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

Title T.B.A.

HG E 1.2

22 May 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 E 1.2

29 May 2024
16:30-17:30

Prof. Dr. Ivan Trapasso
Politecnico di Torino

Event Details

Speaker invited by

Prof. Dr. Rima Alaifari

Abstract

In this talk we provide a concise overview of the fundamental principles underlying harmonic analysis in phase space. The roots of this vibrant field of modern Fourier analysis are to be found at the crossroads of signal analysis, mathematical physics, representation theory and analysis of partial differential equations. The key idea is to exploit a dictionary of oscillating wave packets (or equivalently, the combined structure of translations and modulations or dilations) to investigate properties of functions, distributions and operators in terms of suitable companion phase space representations. Addressing time and frequency/scale on the same level presents both advantages and challenges due to the uncertainty principle. In essence, time and frequency exhibit a somewhat dual nature as variables, hence the efforts to handle them concurrently are ultimately directed to keep track of the multifaceted manifestations of their entanglement. We will delve into these issues, whose origins date back to the foundations of quantum mechanics, and show how they continue to stimulate insightful research in analysis. Lastly, we will offer a taste of applications of these techniques to some problems motivated by the current challenges of data science, mostly in order to convey the message that the principles of time-frequency analysis are ubiquitous, hence adopting a phase space perspective can provide a versatile framework to explore problems from pure and applied mathematics.

Explorations in wave packet analysis

HG E 1.2

25 September 2024
16:30-17:30

Dr. Martin Averseng
Université d’Angers

Event Details

Speaker invited by	Prof. Dr. Ralf Hiptmair

Title T.B.A.

HG ? ?

9 October 2024
16:30-17:30

Prof. Dr. Cristinel Mardare
Sorbonne Université

Event Details

Speaker invited by	Prof. Dr. Stefan Sauter

Title T.B.A.

HG E 1.2

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