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Seminar by Mejdi Azaiez
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    :bdg-primary-line:`Speaker`

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        Mejdi Azaiez (Institut Polytechnique de Bordeaux)

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        An intrinsic PGD for parametric symmetric elliptic problems

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        * May 09, 2023 16:00 CEST+0200 (Europe/Rome)
        * May 09, 2023 10:00 EDT-0400 (US/Eastern)
        * May 09, 2023 09:00 CDT-0500 (US/Central)
        * May 09, 2023 07:00 PDT-0700 (US/Pacific)

    :bdg-primary-line:`Abstract`

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        We introduce a technique for the reduced order approximation of
        parametric symmetric elliptic partial differential equations. This
        method is analogous to the best approximation property of the Proper
        Orthogonal Decomposition (POD) subspaces, excepting that in our case the
        norm, intrinsic to the problem, is parameter-depending, and then the POD
        optimal sub-spaces cannot be characterized by means of a spectral
        problem. We apply a deflation technique to build a series of
        approximating solutions on finite-dimensional optimal subspaces,
        directly in the on-line step. We prove that the partial sums converge to
        the continuous solutions in mean quadratic elliptic norm. We prove the
        linear convergence of the Power Iterate method applied to compute the
        modes of the PGD expansion, for both symmetric and non-symmetric
        problems, when the data are small. We also find a spectral convergence
        ratio of the PGD expansion in the mean parametric norm, for meaningful
        parametric elliptic problems. We end the talk by proving that the PGD
        modes of the discretized problem converge to the PGD modes of the
        continuous problem, with respect to the discretization of both the
        parameter set and the elliptic problem afforded. When possible, we will
        assess our theoretical results by giving some numerical experiments.
        
        This talk is based on references (Azaı̈ez, Chacón Rebollo, and Sanchez
        Muñoz 2023; Azaïez, Chacón Rebollo, and Gómez Mármol 2020; Azaı̈ez et al.
        2018).
        
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              :name: ref-Azaiez2020
        
              Azaïez, M., T. Chacón Rebollo, and M. Gómez Mármol. 2020. “On the
              Computation of Proper Generalized Decomposition Modes of
              Parametric Elliptic Problems.” *SeMA Journal* 77 (1): 59–72.
        
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              :name: ref-Azaiez2018
        
              Azaı̈ez, M., F. Ben Belgacem, J. Casado-Dı́az, T. Chacón Rebollo,
              and F. Murat. 2018. “A New Algorithm of Proper Generalized
              Decomposition for Parametric Symmetric Elliptic Problems.” *SIAM
              Journal on Mathematical Analysis* 50 (5): 5426–45.
        
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              :name: ref-Azaiez2023
        
              Azaı̈ez, M., T. Chacón Rebollo, and S. M. Sanchez Muñoz. 2023.
              “Numerical Approximation of Proper Generalized Decomposition Modes
              to Parameterized Elliptic Problems.” *To Appear in SIAM Journal on
              Scientific Computing*.

    :bdg-primary-line:`Recording`

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        Watch the recording on `our YouTube channel <https://www.youtube.com/watch?v=Q5NAN5Arnac>`_.