Theoretical Guarantees for the Statistical Finite Element Method

TL;DR

This paper provides a theoretical analysis of the statistical finite element method (StatFEM), showing it converges similarly to traditional finite element methods and offering bounds on the Wasserstein-2 distance between prior and posterior distributions.

Contribution

It introduces a new theoretical framework for StatFEM, establishing convergence properties and bounds that were previously uncharacterized, especially in probabilistic terms.

Findings

StatFEM has convergence properties similar to classical FEM.

The Wasserstein-2 distance between prior/posterior and approximation converges at FEM rates.

Numerical examples confirm the theoretical bounds and robustness of StatFEM.

Abstract

The statistical finite element method (StatFEM) is an emerging probabilistic method that allows observations of a physical system to be synthesised with the numerical solution of a PDE intended to describe it in a coherent statistical framework, to compensate for model error. This work presents a new theoretical analysis of the statistical finite element method demonstrating that it has similar convergence properties to the finite element method on which it is based. Our results constitute a bound on the Wasserstein-2 distance between the ideal prior and posterior and the StatFEM approximation thereof, and show that this distance converges at the same mesh-dependent rate as finite element solutions converge to the true solution. Several numerical examples are presented to demonstrate our theory, including an example which test the robustness of StatFEM when extended to nonlinear quantities of interest.