Multilevel Bayesian Quadrature

TL;DR

This paper introduces a Bayesian surrogate modeling approach to enhance multilevel Monte Carlo methods for expensive integrals, demonstrating improved accuracy especially for smooth, low to moderate dimensional problems.

Contribution

It proposes integrating Gaussian process-based Bayesian quadrature into multilevel Monte Carlo to improve efficiency and accuracy for costly scientific model integrations.

Findings

Significant accuracy improvements shown in theory and experiments.

Effective for smooth integrands with low to moderate dimensions.

Case study demonstrates practical impact in tsunami modeling.

Abstract

Multilevel Monte Carlo is a key tool for approximating integrals involving expensive scientific models. The idea is to use approximations of the integrand to construct an estimator with improved accuracy over classical Monte Carlo. We propose to further enhance multilevel Monte Carlo through Bayesian surrogate models of the integrand, focusing on Gaussian process models and the associated Bayesian quadrature estimators. We show, using both theory and numerical experiments, that our approach can lead to significant improvements in accuracy when the integrand is expensive and smooth, and when the dimensionality is small or moderate. We conclude the paper with a case study illustrating the potential impact of our method in landslide-generated tsunami modelling, where the cost of each integrand evaluation is typically too large for operational settings.