Quasi-Monte Carlo algorithms for diffusion equations in high dimensions

Venkiteswaran, Gopalakrishnan; Junk, Michael

Quasi-Monte Carlo algorithms for diffusion equations in high dimensions

Files

There are no files associated with this item.

Date

2005

Authors

Venkiteswaran, Gopalakrishnan

Junk, Michael

Publication type

Journal article

Published in

Mathematics and Computers in Simulation ; 68 (2005), 1. - pp. 23-41. - ISSN 0378-4754. - eISSN 1872-7166

Abstract

Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is N^−1/2 accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve N⁻¹ convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.

Subject (DDC)

510 Mathematics

Cite This

ISO 690

VENKITESWARAN, Gopalakrishnan, Michael JUNK, 2005. Quasi-Monte Carlo algorithms for diffusion equations in high dimensions. In: Mathematics and Computers in Simulation. 68(1), pp. 23-41. ISSN 0378-4754. eISSN 1872-7166. Available under: doi: 10.1016/j.matcom.2004.09.003

BibTex

@article{Venkiteswaran2005Quasi-25405,
  year={2005},
  doi={10.1016/j.matcom.2004.09.003},
  title={Quasi-Monte Carlo algorithms for diffusion equations in high dimensions},
  number={1},
  volume={68},
  issn={0378-4754},
  journal={Mathematics and Computers in Simulation},
  pages={23--41},
  author={Venkiteswaran, Gopalakrishnan and Junk, Michael}
}

RDF

<rdf:RDF
    xmlns:dcterms="http://purl.org/dc/terms/"
    xmlns:dc="http://purl.org/dc/elements/1.1/"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:bibo="http://purl.org/ontology/bibo/"
    xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#"
    xmlns:foaf="http://xmlns.com/foaf/0.1/"
    xmlns:void="http://rdfs.org/ns/void#"
    xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > 
  <rdf:Description rdf:about="https://kops.uni-konstanz.de/server/rdf/resource/123456789/25405">
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:language>eng</dc:language>
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dc:contributor>Venkiteswaran, Gopalakrishnan</dc:contributor>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2013-12-13T10:56:08Z</dc:date>
    <bibo:uri rdf:resource="http://kops.uni-konstanz.de/handle/123456789/25405"/>
    <dc:contributor>Junk, Michael</dc:contributor>
    <dc:rights>terms-of-use</dc:rights>
    <dc:creator>Junk, Michael</dc:creator>
    <dc:creator>Venkiteswaran, Gopalakrishnan</dc:creator>
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2013-12-13T10:56:08Z</dcterms:available>
    <dcterms:bibliographicCitation>Mathematics and Computers in Simulation ; 68 (2005), 1. - S. 23-41</dcterms:bibliographicCitation>
    <dcterms:abstract xml:lang="eng">Diffusion equation posed on a high dimensional space may occur as a sub-problem in advection-diffusion problems (see [G. Venkiteswaran, M. Junk, A QMC approach for high dimensional Fokker–Planck equations modelling polymeric liquids, Math. Comput. Simul. 68 (2005) 43–56.] for a specific application). Although the transport part can be dealt with the method of characteristics, the efficient simulation of diffusion in high dimensions is a challenging task. The traditional Monte Carlo method (MC) applied to diffusion problems converges and is N&lt;sup&gt;−1/2&lt;/sup&gt; accurate, where N is the number of particles. It is well known that for integration, quasi-Monte Carlo (QMC) outperforms Monte Carlo in the sense that one can achieve N&lt;sup&gt;−1&lt;/sup&gt; convergence, up to a logarithmic factor. This is our starting point to develop methods based on Lécot’s approach [C. Lécot, F.E. Khettabi, Quasi-Monte Carlo simulation of diffusion, Journal of Complexity 15 (1999) 342–359.], which are applicable in high dimensions, with a hope to achieve better speed of convergence. Through a number of numerical experiments we observe that some of the QMC methods not only generalize to high dimensions but also show faster convergence in the results and thus, slightly outperform standard MC.</dcterms:abstract>
    <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <dcterms:title>Quasi-Monte Carlo algorithms for diffusion equations in high dimensions</dcterms:title>
    <dcterms:issued>2005</dcterms:issued>
  </rdf:Description>
</rdf:RDF>

Bibliography of Konstanz

Yes