Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

Abstract

We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness B α p,θ (G d ) in the case 0 < p, θ ≤ ∞ and α > 1/p, whereWe prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from B α p,θ (T 2 ) in the case 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, α > 1/p. A non-periodic modification of this classical formula yields upper bounds for B α p,θ (I 2 ) if 1/p < α < 1 + 1/p. In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from B α p,θ (G 2 ) and indicate that a corresponding result is most likely also true in case d > 2. This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.

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

“…Both theorems immediately imply the two-sided relation 10) in the above range of parameters if additionally A s p,θ → C(R d ) holds true.…”

“…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.…”

“…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]. In other words, the cubature formula is optimal for low smoothness but is not able to benefit from higher regularity.…”

“…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.…”

B-Spline Quasi-Interpolation Sampling Representation and Sampling Recovery in Sobolev Spaces of Mixed Smoothness

“…In [38]- [41] and [10]- [12] Smolyak's construction was developed for studying the trigonometric sampling recovery and sampling width for periodic Sobolev classes and Nikol'skii classes having mixed smoothness. Recently, the sampling recovery for Sobolev and Besov classes having mixed smoothness has been investigated in [5,6,17,18,20,34,35,44]. In particular, for non-periodic functions of mixed smoothness linear sampling algorithms on Smolyak grids have been recently studied in [43] (d = 2), [17,18,20,35] using B-spline quasiinterpolation sampling representation.…”