Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Raúl Tempone, Leon Wilkosz
{"title":"通过 McKean-Vlasov 动力学向前传播低差异:从 QMC 到 MLQMC","authors":"Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Raúl Tempone, Leon Wilkosz","doi":"arxiv-2409.09821","DOIUrl":null,"url":null,"abstract":"This work develops a particle system addressing the approximation of\nMcKean-Vlasov stochastic differential equations (SDEs). The novelty of the\napproach lies in involving low discrepancy sequences nontrivially in the\nconstruction of a particle system with coupled noise and initial conditions.\nWeak convergence for SDEs with additive noise is proven. A numerical study\ndemonstrates that the novel approach presented here doubles the respective\nconvergence rates for weak and strong approximation of the mean-field limit,\ncompared with the standard particle system. These rates are proven in the\nsimplified setting of a mean-field ordinary differential equation in terms of\nappropriate bounds involving the star discrepancy for low discrepancy sequences\nwith a group structure, such as Rank-1 lattice points. This construction\nnontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An\nasymptotic error analysis reveals that the proposed approach outperforms\nmethods based on the classic particle system with independent initial\nconditions and noise.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"19 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC\",\"authors\":\"Nadhir Ben Rached, Abdul-Lateef Haji-Ali, Raúl Tempone, Leon Wilkosz\",\"doi\":\"arxiv-2409.09821\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This work develops a particle system addressing the approximation of\\nMcKean-Vlasov stochastic differential equations (SDEs). The novelty of the\\napproach lies in involving low discrepancy sequences nontrivially in the\\nconstruction of a particle system with coupled noise and initial conditions.\\nWeak convergence for SDEs with additive noise is proven. A numerical study\\ndemonstrates that the novel approach presented here doubles the respective\\nconvergence rates for weak and strong approximation of the mean-field limit,\\ncompared with the standard particle system. These rates are proven in the\\nsimplified setting of a mean-field ordinary differential equation in terms of\\nappropriate bounds involving the star discrepancy for low discrepancy sequences\\nwith a group structure, such as Rank-1 lattice points. This construction\\nnontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An\\nasymptotic error analysis reveals that the proposed approach outperforms\\nmethods based on the classic particle system with independent initial\\nconditions and noise.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"19 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09821\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09821","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Forward Propagation of Low Discrepancy Through McKean-Vlasov Dynamics: From QMC to MLQMC
This work develops a particle system addressing the approximation of
McKean-Vlasov stochastic differential equations (SDEs). The novelty of the
approach lies in involving low discrepancy sequences nontrivially in the
construction of a particle system with coupled noise and initial conditions.
Weak convergence for SDEs with additive noise is proven. A numerical study
demonstrates that the novel approach presented here doubles the respective
convergence rates for weak and strong approximation of the mean-field limit,
compared with the standard particle system. These rates are proven in the
simplified setting of a mean-field ordinary differential equation in terms of
appropriate bounds involving the star discrepancy for low discrepancy sequences
with a group structure, such as Rank-1 lattice points. This construction
nontrivially provides an antithetic multilevel quasi-Monte Carlo estimator. An
asymptotic error analysis reveals that the proposed approach outperforms
methods based on the classic particle system with independent initial
conditions and noise.