Departement Mathematik

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Zurich colloquium in applied and computational mathematics

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Herbstsemester 2024

Datum / Zeit Referent:in Titel Ort
25. September 2024
16:00-17:00 		Dr. Martin Averseng
Université d’Angers
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Zurich Colloquium in Applied and Computational Mathematics

Titel Non-manifold boundary element methods
Referent:in, Affiliation Dr. Martin Averseng, Université d’Angers
Datum, Zeit 25. September 2024, 16:00-17:00
Ort HG G 19.1
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 methodsread_more
HG G 19.1
2. Oktober 2024
16:30-17:30 		Prof. Dr. Christiane Helzel
Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf
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Zurich Colloquium in Applied and Computational Mathematics

Titel Active Flux Methods for Hyperbolic Conservation Laws
Referent:in, Affiliation Prof. Dr. Christiane Helzel, Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf
Datum, Zeit 2. Oktober 2024, 16:30-17:30
Ort HG G 19.2
Active Flux Methods for Hyperbolic Conservation Laws
HG G 19.2
9. Oktober 2024
16:30-17:30 		Prof. Dr. Cristinel Mardare
Sorbonne Université
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Zurich Colloquium in Applied and Computational Mathematics

Titel On the divergence equation and its relation to Korn’s inequalities
Referent:in, Affiliation Prof. Dr. Cristinel Mardare, Sorbonne Université
Datum, Zeit 9. Oktober 2024, 16:30-17:30
Ort HG G 19.2
On the divergence equation and its relation to Korn’s inequalities
HG G 19.2
16. Oktober 2024
16:30-17:30 		Prof. Dr. Gigliola Staffilani
Massachusetts Institute of Technology
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Zurich Colloquium in Applied and Computational Mathematics

Titel A curious phenomenon in wave turbulence theory
Referent:in, Affiliation Prof. Dr. Gigliola Staffilani, Massachusetts Institute of Technology
Datum, Zeit 16. Oktober 2024, 16:30-17:30
Ort HG G 19.2
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 theoryread_more
HG G 19.2
23. Oktober 2024
16:30-17:30 		Prof. Dr. Fatih Ecevit
Dept. of Mathematics, Boğaziçi University
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Zurich Colloquium in Applied and Computational Mathematics

Titel High-frequency BEM for sound-soft/hard multiple scattering and applications to the scattering amplitude
Referent:in, Affiliation Prof. Dr. Fatih Ecevit, Dept. of Mathematics, Boğaziçi University
Datum, Zeit 23. Oktober 2024, 16:30-17:30
Ort HG G 19.2
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 amplituderead_more
HG G 19.2
20. November 2024
16:30-17:30 		Prof. Dr. Carlos Jerez-Hanckes
Universidad Adolfo Ibañez, Santiago, Chile
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Zurich Colloquium in Applied and Computational Mathematics

Titel New Insights on Wave Scattering by Multiple Open Arcs: Lightning-Fast Methods and Shape Holomorphy
Referent:in, Affiliation Prof. Dr. Carlos Jerez-Hanckes, Universidad Adolfo Ibañez, Santiago, Chile
Datum, Zeit 20. November 2024, 16:30-17:30
Ort HG G 19.2
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 Holomorphyread_more
HG G 19.2
11. Dezember 2024
16:30-17:30 		Dr. Federico Pichi
SISSA, Trieste, Italy
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Zurich Colloquium in Applied and Computational Mathematics

Titel Graph-based machine learning approaches for model order reduction
Referent:in, Affiliation Dr. Federico Pichi, SISSA, Trieste, Italy
Datum, Zeit 11. Dezember 2024, 16:30-17:30
Ort HG G 19.2
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 reductionread_more
HG G 19.2
18. Dezember 2024
16:30-17:30 		Prof. Dr. Dirk Pauly
TU Dresden
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Zurich Colloquium in Applied and Computational Mathematics

Titel Traces for Hilbert Complexes
Referent:in, Affiliation Prof. Dr. Dirk Pauly, TU Dresden
Datum, Zeit 18. Dezember 2024, 16:30-17:30
Ort HG G 19.2
Abstract We study a new notion of trace and extension operators for abstract Hilbert complexes.
Traces for Hilbert Complexesread_more
HG G 19.2

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