Low-rank approximate inverse for preconditioning tensor-structured linear systems

TL;DR

This paper introduces a greedy algorithm to construct low-rank tensor approximations of inverse operators, serving as efficient preconditioners for high-dimensional linear systems, enhancing solver convergence.

Contribution

It develops a novel greedy method for low-rank inverse approximation in tensor formats, incorporating properties like symmetry and sparsity during correction steps.

Findings

Effective preconditioners for tensor-structured systems

Improved convergence of iterative solvers

High-quality low-rank inverse approximations

Abstract

In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator. It provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators. These subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank-one corrections. In order to handle high-order tensors, approximate projections are computed in low-rank Hierarchical Tucker subsets of the successive subspaces of operators. Some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step, where an optimal rank-one correction is searched as the tensor product of operators with the desired properties. Numerical examples illustrate the ability of this algorithm to provide efficient preconditioners for linear systems in tensor format that improve the convergence of iterative solvers and also the quality of the resulting low-rank approximations of the solution.