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Quasi-Monte Carlo point sets with small t-values and WAFOM

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  • Harase, Shin

Abstract

The t-value of a (t,m,s)-net is an important criterion of point sets for quasi-Monte Carlo integration, and many point sets are constructed in terms of the t-values, as this leads to small integration error bounds. Recently, Matsumoto, Saito, and Matoba proposed the Walsh figure of merit (WAFOM) as a quickly computable criterion of point sets that ensures higher order convergence for function classes of very high smoothness. In this paper, we consider a search algorithm for point sets whose t-value and WAFOM are both small, so as to be effective for a wider range of function classes. For this, we fix digital (t,m,s)-nets with small t-values (e.g., Sobol’ or Niederreiter–Xing nets) in advance, apply random linear scrambling, and select scrambled digital (t,m,s)-nets in terms of WAFOM. Experiments show that the resulting point sets improve the rates of convergence for smooth functions and are robust for non-smooth functions.

Suggested Citation

  • Harase, Shin, 2015. "Quasi-Monte Carlo point sets with small t-values and WAFOM," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 318-326.
    • Handle: RePEc:eee:apmaco:v:254:y:2015:i:c:p:318-326
    DOI: 10.1016/j.amc.2014.12.144
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