Hyperbolic cross approximation in infinite dimensions

TL;DR

This paper establishes tight bounds on the size of hyperbolic cross index sets in infinite dimensions, enabling dimension-independent linear approximation of functions relevant to parametric and stochastic PDEs.

Contribution

It provides the first dimension-independent bounds for hyperbolic crosses in infinite dimensions, facilitating tractable linear approximation in high-dimensional function spaces.

Findings

Bounds are tight and independent of dimension.

Approximation error bounds are dimension-independent.

Rates depend only on smoothness parameters through constants.

Abstract

We give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev-Korobov-type smoothness and mixed Sobolev-analytic-type smoothness in the infinite-dimensional case where specific summability properties of the smoothness indices are fulfilled. These estimates are then applied to the linear approximation of functions from the associated spaces in terms of the $\varepsilon$-dimension of their unit balls. Here, the approximation is based on linear information. Such function spaces appear for example for the solution of parametric and stochastic PDEs. The obtained upper and lower bounds of the approximation error as well as of the associated $\varepsilon$-complexities are completely independent of any dimension. Moreover, the rates are independent of the parameters which define the smoothness properties of the infinite-variate parametric or stochastic part of the solution. These parameters are only contained in the order constants. This way, linear approximation theory becomes possible in the infinite-dimensional case and corresponding infinite-dimensional problems get tractable.