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.