Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

Abstract

We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces S r p,q B(T d ) with dominating mixed smoothness 1/ p < r < 2. We show that order 2 digital nets achieve the optimal rate of convergence N −r (log N ) (d−1)(1−1/q) . The logarithmic term does not depend on r and hence improves the known bound of Dick (SIAM J Numer Anal 45: 2007) for the special case of Sobolev spaces H r mix (T d ). Secondly, the rate of convergence is independent of the integrability p of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of order 2 digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case r = 2.Mathematics Subject Classification 11K06 · 11J71 · 42C10 · 46E35 · 65C05 · 65D30 · 65D32 · 91G60

Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube

“…Such a kink can achieve smoothness s = 2 in case p = 1. The error bounds and numerical experiments in [17,46] show that the convergence rate of the worst-case error for several cubature rules are determined by this regularity.…”

“…Numerical integration in periodic Besov spaces of dominating mixed smoothness B s p,θ has been studied in [7,8,10,17,39], see also Section 8 in the recent survey [9]. There are many results for W s p (T d ) and B s p,∞ (T d ) in this direction, see for instance [38] and the references therein.…”

“…There are many results for W s p (T d ) and B s p,∞ (T d ) in this direction, see for instance [38] and the references therein. The recent contribution [17] especially deals with periodic Besov spaces of dominating mixed smoothness and analyzes cubature formulae on digital nets. Except for the Fibonacci cubature rule [10,[35][36][37][38] in d = 2, the discussed methods in [17,43] are either not optimal or the optimal methods only work for specific (restricted) sets of parameters, for instance small smoothness s < 2 in [17].…”

“…This research area developed into several directions and attracted a lot of interest in the past 50 years, starting with the seminal papers by Korobov [20], Hlawka [18], and Bakhvalov [2]. Afterwards many authors contributed to the construction and analysis of optimal cubature formulae for multivariate functions (with bounded mixed derivative), see, e.g., Frolov [12], Bykovskii [3], Temlyakov [33][34][35][36][37][38][39], Dubinin [7,8], Skriganov [31], Triebel [43], Hinrichs et al [16,17], Markhasin [23], Novak and Woźniakowski [26], Krieg and Novak [21], Dick and Pillichshammer [6], Dũng and Ullrich [10], Goda et al [14,15], and Ullrich [45], to mention just a few. More historical comments and further references can be found at the end of this introduction as well as in the recent survey paper [9,Sect.…”

Characterization of Local Besov Spaces via Wavelet Basis Expansions

“…Besov spaces and their generalizations are particularly suitable for such studies. Recent papers describing the smoothness of functions from these spaces by the decay of the coefficient sequences are e.g., Bazarkhanov [1] for Meyer wavelets, Dinh [2] for mixed B-splines, and Hinrichs et al [3] for Faber-Schauder bases.…”