Statistical Finite Elements via Interacting Particle Langevin Dynamics

TL;DR

This paper introduces a novel interacting particle Langevin algorithm for inverse PDE problems, leveraging statistical finite elements to jointly estimate parameters and solutions with improved computational efficiency.

Contribution

It develops a new joint estimation method combining statFEM and IPLA for inverse PDE problems, with comprehensive numerical validation and preconditioning strategies.

Findings

Effective parameter estimation in PDE models demonstrated.

Significant computational efficiency improvements shown.

Versatile approach for linear and nonlinear PDEs.

Abstract

In this paper, we develop a class of interacting particle Langevin algorithms to solve inverse problems for partial differential equations (PDEs). In particular, we leverage the statistical finite elements (statFEM) formulation to obtain a finite-dimensional latent variable statistical model where the parameter is that of the (discretised) forward map and the latent variable is the statFEM solution of the PDE which is assumed to be partially observed. We then adapt a recently proposed expectation-maximisation like scheme, interacting particle Langevin algorithm (IPLA), for this problem and obtain a joint estimation procedure for the parameters and the latent variables. We consider three main examples: (i) estimating the forcing for linear Poisson PDE, (ii) estimating diffusivity for linear Poisson PDE, and (iii) estimating the forcing for nonlinear Poisson PDE. We provide computational complexity estimates for forcing estimation in the linear case. We also provide comprehensive numerical experiments and preconditioning strategies that significantly improve the performance, showing that the proposed class of methods can be the choice for parameter inference in PDE models.