Application of adaptive ANOVA and reduced basis methods to the stochastic Stokes-Brinkman problem

TL;DR

This paper combines adaptive ANOVA, stochastic collocation, and reduced basis methods to efficiently solve the stochastic Stokes-Brinkman equations modeling fluid flow in heterogeneous porous media, reducing computational costs while maintaining accuracy.

Contribution

It introduces an adaptive ANOVA approach with reduced basis methods for the stochastic Stokes-Brinkman problem, enabling efficient computation of statistical moments in high-dimensional stochastic spaces.

Findings

Adaptive ANOVA reduces collocation points needed.

Reduced basis methods significantly lower computational cost.

Effective for both isotropic and anisotropic permeabilities.

Abstract

The Stokes-Brinkman equations model fluid flow in highly heterogeneous porous media. In this paper, we consider the numerical solution of the Stokes-Brinkman equations with stochastic permeabilities, where the permeabilities in subdomains are assumed to be independent and uniformly distributed within a known interval. We employ a truncated anchored ANOVA decomposition alongside stochastic collocation to estimate the moments of the velocity and pressure solutions. Through an adaptive procedure selecting only the most important ANOVA directions, we reduce the number of collocation points needed for accurate estimation of the statistical moments. However, for even modest stochastic dimensions, the number of collocation points remains too large to perform high-fidelity solves at each point. We use reduced basis methods to alleviate the computational burden by approximating the expensive high-fidelity solves with inexpensive approximate solutions on a low-dimensional space. We furthermore develop and analyze rigorous a posteriori error estimates for the reduced basis approximation. We apply these methods to 2D problems considering both isotropic and anisotropic permeabilities.