Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube

TL;DR

This paper extends the understanding of numerical integration for multivariate functions with mixed smoothness, showing that boundary conditions do not affect the asymptotic error rate and proposing modifications to Frolov's cubature formulae for better applicability.

Contribution

It introduces two modifications of Frolov's cubature formulae suitable for functions with boundary support, maintaining optimal error rates and extending classical boundedness results for change of variable mappings.

Findings

Asymptotic integration error rate is unaffected by boundary conditions.

Two tailored modifications of Frolov's cubature formulae are proposed.

The second modification is more suitable for applications due to its simplicity.

Abstract

In a recent article by two of the present authors it turned out that Frolov's cubature formulae are optimal and universal for various settings (Besov-Triebel-Lizorkin spaces) of functions with dominating mixed smoothness. Those cubature formulae go well together with functions supported inside the unit cube $[0,1]^d$. The question for the optimal numerical integration of multivariate functions with non-trivial boundary data, in particular non-periodic functions, arises. In this paper we give a general result that the asymptotic rate of the minimal worst-case integration error is not affected by boundary conditions in the above mentioned spaces. In fact, we propose two tailored modifications of Frolov's cubature formulae suitable for functions supported on the cube (not in the cube) which provide the same minimal worst-case error up to a constant. This constant involves the norms of a ``change of variable'' and a ``pointwise multiplication'' mapping, respectively, between the function spaces of interest. In fact, we complement, extend and improve classical results by Bykovskii, Dubinin and Temlyakov on the boundedness of change of variable mappings in Besov-Sobolev spaces of mixed smoothness. Our proof technique relies on a new characterization via integral means of mixed differences and maximal function techniques, general enough to treat Besov and Triebel-Lizorkin spaces at once. The second modification, which only tackles the case of periodic functions, is based on a pointwise multiplication and is therefore most likely more suitable for applications than the (traditional) ``change of variable'' approach. These new theoretical insights are expected to be useful for the design of new (and robust) cubature rules for multivariate functions on the cube.