Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions

TL;DR

This paper introduces tent-transformed lattice rules for efficient multivariate integration and approximation of smooth non-periodic functions, leveraging connections to periodic function spaces for improved error bounds.

Contribution

The paper presents a novel approach using tent-transformed lattice points for non-periodic functions, with theoretical error bounds derived from connections to periodic function spaces.

Findings

Constructive worst-case error bounds with good convergence rates

Effective algorithms for multivariate integration and approximation

Connection established between cosine and Korobov spaces

Abstract

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space.