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dc.contributor.authorHaji Ali, Abdul Lateef
dc.contributor.authorTempone, Raul
dc.date.accessioned2017-09-21T09:25:34Z
dc.date.available2017-09-21T09:25:34Z
dc.date.issued2017-09-12
dc.identifier.citationHaji-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.issn0960-3174
dc.identifier.issn1573-1375
dc.identifier.doi10.1007/s11222-017-9771-5
dc.identifier.urihttp://hdl.handle.net/10754/625499
dc.description.abstractWe 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.sponsorshipR. 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.publisherSpringer Nature
dc.relation.urlhttp://link.springer.com/article/10.1007/s11222-017-9771-5
dc.rightsThis 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.urihttp://creativecommons.org/licenses/by/4.0
dc.subjectMulti-index Monte Carlo
dc.subjectMultilevel Monte Carlo
dc.subjectMonte Carlo
dc.subjectParticle systems
dc.subjectMcKean–Vlasov
dc.subjectMean-field
dc.subjectStochastic differential equations
dc.subjectWeak approximation
dc.subjectSparse approximation
dc.subjectCombination technique
dc.titleMultilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation
dc.typeArticle
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
dc.identifier.journalStatistics and Computing
dc.eprint.versionPublisher's Version/PDF
dc.contributor.institutionMathematical Institute, University of Oxford, Oxford, UK
dc.identifier.arxividarXiv:1610.09934
kaust.personTempone, Raul
refterms.dateFOA2018-06-13T12:16:27Z


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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.
Except where otherwise noted, this item's license is described as 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.