Geometrical inverse preconditioning for symmetric positive definite matrices

TL;DR

This paper introduces a geometrical inverse preconditioning method for symmetric positive definite matrices by minimizing a cosine-based function, utilizing gradient methods and geometric properties, with promising numerical results.

Contribution

It proposes a novel inverse preconditioning approach based on geometric properties and gradient methods for symmetric positive definite matrices.

Findings

Effective gradient-based algorithms developed

Preliminary numerical results show promising performance

Applicable to both dense and sparse matrices

Abstract

We focus on inverse preconditioners based on minimizing $F(X) = 1-\cos(XA,I)$, where $XA$ is the preconditioned matrix and $A$ is symmetric and positive definite. We present and analyze gradient-type methods to minimize $F(X)$ on a suitable compact set. For that we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of $F(X)$ on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included.