Sparse-grid sampling recovery and numerical integration of functions having mixed smoothness

TL;DR

This paper surveys recent advances in sparse-grid algorithms for the approximation and integration of functions with mixed smoothness, focusing on their sampling methods and extensions to broader cases.

Contribution

It provides a concise overview of recent developments in sparse-grid techniques for functions with mixed smoothness, including extensions to more general scenarios.

Findings

Sparse-grid algorithms effectively approximate functions with mixed smoothness.

Extensions of these methods accommodate more general function classes.

The survey highlights key recent results in the field.

Abstract

We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.