Bayesian Quadrature for Multiple Related Integrals
This work addresses the need for more efficient and accurate uncertainty quantification in numerical integration for domains like engineering and graphics, though it is incremental as it builds on existing Bayesian quadrature.
The paper tackles the problem of computing integrals for multiple related functions by extending Bayesian quadrature, enabling information transfer between numerical methods to improve uncertainty representation and numerical efficiency. It demonstrates the method's effectiveness in multi-fidelity engineering models and computer graphics, with proven convergence rates.
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.