Seminar by Yvon Maday
Speaker
Yvon Maday (Sorbonne Université)
Title
Nonlinear compressive reduced basis approximation for PDE’s
Date
May 23, 2023 16:00 CEST+0200 (Europe/Rome)
May 23, 2023 10:00 EDT-0400 (US/Eastern)
May 23, 2023 09:00 CDT-0500 (US/Central)
May 23, 2023 07:00 PDT-0700 (US/Pacific)
Abstract
Linear model reduction techniques design offline low-dimensional subspaces that are tailored to the approximation of solutions to a parameterized partial differential equation, for the purpose of fast online numerical simulations. These methods, such as the Proper Orthogonal Decomposition (POD) or Reduced Basis (RB) methods, are very effective when the family of solutions has fast-decaying Karhunen-Loève eigenvalues or Kolmogorov m-widths, reflecting the approximability by finite-dimensional linear spaces. On the other hand, they become ineffective when these quantities have a slow decay,in particular for families of solutions to hyperbolic transport equations with parameter-dependent shock positions. The objective of this work is to explore the ability of nonlinear model reduction to circumvent this particular situation via a nonlinear reconstruction. Based on the recent paper (Cohen et al. 2023), we first review the different notions (linear and nonlinear) of m-widths for compact sets, and in particular the Kolmogorov and Gelfand m-width. Built on these that can be very different we then propose a new non-linear approach (Barnett, Farhat, and Maday 2023) to take advantage of the small Gelfand m-width in a context where the Kolmogorov m-width is large.
Barnett, Joshua L, Charbel Farhat, and Yvon Maday. 2023. “Mitigating the Kolmogorov Barrier for the Reduction of Aerodynamic Models Using Neural-Network-Augmented Reduced-Order Models.” In AIAA SCITECH 2023 Forum, 0535.
Cohen, Albert, Charbel Farhat, Agustı́n Somacal, and Yvon Maday. 2023. “Nonlinear Compressive Reduced Basis Approximation for PDE’s.” Comptes Rendus Académie Des Sciences.
Recording
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