Seminar by Mejdi Azaiez

Speaker

Mejdi Azaiez (Institut Polytechnique de Bordeaux)

Title

An intrinsic PGD for parametric symmetric elliptic problems

Date

  • 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)

Abstract

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).

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.

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.

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.

Recording

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