# Abdul-Lateef Haji-Ali: Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation

**Time: **
Wed 2018-08-08 10.15 - 11.00

**Lecturer: **
Abdul-Lateef Haji-Ali, University of Oxford

**Location: ** KTH Mathematics, Lindstedtsvägen 25, floor 4, room 3418

Abstract:

In this talk, I will talk about our recent work where we address the approximation of functionals depending on a system of particles, described by stochastic differential equations (SDEs), in the mean-field limit when the number of particles is infinite. This problem is equivalent to estimating the weak solution of the limiting McKean-Vlasov SDE. To that end, our approach uses systems with finite numbers of particles and a time-stepping scheme. In this setting, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo, Multilevel Monte Carlo (MLMC) and Multi-Index Monte Carlo methods and show that, based on some assumptions that are verified numerically, we are able to achieve a near-optimal work complexity in a typical setting. I will also discuss the theoretical challenges involved when proving the necessary assumptions for our methods and some more recent results in that direction.

References:

— "Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation", by A. L. Haji Ali and R. Tempone. *arXiv:1610.09934*, October 2016. Statistics and Computing, 2017.

— "Multi Index Monte Carlo: When Sparsity Meets Sampling", by A.-L. Haji-Ali, F. Nobile, and R. Tempone. Numerische Mathematik, Vol. 132(4), Pages 767--806, 2016.