Quasi-Monte Carlo for Bayesian design of experiment problems governed by parametric PDEs

TL;DR

This paper develops and analyzes quasi-Monte Carlo methods for Bayesian optimal experimental design problems governed by PDEs, demonstrating improved convergence rates through sparse tensor product approaches.

Contribution

It introduces a detailed analysis of QMC cubature rules for high-dimensional Bayesian PDE design problems, showing enhanced convergence with sparse tensor methods.

Findings

Sparse tensor product QMC significantly improves convergence rates.

Numerical experiments confirm theoretical convergence predictions.

Method is practically applicable to elliptic PDE problems with unknown coefficients.

Abstract

This paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration problems over high-dimensional parameter and data domains arising in Bayesian optimal design problems. We provide a detailed analysis for these double integration problems using two approaches: a full tensor product and a sparse tensor product combination of quasi-Monte Carlo (QMC) cubature rules over the parameter and data domains. Specifically, we show that the latter approach significantly improves the convergence rate, exhibiting performance comparable to that of QMC integration of a single high-dimensional integral. Furthermore, we numerically verify the predicted convergence rates for an elliptic PDE problem with an unknown diffusion coefficient in two spatial dimensions, offering empirical evidence supporting the theoretical results and highlighting practical applicability.