Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms

Stochastic Galerkin methods have become a significant tool for the resolution of stochastic partial differential equations (SPDE). However, they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality. Some alternative techniques, based on the construction of suitable reduced deterministic or stochastic bases, have been proposed in order to reduce these computational costs. Recently, a new approach, based on the concept of generalized spectral decomposition (GSD), has been introduced for the definition and the automatic construction of reduced bases. In this paper, the concept of GSD, initially introduced for a class of linear elliptic SPDE, is extended to a wider class of stochastic problems. The proposed definition of the GSD leads to the resolution of an invariant subspace problem, which is interpreted as an eigen-like problem. This interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases, which are associated with dominant generalized eigenspaces. The proposed algorithms, by separating the resolution of reduced stochastic and deterministic problems, lead to drastic computational savings. Their efficiency is illustrated on several examples, where they are compared to classical resolution techniques.