Anchored ANOVA Petrov–Galerkin projection schemes for parabolic stochastic partial differential equations

Abstract

We present a numerical scheme based on the combination of Hoeffding’s functional analysis of variance (ANOVA) decomposition with stochastic Galerkin projection for solving a class of high-dimensional parabolic stochastic partial differential equations (SPDEs). The central idea underpinning the proposed approach is to approximate the SPDE solution using a Hoeffding-ANOVA decomposition whose component functions are constrained to be orthogonal with respect to an appropriate measure. We show that when a Dirac product measure is chosen to enforce the orthogonality constraints and the test functions are chosen appropriately, the original stochastic weak formulation can be decoupled into low-dimensional subproblems that can be efficiently solved in parallel using polynomial chaos based stochastic projection schemes. As a result, the proposed approach scales very well to SPDEs with large number of random variables. We theoretically analyze the proposed formulation and provide a priori error estimates, as a function of the spatial, stochastic and temporal discretization parameters and the ANOVA expansion order, that hold under appropriate stochastic regularity assumptions. Numerical studies are presented for a set of time-dependent stochastic diffusion problems with up to 50 random variables to demonstrate the effectiveness of the proposed approach and comparisons are made against classical polynomial chaos stochastic Galerkin projection scheme (gPC) and the generalized spectral decomposition scheme. These studies show that the proposed approach provides accuracy comparable to classical gPC Galerkin projection schemes while incurring significantly lower computational cost.