An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands

Abstract

In a recent paper by the authors, it is shown that there exists a quasi-Monte Carlo (QMC) rule which achieves the best possible rate of convergence for numerical integration in a reproducing kernel Hilbert space consisting of smooth functions. In this paper we provide an explicit construction of such an optimal order QMC rule. Our approach is to exploit both the decay and the sparsity of the Walsh coefficients of the reproducing kernel simultaneously. This can be done by applying digit interlacing composition due to Dick to digital nets with large minimum Hamming and Niederreiter-Rosenbloom-Tsfasman metrics due to Chen and Skriganov. To our best knowledge, our construction gives the first QMC rule which achieves the best possible convergence in this function space.

References in corpus (5)

• The role of Frolov's cubature formula for functions with bounded mixed derivative
• The decay of the Walsh coefficients of smooth functions
• Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions
• Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness
• Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients

Cited by in corpus (6)

• Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube
• Lattice rules with random $n$ achieve nearly the optimal $\mathcal{O}(n^{-\alpha -1/2})$ error independently of the dimension
• Optimal order quadrature error bounds for infinite-dimensional higher order digital sequences
• Quasi-Monte Carlo integration using digital nets with antithetics
• Recent advances in higher order quasi-Monte Carlo methods
• Digital net properties of a polynomial analogue of Frolov's construction