dc.contributor.author Haji Ali, Abdul Lateef dc.contributor.author Tempone, Raul dc.date.accessioned 2017-09-21T09:25:34Z dc.date.available 2017-09-21T09:25:34Z dc.date.issued 2017-09-12 dc.identifier.citation Haji-Ali A-L, Tempone R (2017) Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation. Statistics and Computing. Available: http://dx.doi.org/10.1007/s11222-017-9771-5. dc.identifier.issn 0960-3174 dc.identifier.issn 1573-1375 dc.identifier.doi 10.1007/s11222-017-9771-5 dc.identifier.uri http://hdl.handle.net/10754/625499 dc.description.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 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. dc.description.sponsorship 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. dc.publisher Springer Nature dc.relation.url http://link.springer.com/article/10.1007/s11222-017-9771-5 dc.rights This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. dc.rights.uri http://creativecommons.org/licenses/by/4.0 dc.subject Multi-index Monte Carlo dc.subject Multilevel Monte Carlo dc.subject Monte Carlo dc.subject Particle systems dc.subject McKean–Vlasov dc.subject Mean-field dc.subject Stochastic differential equations dc.subject Weak approximation dc.subject Sparse approximation dc.subject Combination technique dc.title Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation dc.type Article dc.contributor.department Applied Mathematics and Computational Science Program dc.contributor.department Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division dc.identifier.journal Statistics and Computing dc.eprint.version Publisher's Version/PDF dc.contributor.institution Mathematical Institute, University of Oxford, Oxford, UK dc.identifier.arxivid arXiv:1610.09934 kaust.person Tempone, Raul refterms.dateFOA 2018-06-13T12:16:27Z
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