Sequential Monte Carlo Methods for Bayesian Elliptic Inverse Problems

TL;DR

This paper develops and analyzes a dimension-independent Sequential Monte Carlo method for Bayesian inverse problems involving elliptic PDEs, demonstrating its effectiveness through theoretical proofs and numerical experiments.

Contribution

It introduces an enhanced SMC algorithm tailored for elliptic inverse problems and proves its convergence rate is independent of discretization dimension.

Findings

SMC method converges at a Monte Carlo rate with dimension-independent constants.

The enhanced SMC effectively handles the complexity of elliptic inverse problems.

Numerical experiments confirm the theoretical properties and practical efficiency.

Abstract

In this article we consider a Bayesian inverse problem associated to elliptic partial differential equations (PDEs) in two and three dimensions. This class of inverse problems is important in applications such as hydrology, but the complexity of the link function between unknown field and measurements can make it difficult to draw inference from the associated posterior. We prove that for this inverse problem a basic SMC method has a Monte Carlo rate of convergence with constants which are independent of the dimension of the discretization of the problem; indeed convergence of the SMC method is established in a function space setting. We also develop an enhancement of the sequential Monte Carlo (SMC) methods for inverse problems which were introduced in \cite{kantas}; the enhancement is designed to deal with the additional complexity of this elliptic inverse problem. The efficacy of the methodology, and its desirable theoretical properties, are demonstrated on numerical examples in both two and three dimensions.