High-dimensional integration: The quasi-Monte Carlo way

DOI Web Site 被引用文献15件

書誌事項

公開日
2013-04-02
権利情報
  • https://www.cambridge.org/core/terms
DOI
  • 10.1017/s0962492913000044
公開者
Cambridge University Press (CUP)

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説明

<jats:p>This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]<jats:italic><jats:sup>s</jats:sup></jats:italic>, where<jats:italic>s</jats:italic>may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.</jats:p>

収録刊行物

  • Acta Numerica

    Acta Numerica 22 133-288, 2013-04-02

    Cambridge University Press (CUP)

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