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A Particle Method for Fokker-Planck Equations in High Dimensions

  • The dissertation is concerned with the numerical solution of Fokker-Planck equations in high dimensions arising in the study of dynamics of polymeric liquids. Traditional methods based on tensor product structure are not applicable in high dimensions for the number of nodes required to yield a fixed accuracy increases exponentially with the dimension; a phenomenon often referred to as the curse of dimension. Particle methods or finite point set methods are known to break the curse of dimension. The Monte Carlo method (MCM) applied to such problems are 1/sqrt(N) accurate, where N is the cardinality of the point set considered, independent of the dimension. Deterministic version of the Monte Carlo method called the quasi Monte Carlo method (QMC) are quite effective in integration problems and accuracy of the order of 1/N can be achieved, up to a logarithmic factor. However, such a replacement cannot be carried over to particle simulations due to the correlation among the quasi-random points. The method proposed by Lecot (C.Lecot and F.E.Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity, 15 (1999), pp.342-359) is the only known QMC approach, but it not only leads to large particle numbers but also the proven order of convergence is 1/N^(2s) in dimension s. We modify the method presented there, in such a way that the new method works with reasonable particle numbers even in high dimensions and has better order of convergence. Though the provable order of convergence is 1/sqrt(N), the results show less variance and thus the proposed method still slightly outperforms standard MCM.
  • Eine Partikel Methode für Fokker-Planck Gleichungen in hohen Dimensionen

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Verfasserangaben:Venkiteswaran Gopalakrishnan
URN (Permalink):urn:nbn:de:bsz:386-kluedo-15786
Betreuer:Michael Junk
Sprache der Veröffentlichung:Englisch
Jahr der Fertigstellung:2002
Jahr der Veröffentlichung:2002
Veröffentlichende Institution:Technische Universität Kaiserslautern
Titel verleihende Institution:Technische Universität Kaiserslautern
Datum der Annahme der Abschlussarbeit:31.01.2003
Datum der Publikation (Server):16.06.2003
Freies Schlagwort / Tag:Diffusion; Partikel Methoden; QMC
Diffusion; Monte Carlo; discrepancy; quasi-Monte Carlo
GND-Schlagwort:Fokker-Planck-Gleichung; Monte Carlo
Fachbereiche / Organisatorische Einheiten:Fachbereich Mathematik
DDC-Sachgruppen:5 Naturwissenschaften und Mathematik / 510 Mathematik
Lizenz (Deutsch):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011