Abstract

Multilevel Monte Carlo (MLMC) is a variance reduction technique for stochastic simulation and Bayesian inference which greatly reduces the computational cost of standard Monte Carlo approaches by employing cheap, coarse-scale models with lower fidelity to carry out the bulk of the stochastic simulations, while maintaining the overall accuracy of the fine scale model through a small number of well-chosen high fidelity simulations. In this talk, I will first review the ideas behind the approach and discuss a number of applications and extensions that illustrate the generality of the approach. The multilevel Monte Carlo method (in its practical form) has originally been introduced and popularised about 10 years ago by Mike Giles for stochastic differential equations in mathematical finance and has attracted a lot of interest in the context of uncertainty quantification of physical systems modelled by partial differential equations (PDEs). The underlying idea had actually been discovered 10 years earlier in 1998, in an information-theoretical paper by Stefan Heinrich, but had remained largely unknown until 2008. In recent years, there has been an explosion of activity and its application has been extended, among others, to biological/chemical reaction networks, plasma physics, interacting particle systems as well as to nested simulations. More importantly for this community, the approach has also been extended to Markov chain Monte Carlo, sequential Monte Carlo and other filtering techniques. In the second part of the talk, I will describe in more detail how the MLMC framework can provide a computationally tractable methodology for Bayesian inference in high-dimensional models constrained by PDEs and demonstrate the potential on a toy problem in the context of Metropolis-Hastings MCMC . Finally, I will finish the talk with some perspectives beyond the classical MLMC framework, in particular using sample-dependent model hierarchies and a posteriori error estimators and extending the classical discrete, level-based approach to a new Continuous Level Monte Carlo method.