Monte Carlo methods are one of the main tools of modern statistics and applied mathematics. They are commonly used to approximate integrals, which allows statisticians to solve many tasks of interest such as making predictions or inferring parameter values of a given model. However, the recent surge in data available to scientists has led to an increase in the complexity of mathematical models, rendering them much more computationally expensive to evaluate. This has a particular bearing on Monte Carlo methods, which will tend to be much slower due to the high computational costs.
This talk will introduce a Monte Carlo integration scheme which makes use of properties of the integrand (e.g. smoothness or periodicity) in order to obtain fast convergence rates in the number of integrand evaluations. This will allow users to obtain much more precise estimates of integrals for a given number of model evaluations. Both theoretical properties of the methodology, including convergence rates, and practical issues, such as the tuning of parameters, will be discussed. Finally, the proposed algorithm will be illustrated on a Bayesian inverse problem for a PDE model of subsurface flow.