High-dimensional nonlinear approximation by parametric manifolds in H\"older-Nikol'skii spaces of mixed smoothness

TL;DR

This paper develops explicit nonlinear approximation methods for high-dimensional functions with mixed smoothness in H"older-Nikol'skii spaces, providing dimension-dependent error estimates and novel asymptotic results for the 2D case.

Contribution

It introduces new explicit nonlinear approximation techniques using tensor product Faber series and sparse grids, with detailed error bounds depending on dimension and complexity.

Findings

Derived dimension-dependent approximation error estimates.

Constructed explicit nonlinear approximation methods.

Established new asymptotic order of N-widths for 2D case.

Abstract

We study high-dimensional nonlinear approximation of functions in H\"older-Nikol'skii spaces $H^\alpha_\infty(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$ having mixed smoothness, by parametric manifolds. The approximation error is measured in the $L_\infty$-norm. In this context, we explicitly constructed methods of nonlinear approximation, and give dimension-dependent estimates of the approximation error explicitly in dimension $d$ and number $N$ measuring computation complexity of the parametric manifold of approximants. For $d=2$, we derived a novel right asymptotic order of noncontinuous manifold $N$-widths of the unit ball of $H^\alpha_\infty(\mathbb{I}^2)$ in the space $L_\infty(\mathbb{I}^2)$. In constructing approximation methods, the function decomposition by the tensor product Faber series and special representations of its truncations on sparse grids play a central role.