rémi Abgrall

Abstract

Scientific computing is routinely assisting in the design of systems or components, which have potentially very complex shapes. In these situations, it is often underestimated that the mesh generation process takes the overwhelming portion of the overall analysis and design cycle. If high order discretizations are sought, the situation is even more critical. Methods that could ease these limitations are of great importance, since they could more effectively interface with meta-algorithms from Optimization, Uncertainty Quantification, Reduced Order Modeling, Machine Learning, and Artificial Neural Networks, in large-scale applications. Recently, immersed/embedded/unfitted boundary finite element methods (cutFEM, Finite Cell Method, Immerso-Geometric Analysis, etc.) have been proposed for this purpose, since they obviate the burden of body-fitted meshing. Unfortunately, most unfitted finite element methods are also difficult to implement due to: (a) the need to perform complex cell cutting operations at boundaries, (b) the necessity of specialized quadrature formulas on cut elements, and (c) the consequences that these operations may have on the overall conditioning/stability of the ensuing algebraic problems. This talk introduces a simple, stable, and accurate unfitted boundary method, named “Shifted Boundary Method” (SBM), which eliminates the need to perform cell cutting operations. Boundary conditions are imposed on the boundary of a “surrogate” discrete computational domain, specifically constructed to avoid cut elements. Appropriate field extension operators are then constructed by way of Taylor expansions (or similar operators), with the purpose of preserving accuracy when imposing boundary conditions. An extension of the SBM to higher order discretizations will also be presented, together with a summary of the numerical analysis results. The SBM belongs to the broader class of Approximate Boundary Methods, a less explored or somewhat forgotten class of algorithms, which however might have an important role in the future of scientific computing. The performance of the SBM is tested on large-scale problems selected from linear and nonlinear elasticity, fluid mechanics, shallow water flows, thermos-mechanics, porous media flow, and fracture mechanics.

The Shifted Boundary Method: How Approximate Boundaries Can Help in Complex-Geometry Computations

HG E 1.2

8 May 2024
16:30-17:30

Prof. Dr. Maarten de Hoop
Rice University

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

We present results pertaining to selected inverse problems associated with seismology on Earth, Mars and Saturn. We focus on geometrical or travel time data originating from the propagation of singularities and the spectra corresponding with normal modes. For terrestrial or rocky planets we highlight recent insights with generic anisotropic elasticity, and for gas giants we reveal the accommodation of the equations of state all the way up to their boundaries. We briefly touch upon whether information on uniqueness of inverse problems is encoded in the data.

Geometric and spectral inverse problems for terrestrial planets and gas giants

HG E 1.2

15 May 2024
16:30-17:30

Dr. Leonardo Zepeda-Nunez
Google Research, USA

Event Details

Speaker invited by

Prof. Dr. Siddhartha Mishra

Abstract

The advent of generative AI has turbocharged the development of a myriad of commercial applications, and it has slowly started to permeate to scientific computing. In this talk we discussed how recasting the formulation of old and new problems within a probabilistic approach opens the door to leverage and tailor state-of-the-art generative AI tools. As such, we review recent advancements in Probabilistic SciML – including computational fluid dynamics, inverse problems, and particularly climate sciences, with an emphasis on statistical downscaling. Statistical downscaling is a crucial tool for analyzing the regional effects of climate change under different climate models: it seeks to transform low-resolution data from a (potentially biased) coarse-grained numerical scheme (which is computationally inexpensive) into high-resolution data consistent with high-fidelity models. We recast this problem in a two-stage probabilistic framework using unpaired data by combining two transformations: a debiasing step performed by an optimal transport map, followed by an upsampling step achieved through a probabilistic conditional diffusion model. Our approach characterizes conditional distribution without requiring paired data and faithfully recovers relevant physical statistics, even from biased samples. We will show that our method generates statistically correct high-resolution outputs from low-resolution ones, for different chaotic systems, including well known climate models and weather data. We show that the framework is able to upsample resolutions by 8x and 16x while accurately matching the statistics of physical quantities – even when the low-frequency content of the inputs and outputs differs. This is a crucial yet challenging requirement that existing state-of-the-art methods usually struggle with.

