Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness

Abstract

We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness α ∈ N, α ≥ 2, defined over the s-dimensional unit cube. We prove that randomly digitally shifted order β digital nets can achieve the convergence of the root mean square worst-case error of order N −α (log N ) (s−1)/2 when β ≥ 2α. The exponent of the logarithmic term, i.e., (s − 1)/2, is improved compared to the known result by Baldeaux and Dick, in which the exponent is sα/2. Our result implies the existence of a digitally shifted order β digital net achieving the convergence of the worst-case error of order N −α (log N ) (s−1)/2 , which matches a lower bound on the convergence rate of the worst-case error for any cubature rule using N function evaluations and thus is best possible.

4. Recent advances in higher order quasi-Monte Carlo methods

“…By using this inequality together with the propagation rule of higher order digital nets (Lemma 2), we can improve the order of the shift-averaged worstcase error to best possible. The following result is from [36].…”

“…Considering that the best possible exponent of the log N term for the present integration problem is the same as that of the L 2discrepancy, which is (s − 1)/2, one idea is to switch the weight function from µ α to µ 1 in the error analysis. The following interpolation inequality was shown in [36] to realize this:…”

“…Another major step was made in a series of papers [36,37,38], where the authors refined the integration error analysis for smooth functions due to Dick [7,8] and proved that order (2α + 1) digital nets and sequences achieve the best possible order of the worst-case error for a reproducing kernel Hilbert space with dominating mixed smoothness α, which is (log N ) (s−1)/2 /N α . Note that the original work by Dick [8] proves the worst-case error of order (log N ) sα /N α for order α digital nets and sequences, see also [3].…”

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

“…In other words, the cubature formula is optimal for low smoothness but is not able to benefit from higher regularity. Recently, the authors in [14,15] proposed new Quasi-Monte Carlo methods for W s 2 ([0, 1] d ) that are optimal in order for all s > 1/2 but heavily depend on the smoothness parameter s. One goal of this paper was to get rid (or at least weaken) such dependencies.…”

“…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.…”