An intrinsic Proper Generalized Decomposition for parametric symmetric elliptic problems

TL;DR

This paper presents a novel intrinsic Proper Generalized Decomposition method for efficiently approximating solutions to parametric symmetric elliptic PDEs, with proven optimality and convergence properties.

Contribution

It introduces an intrinsic PGD approach that constructs optimal subspaces for parametric elliptic problems without spectral characterization, enabling efficient online approximation.

Findings

Existence of optimal subspaces for best mean approximation

Convergence of partial sums to exact solutions in quadratic norm

Application of deflation technique for online solution building

Abstract

We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation in mean of the error with respect to the parameter in the quadratic norm associated to the elliptic operator, between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the Proper Orthogonal Decomposition (POD) subspaces, excepting that in our case the norm 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.