Numerical computation of intractable integrals is an ubiquitous challenge in all areas of scientific computing, statistics, machine learning, and engineering simulation. This project develops methodology and theory for a class of numerical integration algorithms known as Bayesian cubatures, which are probabilistic integration methods that model the integrand function as a Gaussian process and use statistical inference to assign a Gaussian probability distribution for the integral. The standard deviation of this distribution provides quantification of epistemic uncertainty for the unknown true value of the integral and can be propagated and combined with other sources of uncertainty in computational pipelines. The main objectives are to (a) develop and implement scalable and adaptive Bayesian cubature methods not bound by cubic computational complexity of Gaussian process methods and (b) prove mathematically rigorous results on reliability of uncertainty quantification they provide.

2021

2024