Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation
Bibliography:
Bibliography
A. Haji-Ali, R. Tempone, "Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov
equation", has been accepted for publication in Statistics and
Computing. 2017
Authors:
A. Haji-Ali, R. Tempone
Keywords:
Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation
Abstract:
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 approaches infinity. 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 an 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 parameters, we consider different variants of the
Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that, in the
best case, the optimal work complexity of MLMC to estimate the functional in
one typical setting with an error tolerance of TOL is O(TOL−3). We also consider a method that uses the
recent Multi-index Monte Carlo method and show an improved work complexity in
the same typical setting of O(TOL−2log(TOL−1)2) when using a partitioning
estimator. Our numerical experiments are carried out on the so-called Kuramoto
model, a system of coupled oscillators.
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