Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices

Abstract

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 this, 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.

Geometrical inverse matrix approximation for least-squares problems and acceleration strategies

“…We have extended the MinCos iterative method, originally developed in [12] for symmetric and positive definite matrices, to approximate the inverse of the matrices associated with linear least-squares problems, and we have also described and adapted three different possible acceleration schemes to the generated convergent matrix sequences. Our experiments have shown that the geometrical MinCos scheme is also a robust option to approximate the inverse of matrices of the form A T A, associated with least-squares problems, without requiring the explicit knowledge of A T .…”

“…In here PSD refers to the positive semi-definite closed cone of square matrices which possesses a rich geometrical structure; see, e.g., [1,11]. In Remark 2.1 we summarize the most important properties of Algorithm 2 (see [12]).…”

“…This method has been successfully introduced in [12], and can be seen as an improved version of the Cauchy Method applied to the minimization of the merit function…”

“…The sequence of matrices {X (k) } ⊂ IR n×n generated either by Algorithm 1 or by Algorithm 2 can be viewed as simplified and improved versions of the CauchyCos algorithm developed in [12], which is a specialized version of the Cauchy (steepest descent) method for minimizing F (X). Nevertheless, both algorithm are gradient-type methods, and as a consequence they can be accelerated using some wellknown acceleration strategies which extend effective scalar and vector modern sequence acceleration techniques; see, e.g., [2,3].…”

“…For the minimization of F (X) we will extend and adapt the simplified gradient-type scheme Min-Cos, introduced in [12], to the linear least-squares scenario. Moreover, we will adapt and apply some well-known modern matrix acceleration strategies to the generated convergent sequences.…”