Bayesian Quadrature for Multiple Related Integrals

TL;DR

This paper introduces a Bayesian quadrature method for efficiently computing multiple related integrals, enabling better uncertainty quantification and information transfer across related numerical problems.

Contribution

It extends Bayesian quadrature to handle multiple related integrals, providing convergence guarantees and demonstrating improved efficiency in engineering and graphics applications.

Findings

Enhanced numerical efficiency in multi-fidelity models

Convergence rates established for the proposed method

Effective uncertainty representation across related integrals

Abstract

Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to incomplete/finite information about the continuous mathematical problem being approximated. In this paper, we demonstrate that this paradigm can provide additional advantages, such as the possibility of transferring information between several numerical methods. This allows users to represent uncertainty in a more faithful manner and, as a by-product, provide increased numerical efficiency. We propose the first such numerical method by extending the well-known Bayesian quadrature algorithm to the case where we are interested in computing the integral of several related functions. We then prove convergence rates for the method in the well-specified and misspecified cases, and demonstrate its efficiency in the context of multi-fidelity models for complex engineering systems and a problem of global illumination in computer graphics.