Hyperbolic Cross Approximation

TL;DR

This survey reviews hyperbolic cross approximation, a multivariate technique effective in high dimensions, highlighting its theoretical foundations, practical challenges, and recent research directions to stimulate further study.

Contribution

It provides a comprehensive overview of classical and recent developments in hyperbolic cross approximation, emphasizing open problems and connections to other mathematical fields.

Findings

Hyperbolic cross approximation is effective for moderate to high dimensions.

Many fundamental problems in the area remain unsolved.

Recent connections to probability, discrepancy, and numerical integration are highlighted.

Abstract

Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of these problems is high dimensions. We present here a survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions. The theory of hyperbolic cross approximation and related theory of functions with mixed smoothness are under detailed study for more than 50 years. It is now well understood that this theory is important both for theoretical study and for practical applications. It is also understood that both theoretical analysis and construction of practical algorithms are very difficult problems. This explains why many fundamental problems in this area are still unsolved. Only a few survey papers and monographs on the topic are published. This and recently discovered deep connections between the hyperbolic cross approximation (and related sparse grids) and other areas of mathematics such as probability, discrepancy, and numerical integration motivated us to write this survey. We try to put emphases on the development of ideas and methods rather than list all the known results in the area. We formulate many problems, which, to our knowledge, are open problems. We also include some very recent results on the topic, which sometimes highlight new interesting directions of research. We hope that this survey will stimulate further active research in this fascinating and challenging area of approximation theory and numerical analysis.