Quasi-Monte Carlo for Bayesian shape inversion governed by the Poisson problem subject to Gevrey regular domain deformations

TL;DR

This paper develops a quasi-Monte Carlo method for efficient Bayesian shape inversion governed by the Poisson problem, leveraging Gevrey regularity to achieve faster convergence in high-dimensional integrals.

Contribution

It introduces a novel quasi-Monte Carlo cubature rule tailored for Bayesian shape inversion with Gevrey regular domain deformations, improving convergence rates over traditional Monte Carlo methods.

Findings

Achieves dimension-independent, faster convergence rates for high-dimensional integrals.

Validates theoretical convergence results through numerical experiments.

Analyzes effects of dimension truncation and finite element errors.

Abstract

We consider the application of a quasi-Monte Carlo cubature rule to Bayesian shape inversion subject to the Poisson equation under Gevrey regular parameterizations of domain uncertainty. We analyze the parametric regularity of the associated posterior distribution and design randomly shifted rank-1 lattice rules which can be shown to achieve dimension-independent, faster-than-Monte Carlo cubature convergence rates for high-dimensional integrals over the posterior distribution. In addition, we consider the effect of dimension truncation and finite element discretization errors for this model. Finally, a series of numerical experiments are presented to validate the theoretical results.