Quasi-Monte Carlo and Multilevel Monte Carlo Methods for Computing Posterior Expectations in Elliptic Inverse Problems

TL;DR

This paper analyzes the use of Quasi-Monte Carlo and Multilevel Monte Carlo methods to efficiently compute posterior expectations in elliptic inverse problems, providing convergence analysis and numerical validation.

Contribution

It offers a comprehensive convergence and complexity analysis of ratio estimators using QMC and MLMC for Bayesian PDE inverse problems, with numerical demonstrations.

Findings

Complexity of ratio estimator matches individual estimators.

Numerical simulations confirm effectiveness of the methods.

Analysis applies to model elliptic problems.

Abstract

We are interested in computing the expectation of a functional of a PDE solution under a Bayesian posterior distribution. Using Bayes' rule, we reduce the problem to estimating the ratio of two related prior expectations. For a model elliptic problem, we provide a full convergence and complexity analysis of the ratio estimator in the case where Monte Carlo, quasi-Monte Carlo or multilevel Monte Carlo methods are used as estimators for the two prior expectations. We show that the computational complexity of the ratio estimator to achieve a given accuracy is the same as the corresponding complexity of the individual estimators for the numerator and the denominator. We {also include numerical simulations, in the context of the model elliptic problem, which demonstrate the effectiveness of the approach.