Greedy algorithms for high-dimensional non-symmetric linear problems

TL;DR

This paper reviews and proposes numerical greedy algorithms for solving high-dimensional non-symmetric linear problems, extending existing methods beyond symmetric cases with theoretical and practical insights.

Contribution

It introduces new approaches and reviews existing algorithms for high-dimensional non-symmetric problems, including convergence analysis and implementation details.

Findings

Algorithms effectively approximate high-dimensional functions

Convergence results are established for new methods

Numerical examples demonstrate practical performance

Abstract

In this article, we present a family of numerical approaches to solve high-dimensional linear non-symmetric problems. The principle of these methods is to approximate a function which depends on a large number of variates by a sum of tensor product functions, each term of which is iteratively computed via a greedy algorithm. There exists a good theoretical framework for these methods in the case of (linear and nonlinear) symmetric elliptic problems. However, the convergence results are not valid any more as soon as the problems considered are not symmetric. We present here a review of the main algorithms proposed in the literature to circumvent this difficulty, together with some new approaches. The theoretical convergence results and the practical implementation of these algorithms are discussed. Their behaviors are illustrated through some numerical examples.