On the computation of Proper Generalized Decomposition modes of parametric elliptic problems

Abstract

In a recent paper (Azaiez et al. in SIAM J Math Anal 50(5):5426–5445, https://doi.org/10.1137/17m1137164, 2018) a new algorithm of Proper Generalized Decomposition for parametric symmetric elliptic partial differential equations has been introduced. For any given dimension, this paper proves the existence of an optimal subspace of at most that dimension which realizes the best approximation—in mean parametric norm associated to the elliptic operator—of the error between the exact solution and the Galerkin solution calculated on the subspace. When the dimension is equal one and making use of a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, the method turns to be a classical progressive proper generalized decomposition. In this contribution 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.