TY - JOUR

T1 - Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

AU - Haji Ali, Abdul Lateef

AU - Tempone, Raul

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: R. Tempone is a member of the KAUST Strategic Research Initiative, Center for Uncertainty Quantification in Computational Sciences and Engineering. R. Tempone received support from the KAUST CRG3 Award Ref: 2281 and the KAUST CRG4 Award Ref: 2584. The authors would like to thank Lukas Szpruch for the valuable discussions regarding the theoretical foundations of the methods.

PY - 2017/9/12

Y1 - 2017/9/12

N2 - 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 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 $$\mathrm {TOL}$$TOL, is 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 . Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.

AB - 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 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 $$\mathrm {TOL}$$TOL, is 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 . Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.

UR - http://hdl.handle.net/10754/625499

UR - http://link.springer.com/article/10.1007/s11222-017-9771-5

UR - http://www.scopus.com/inward/record.url?scp=85029165392&partnerID=8YFLogxK

U2 - 10.1007/s11222-017-9771-5

DO - 10.1007/s11222-017-9771-5

M3 - Article

SN - 0960-3174

VL - 28

SP - 923

EP - 935

JO - Statistics and Computing

JF - Statistics and Computing

IS - 4

ER -