Examples of computational approaches to accommodate randomness in elliptic PDEs

TL;DR

This paper reviews recent computational methods for solving linear elliptic PDEs with randomness, focusing on variance reduction and flexible finite element techniques for multiscale and fluctuating geometries.

Contribution

It introduces and evaluates new numerical approaches for efficiently simulating elliptic PDEs with random coefficients and geometries, emphasizing their practical effectiveness.

Findings

Variance reduction techniques improve accuracy at lower computational costs.

Finite element methods can be adapted to random geometries.

Numerical experiments demonstrate the efficiency of proposed methods.

Abstract

We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of multiscale problems, but the purpose is more general. On a set of prototypical situations, we investigate two critical issues present in many settings: variance reduction techniques to obtain sufficiently accurate results at a limited computational cost when solving PDEs with random coefficients, and finite element techniques that are sufficiently flexible to carry over to geometries with random fluctuations. Some elements of theoretical analysis and numerical analysis are briefly mentioned. Numerical experiments, although simple, provide convincing evidence of the efficiency of the approaches.