Geometrical inverse preconditioning for symmetric positive definite matrices
This work addresses the problem of constructing effective inverse preconditioners for symmetric positive definite linear systems, but the results are preliminary and the approach is incremental.
The paper develops gradient-type methods for minimizing a cosine-based objective function to compute inverse preconditioners for symmetric positive definite matrices, and presents preliminary numerical results showing dense and sparse approximations.
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.