Recent Advances in Probabilistic Scientific Machine learning

HG E 1.2

25 September 2024
16:00-17:00

Dr. Martin Averseng
Université d’Angers

Event Details

Speaker invited by

Prof. Dr. Ralf Hiptmair

Abstract

The Boundary Element Method (BEM) is a discretization technique commonly employed for the accurate and rapid numerical solution of constant-coefficient second-order partial differential equations (PDEs) in the complement of an obstacle, e.g., electromagnetic scattering problems. The BEM exploits the fundamental solution of the PDE to reduce the number of unknowns compared to the Finite Element Method for the same level of error. However, it is a non-local method and thus leads to full linear systems. For this reason, the BEM linear systems are often solved iteratively, and good preconditioners often turn out to be a key ingredient to ensure a fast resolution. In this talk, we present the motivation and the practical and mathematical challenges for extending the BEM to geometric settings involving non-manifold boundaries. We will first summarize the recent advances in the mathematical formalization of this problem. We will then present a particular preconditioning technique for "multi-screen" obstacles based on substructuring (domain decomposition). This work is in collaboration with Xavier Claeys and Ralf Hiptmair.

Non-manifold boundary element methods

HG G 19.1

2 October 2024
16:30-17:30

Prof. Dr. Christiane Helzel
Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf

Event Details

Speaker invited by	Prof. Dr. Rémi Abgrall

Active Flux Methods for Hyperbolic Conservation Laws

HG G 19.2

9 October 2024
16:30-17:30

Prof. Dr. Cristinel Mardare
Sorbonne Université

Event Details

Speaker invited by	Prof. Dr. Stefan Sauter

On the divergence equation and its relation to Korn’s inequalities

HG G 19.2

16 October 2024
16:30-17:30

Prof. Dr. Gigliola Staffilani
Massachusetts Institute of Technology

Event Details

Speaker invited by

Prof. Dr. Habib Ammari

Abstract

In this talk we will use the periodic cubic nonlinear Schrödinger equation to present some estimates of the long time dynamics of the energy spectrum, a fundamental object in the study of wave turbulence theory. Going back to Bourgain, one possible way to conduct the analysis is to look at the growth of high Sobolev norms. It turns out that this growth is sensitive to the nature of the space periodicity of the system. I will present a combination of old and very recent results in this direction.

A curious phenomenon in wave turbulence theory

HG G 19.2

23 October 2024
16:30-17:30

Prof. Dr. Fatih Ecevit
Dept. of Mathematics, Boğaziçi University

Event Details

Speaker invited by

Prof. Dr. Stefan Sauter

Abstract

We present our recent developments on the asymptotic expansions of high-frequency multiple scattering iterations in the exterior of sound-hard scatterers. As in the sound-soft case, these expansions lead into wavenumber dependent estimates on the derivatives (of all orders) of the multiple scattering iterations which, in turn, allow for the design and analysis of Galerkin boundary element methods (BEM) for their frequency independent approximation. We also present preliminary theoretical developments related to the accurate approximation of the remaining infinite tail in the Neumann series formulation of multiple scattering problems. Time permitting, in the second part of the talk, we present our preliminary results on the frequency independent approximation of the sound-soft scattering amplitude based on Bayliss-Turkel type local approximations to the Dirichlet-to-Neumann operator. Joint with: Y. Boubendir (NJIT) and S. Lazergui (NJIT)

High-frequency BEM for sound-soft/hard multiple scattering and applications to the scattering amplitude

