Mathematical Colloquium

Friday, June 30 2017, 2:15pm

Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation

Raul Tempone (King Abdullah University of Science and Technology)

Abstract:
Our goal here is to approximate functionals of a system of a large number of particles, described by a coupled system of Ito stochastic differential equations (SDEs).
To this end, our Monte Carlo simulations use systems with finite numbers of particles and the Euler-Maruyama time-stepping scheme. In this case, there are two discretization parameters: the number of time steps and the number of particles.
Based on these two discretization parameters, we consider different variants of the Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that the optimal work complexity of MLMC to estimate a given smooth functional in a standard setting with an error tolerance of $\tol$ is $\Order{\tol^{-3}}$. We also propose a partitioning estimator that applies our novel Multi-index Monte Carlo method and show an improved work complexity in the same typical setting of $\Order{\tol^{-2}\log(\tol)^2}$. Our numerical results with a Kuramoto system of oscillators provide a complete agreement with the outlined theory.
Keywords: Multilevel Monte Carlo, Monte Carlo, Particle systems, McKean-Vlasov, Mean-field, Stochastic Differential Equations, Weak Approximation, Sparse Approximation, Combination technique
Subclass: 65C05, 65N30, 65N22
References:
"Multilevel and Multi-index Monte Carlo methods for the McKean -Vlasov equation," by  A. L. Haji Ali and R. Tempone. \emph{arXiv:1610.09934 }, October 2016.

Ort:  Raum 008/SeMath, Pontdriesch 14-16, 52062 Aachen