数学物理学报(英文版)

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TRACTABILITY OF MULTIVARIATE INTEGRATION PROBLEM FOR PERIODIC CONTINUOUS FUNCTIONS

房艮孙; 龙晶凡   

  1. 北京师范大学数学学院, 北京 100875
  • 收稿日期:2005-10-19 修回日期:2006-03-10 出版日期:2007-10-20 发布日期:2007-10-20
  • 通讯作者: 房艮孙
  • 基金资助:

    Project supported by the National Natural Science Foundation of China (10671019), Research Fund for the Doctoral Program Higher Education(20050027007), and Beijing Educational Committee (2002Kj112)

TRACTABILITY OF MULTIVARIATE INTEGRATION PROBLEM FOR PERIODIC CONTINUOUS FUNCTIONS

Fang Gensun; Long Jingfan   

  1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
  • Received:2005-10-19 Revised:2006-03-10 Online:2007-10-20 Published:2007-10-20
  • Contact: Fang Gensun

摘要:

The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted 1-periodic continuous functions spaces of d coordinates with absolutely convergent Fourier series. The authors reduce the initial error by a factor ε for functions from the unit ball of the weighted periodic continuous functions spaces. Tractability is the minimal number of function samples required to solve the problem in polynomial in ε-1 and d, and the strong tractability is the presence of only a polynomial dependence in ε-1. This problem has been recently studied for quasi-Monte Carlo quadrature rules, quadrature rules with non-negative coefficients, and rules for which all quadrature weights are arbitrary for weighted Korobov spaces of smooth periodic functions of d variables. The authors show that the tractability and strong tractability of a multivariate integration problem in worst case setting hold for the weighted periodic
continuous functions spaces with absolutely convergent Fourier series under the same assumptions as in Ref.[14] on the weights of the Korobov space for quasi-Monte Carlo rules and rules for which all quadrature weights are non-negative. The arguments are not constructive.

关键词: Information-based complexity, tractability, Monte Carlo methods, multivariate integration

Abstract:

The authors study the tractability and strong tractability of a multivariate integration problem in the worst case setting for weighted 1-periodic continuous functions spaces of d coordinates with absolutely convergent Fourier series. The authors reduce the initial error by a factor ε for functions from the unit ball of the weighted periodic continuous functions spaces. Tractability is the minimal number of function samples required to solve the problem in polynomial in ε-1 and d, and the strong tractability is the presence of only a polynomial dependence in ε-1. This problem has been recently studied for quasi-Monte Carlo quadrature rules, quadrature rules with non-negative coefficients, and rules for which all quadrature weights are arbitrary for weighted Korobov spaces of smooth periodic functions of d variables. The authors show that the tractability and strong tractability of a multivariate integration problem in worst case setting hold for the weighted periodic
continuous functions spaces with absolutely convergent Fourier series under the same assumptions as in Ref.[14] on the weights of the Korobov space for quasi-Monte Carlo rules and rules for which all quadrature weights are non-negative. The arguments are not constructive.

Key words: Information-based complexity, tractability, Monte Carlo methods, multivariate integration

中图分类号: 

  • 41A55