The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative

Abstract

We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov B s p,θ and Triebel-Lizorkin spaces F s p,θ and our results treat the whole range of admissible parameters (s ≥ 1/p). In particular, we obtain upper bounds for the difficult the case of small smoothness which is given for Triebel-Lizorkin spaces F s p,θ in case 1 < θ < p < ∞ with 1/p < s ≤ 1/θ. The presented upper bounds on the worst-case error show a completely different behavior compared to "large" smoothness s > 1/θ. In the latter case the presented upper bounds are optimal, i.e., they can not be improved by any other cubature formula. The optimality for "small" smoothness is open.

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

“…The following two approaches reduce Problem (I) to the problem of numerical integration of functions with homogeneous boundary, where we know that the Frolov cubature formulae perform optimally in various settings [46]. The first approach being proposed by Bykovskii [3] shows that numerical integration in spaces with bounded mixed derivative is asymptotically not "harder" than the integration of functions with homogeneous boundary.…”

“…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.…”

“…Note that this includes the case of small smoothness, which is present if 2 < p < ∞ and 1/ p < s ≤ 1/2, see [35,46].…”

“…To gain this, we use a reduction of the problem to theÅ s p,θ setting via tailored transformations in connection with the Frolov cubature formulae [12,36], which recently attracted significant interest [19,21,[44][45][46]. It has been already proved by Bykovskii [3] …”

“…Frolov's construction [12] dates back to the 1970s and gives a simple construction of admissible lattices that perform optimally with respect to several function classes of functions supported in the unit cube with mixed smoothness properties [46]. This will be the starting point for the construction and analysis of cubature formulae for the numerical integration of functions with nontrivial boundary data in this paper.…”