An adaptive reduced basis collocation method based on PCM ANOVA decomposition for anisotropic stochastic PDEs

TL;DR

This paper introduces an adaptive reduced basis collocation method combining PCM and ANOVA decomposition to efficiently solve high-dimensional anisotropic stochastic PDEs, improving accuracy and computational efficiency.

Contribution

It proposes a novel two-stage adaptive algorithm integrating PCM and ANOVA for dimension reduction and solution refinement in stochastic PDEs.

Findings

Effective identification of important dimensions using ANOVA decomposition.

Adaptive polynomial order increase improves solution accuracy.

Method successfully applied to a benchmark convection-diffusion problem.

Abstract

The combination of reduced basis and collocation methods enables efficient and accurate evaluation of the solutions to parameterized PDEs. In this paper, we study the stochastic collocation methods that can be combined with reduced basis methods to solve high-dimensional parameterized stochastic PDEs. We also propose an adaptive algorithm using a probabilistic collocation method (PCM) and ANOVA decomposition. This procedure involves two stages. First, the method employs an ANOVA decomposition to identify the effective dimensions, i.e., subspaces of the parameter space in which the contributions to the solution are larger, and sort the reduced basis solution in a descending order of error. Then, the adaptive search refines the parametric space by increasing the order of polynomials until the algorithm is terminated by a saturation constraint. We demonstrate the effectiveness of the proposed algorithm for solving a stationary stochastic convection-diffusion equation, a benchmark problem chosen because solutions contain steep boundary layers and anisotropic features. We show that two stages of adaptivity are critical in a benchmark problem with anisotropic stochasticity.