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Seminar by Yvon Maday
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    :bdg-primary-line:`Speaker`

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        Yvon Maday (Sorbonne Université)

    :bdg-primary-line:`Title`

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        Nonlinear compressive reduced basis approximation for PDE's

    :bdg-primary-line:`Date`

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        * 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)

    :bdg-primary-line:`Abstract`

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        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.
        
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              :name: ref-Barnett2023
        
              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.
        
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              :name: ref-Cohen2023
        
              Cohen, Albert, Charbel Farhat, Agustı́n Somacal, and Yvon Maday.
              2023. “Nonlinear Compressive Reduced Basis Approximation for
              PDE’s.” *Comptes Rendus Académie Des Sciences*.

    :bdg-primary-line:`Recording`

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        Watch the recording on `our YouTube channel <https://www.youtube.com/watch?v=k7uzIyROHQM>`_.