A Monte Carlo approach to computing stiffness matrices arising in polynomial chaos approximations

TL;DR

This paper introduces a Monte Carlo-based method for assembling finite element matrices in polynomial chaos approximations of elliptic equations with random coefficients, simplifying computations by using sparse matrices.

Contribution

It presents a novel Monte Carlo approach that replaces full matrices with sparse block-diagonal matrices, generalizing classical Monte Carlo and collocation methods for stochastic elliptic problems.

Findings

Efficient assembly of finite element matrices using Monte Carlo

Reduction in computational complexity with sparse matrices

Generalization of classical Monte Carlo and collocation methods

Abstract

We use a Monte Carlo method to assemble finite element matrices for polynomial Chaos approximations of elliptic equations with random coefficients. In this approach, all required expectations are approximated by a Monte Carlo method. The resulting methodology requires dealing with sparse block-diagonal matrices instead of block-full matrices. This leads to the solution of a coupled system of elliptic equations where the coupling is given by a Kronecker product matrix involving polynomial evaluation matrices. This generalizes the Classical Monte Carlo approximation and Collocation method for approximating functionals of solutions of these equations.