Multilevel Monte Carlo and beyond

Multilevel Monte Carlo (MLMC)  is a recently introduced variance reduction technique for stochastic simulation which greatly reduces the computational cost 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 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 PDEs. (The first, theoretical paper was by Stefan Heinrich in 1998.)  The approach has been extended to Markov chain Monte Carlo, sequential Monte Carlo and other filtering techniques. Among others, its application has been extended to biological/chemical reaction networks, plasma physics, interacting particle systems and more recently to nested simulations.

In the second half of the talk, I will go beyond the classical MLMC framework and use sample-dependent model hierarchies and a posteriori error estimators to efficiently estimate rare events (Multilevel Subset Simulation) as well as to extend the framework from the discrete, level-based approach to a new Continuous Level Monte Carlo (CLMC) method. These latter extensions are work in progress in collaboration with Gianluca Detommaso (Bath), Tim Dodwell (Exeter) and Daniel Elfverson (Umea).