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

The t-value of a (t,m, s)-net is an important criterion of point sets for quasiMonte Carlo integration, and many point sets are constructed in terms of tvalues, 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 ensure 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 obtained point sets improve the rates of convergence for smooth functions and are robust for non-smooth functions.