The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets

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• 作者：Ligia L. Cristea ; Josef Dick ; Gunther Leobacher and Friedrich Pillichshammer
• 关键词：Mathematics Subject Classification (1991) 11K38– ; 11K06
• 刊名：Numerische Mathematik
• 出版年：2007
• 出版时间：January, 2007
• 年：2007
• 卷：105
• 期：3
• 页码：413-455
• 全文大小：725 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245

文摘

In this paper we investigate multivariate integration in reproducing kernel Sobolev spaces for which the second partial derivatives are square integrable. As quadrature points for our quasi-Monte Carlo algorithm we use digital (t,m,s)-nets over \mathbbZ₂\mathbb{Z}_2 which are randomly digitally shifted and then folded using the tent transformation. For this QMC algorithm we show that the root mean square worst-case error converges with order 2m(-2+e)2^{m(-2+\varepsilon)} for any ɛ > 0, where 2 m is the number of points. A similar result for lattice rules has previously been shown by Hickernell.