Zurich Colloquium in Applied and Computational Mathematics

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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

26 February 2025
16:30-17:30

Prof. Dr. Philippe Ciarlet
Institut für Mathematik, Universität Zürich

Event Details

Speaker invited by	Prof. Dr. Stefan Sauter

Title T.B.A.

HG G 19.2

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