Efficient multivariate approximation on the cube

TL;DR

This paper introduces a method combining periodization and approximation techniques to efficiently approximate high-dimensional non-periodic functions on the cube, leveraging transformations to Sobolev spaces and adapting lattice approximation algorithms.

Contribution

It develops conditions for torus-to-cube transformations that make non-periodic functions smooth in Sobolev spaces, enabling efficient high-dimensional approximation.

Findings

Effective approximation in up to 5 dimensions.

Conditions for smooth transformations to Sobolev spaces.

Adapted algorithms for fast evaluation and reconstruction.

Abstract

We combine a periodization strategy for weighted $L_{2}$-integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube $\left[-\frac{1}{2},\frac{1}{2}\right]^{d}$. Our concept allows to determine conditions on the $d$-variate torus-to-cube transformations ${\psi:\left[-\frac{1}{2},\frac{1}{2}\right]^{d}\to\left[-\frac{1}{2},\frac{1}{2}\right]^{d}}$ such that a non-periodic function is transformed into a smooth function in the Sobolev space $\mathcal H^{m}(\mathbb{T}^{d})$ when applying $\psi$. We adapt some $L_{\infty}(\mathbb{T}^{d})$- and $L_{2}(\mathbb{T}^{d})$-approximation error estimates for single rank-$1$ lattice approximation methods and adjust algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting. We illustrate the theoretical findings by means of numerical tests in up to $d=5$ dimensions.