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

Anchored analysis of variance Petrov–Galerkin projection schemes for linear stochastic structural dynamics

“…Here, we introduce the stochastic-weighted residual form of (1.1) and show that by defining an appropriate test space, the original high-dimensional problem can be decoupled into stochastic low-dimensional subproblems that can be solved independently of each other. This is an important property of AAPG projection schemes that enables the development of parallel numerical implementations that scale very well to high-dimensional problems [40].…”

“…The test space V that we shall use was originally proposed in reference [40] in the context of parabolic SPDEs. The main difference compared with reference [40] is that because we are dealing with SODEs, the trial and test spaces do not depend on spatial coordinates. The space of test functions corresponding to the Lth-order truncated ANOVA decomposition is defined as…”

“…Another function decomposition scheme that has been widely studied in various fields is the Hoeffding functional analysis of variance (ANOVA) representation [31][32][33][34]. This approach has been successfully applied within both non-intrusive [35][36][37][38] and intrusive settings [39,40] to solve uncertainty quantification problems. More recently, Audouze & Nair [40] proposed the anchored ANOVA Petrov-Galerkin (AAPG) projection scheme for solving high-dimensional parabolic SPDEs.…”

“…This approach has been successfully applied within both non-intrusive [35][36][37][38] and intrusive settings [39,40] to solve uncertainty quantification problems. More recently, Audouze & Nair [40] proposed the anchored ANOVA Petrov-Galerkin (AAPG) projection scheme for solving high-dimensional parabolic SPDEs. In this work, it was shown that by using an anchored ANOVA decomposition in conjunction with an appropriate test space, the original high-dimensional weak form can be decoupled into low-dimensional subproblems that can be solved independently of each other.…”

“…The remainder of this paper is organized as follows: §2 outlines the anchored ANOVA decomposition scheme and some of its theoretical properties. Section 3 provides the derivation of the AAPG projection scheme [40] in the special context of linear stochastic structural dynamical systems. This is followed by §4 where we present numerical results obtained using MCS, gPC, GSD and AAPG schemes for two case studies providing accuracy and performance comparisons.…”