HG G 19.2

20 November 2024
16:30-17:30

Prof. Dr. Carlos Jerez-Hanckes
Universidad Adolfo Ibañez, Santiago, Chile

Event Details

Speaker invited by

Prof. Dr. Ralf Hiptmair

Abstract

In this talk, we will focus on solving time-harmonic, acoustic, elastic and polarized electromagnetic waves scattered by multiple finite-length open arcs in unbounded two-dimensional domain. We will first recast the corresponding boundary value problems with Dirichlet or Neumann boundary conditions, as weakly- and hyper-singular boundary integral equations (BIEs), respectively. Then, we will introduce a family of fast spectral Galerkin methods for solving the associated BIEs. Discretization bases of the resulting BIEs employ weighted Chebyshev polynomials that capture the solutions' edge behavior. We will show that these bases guarantee exponential convergence in the polynomial degree when assuming analyticity of sources and arc geometries. Numerical examples will demonstrate the accuracy and robustness of the proposed methods with respect to number of arcs and wavenumber. Moreover, we will show that for general weakly- and hyper-singular boundary integral equations their solutions depend holomorphically upon perturbations of the arcs' parametrizations. These results are key to prove the shape holomorphy of domain-to-solution maps associated to BIEs appearing in uncertainty quantification, inverse problems and deep learning, to name a few applications. Also, they pose new questions you may have the answer to!

New Insights on Wave Scattering by Multiple Open Arcs: Lightning-Fast Methods and Shape Holomorphy

HG G 19.2

4 December 2024
16:30-17:30

Prof. Dr. Gui-Qiang G. Chen
University of Oxford

Event Details

Speaker invited by	Prof. Dr. Habib Ammari

Title T.B.A.

HG G 19.2

11 December 2024
16:30-17:30

Dr. Federico Pichi
SISSA, Trieste, Italy

Event Details

Speaker invited by

Prof. Dr. Christoph Schwab

Abstract

The development of efficient reduced order models (ROMs) from a deep learning perspective enables users to overcome the limitations of traditional approaches [1, 2]. One drawback of the techniques based on convolutional autoencoders is the lack of geometrical information when dealing with complex domains defined on unstructured meshes. The present work proposes a framework for nonlinear model order reduction based on Graph Convolutional Autoencoders (GCA) to exploit emergent patterns in different physical problems, including those showing bifurcating behavior, high-dimensional parameter space, slow Kolmogorov-decay, and varying domains [3]. Our methodology extracts the latent space’s evolution while introducing geometric priors, possibly alleviating the learning process through up- and down-sampling operations. Among the advantages, we highlight the high generalizability in the low-data regime and the great speedup. Moreover, we will present a novel graph feedforward network (GFN), extending the GCA approach to exploit multifidelity data, leveraging graph-adaptive weights, enabling large savings, and providing computable error bounds for the predictions [4]. This way, we overcome the limitations of the up- and down-sampling procedures by building a resolution-invariant GFN-ROM strategy capable of training and testing on different mesh sizes, resulting in a more lightweight and flexible architecture. References [1] Lee, K. and Carlberg, K.T. (2020) ‘Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders’, Journal of Computational Physics, 404, p. 108973. Available at: https://doi.org/10.1016/j.jcp.2019.108973. [2] Fresca, S., Dede’, L. and Manzoni, A. (2021) ‘A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs’, Journal of Scientific Computing, 87(2), p. 61. Available at: https://doi.org/10.1007/s10915-021-01462-7. [3] Pichi, F., Moya, B. and Hesthaven, J.S. (2024) ‘A graph convolutional autoencoder approach to model order reduction for parametrized PDEs’, Journal of Computational Physics, 501, p. 112762. Available at: https://doi.org/10.1016/j.jcp.2024.112762. [4] Morrison, O.M., Pichi, F. and Hesthaven, J.S. (2024) ‘GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications’, Computer Methods in Applied Mechanics and Engineering, 432, p. 117458. Available at: https://doi.org/10.1016/j.cma.2024.117458.

Graph-based machine learning approaches for model order reduction

HG G 19.2

18 December 2024
16:30-17:30

Prof. Dr. Dirk Pauly
TU Dresden

Event Details

Speaker invited by	Prof. Dr. Ralf Hiptmair
Abstract	We study a new notion of trace and extension operators for abstract Hilbert complexes.

Traces for Hilbert Complexes

HG G 19.2

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