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 a timestepping 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 when using the partitioning estimator and the Milstein time-stepping scheme. 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 −2 log(TOL −1 ) 2 . Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.

A Fourier-based Picard-iteration approach for a class of McKean–Vlasov SDEs with Lévy jumps

“…We also have the following estimates on the quantities arising inμ X (α,β) t , which will be used repeatedly throughout the paper. The first estimate (12) for Φ can be obtained immediately from Grönwall's lemma and the definition of Φ in (11).The other results can be deduced similarly and the details are postponed until Appendix A.…”

“…It is usually shown that, in the limit N → ∞, the empirical law of a Markovian system with N interacting particles, converges to the law of the solution to the MK-V SDE under study, which can be then approximated via timediscretization and simulation. Following the work by Sznitman in [21], where the first propagation of chaos result was proved, many authors have contributed to this stream of literature, see, for example, [2], [23], [11], and [20], by supposing different forms and regularity assumptions on the MK-V SDE coefficients. Although a very powerful approximating tool which is applicable in many settings, the simulation of large-particle systems can be computationally very expensive.…”

New particle representations for ergodic McKean-Vlasov SDEs

“…The particle systems j and j ′ , j = j ′ , driven by (w i,j ) i,j and (w i,j ′ ) i,j respectively, are independent. This idea for finite time simulations has been proposed in [HAT16]. Another approach that we investigate are self-interacting diffusion…”

Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators

“…Second, imposing slightly stricter regularity assumptions, we can prove better asymptotic efficiency results for this method than for the original one. And third, creating a rigorous convergence theory for the new MLEnKF method is a stepping stone towards a multi-index Ensemble Kalman filtering (MIEnKF) method; See [19] for highly efficient approximations of McKean-Vlasov dynamics by the multi-index Monte Carlo method, and Appendix D for a sketch of the said extension to MIEnKF.…”

“…Following the construction of multi-index Monte Carlo mehods for mean-field dynamics approximations [19], we sketch an extension from MLEnKF to MIEnKF. On the particle level, this extension may be viewed as an extension from two-particle coupling to four-particle coupling.…”