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

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.