Greedy algorithms for high-dimensional non-symmetric linear problems
For researchers working on high-dimensional linear problems, this paper addresses the challenge of non-symmetry where existing convergence theory fails.
The paper presents greedy algorithms for solving high-dimensional non-symmetric linear problems, reviewing existing methods and proposing new approaches. Numerical examples illustrate the behavior of these algorithms.
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.