High-discrepancy sequences

Shu Tezuka

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

First, it is pointed out that the uniform distribution of points in [0, 1]d is not always a necessary condition for every function in a proper subset of the class of all Riemann integrable functions to have the arithmetic mean of function values at the points converging to its integral over [0, 1]d as the number of points goes to infinity. We introduce a formal definition of the d-dimensional high-discrepancy sequences, which are not uniformly distributed in [0, 1]d, and present motivation for the application of these sequences to high-dimensional numerical integration. Then, we prove that there exist non-uniform (∞, d)-sequences which provide the convergence rate O(N-1) for the integration of a certain class of d-dimensional Walsh function series, where N is the number of points.

Original languageEnglish
Pages (from-to)431-441
Number of pages11
JournalKyushu Journal of Mathematics
Volume61
Issue number2
DOIs
Publication statusPublished - Jan 1 2007

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Discrepancy
High-dimensional
Walsh Functions
Proper subset
Uniform distribution
Value Function
Numerical integration
Convergence Rate
Infinity
Necessary Conditions
Series
Class

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

High-discrepancy sequences. / Tezuka, Shu.

In: Kyushu Journal of Mathematics, Vol. 61, No. 2, 01.01.2007, p. 431-441.

Research output: Contribution to journalArticle

Tezuka, Shu. / High-discrepancy sequences. In: Kyushu Journal of Mathematics. 2007 ; Vol. 61, No. 2. pp. 431-441.
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