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

Handle URI:
http://hdl.handle.net/10754/625499
Title:
Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation
Authors:
Haji-Ali, Abdul-Lateef ( 0000-0002-6243-0335 ) ; Tempone, Raul ( 0000-0003-1967-4446 )
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.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
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.
Publisher:
Springer Nature
Journal:
Statistics and Computing
Issue Date:
12-Sep-2017
DOI:
10.1007/s11222-017-9771-5
Type:
Article
ISSN:
0960-3174; 1573-1375
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.
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

DC FieldValue Language
dc.contributor.authorHaji-Ali, Abdul-Lateefen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2017-09-21T09:25:34Z-
dc.date.available2017-09-21T09:25:34Z-
dc.date.issued2017-09-12en
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.en
dc.identifier.issn0960-3174en
dc.identifier.issn1573-1375en
dc.identifier.doi10.1007/s11222-017-9771-5en
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.en
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.en
dc.publisherSpringer Natureen
dc.subjectMulti-index Monte Carloen
dc.subjectMultilevel Monte Carloen
dc.subjectMonte Carloen
dc.subjectParticle systemsen
dc.subjectMcKean–Vlasoven
dc.subjectMean-fielden
dc.subjectStochastic differential equationsen
dc.subjectWeak approximationen
dc.subjectSparse approximationen
dc.subjectCombination techniqueen
dc.titleMultilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equationen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalStatistics and Computingen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionMathematical Institute, University of Oxford, Oxford, UKen
kaust.authorTempone, Raulen