Global iterative methods for sparse approximate inverses of symmetric positive-definite matrices

TL;DR

This paper introduces and analyzes iterative methods for computing sparse approximate inverses of SPD matrices, demonstrating that the locally optimal (preconditioned) minimal residual method outperforms other approaches in convergence and robustness.

Contribution

The paper presents new iterative projection methods, N(P)CG and LO(P)MR, tailored for efficient sparse approximate inverse computation, with analysis and practical dropping strategies.

Findings

LO(P)MR outperforms (P)MR and (P)CG in convergence speed.

N(P)CG offers slight improvements over (P)SD but less than (P)MR.

LO(P)MR is more robust, converging faster to sparser approximations.

Abstract

The nonlinear (preconditioned) conjugate gradient N(P)CG method and the locally optimal (preconditioned) minimal residual LO(P)MR method, both of which are used for the iterative computation of sparse approximate inverses (SPAIs) of symmetric positive-definite (SPD) matrices, are introduced and analyzed. The (preconditioned) conjugate gradient (P)CG method is also employed and presented for comparison. The N(P)CG method is defined as a one-dimensional projection with residuals made orthogonal to the current search direction, itself made $A$-orthogonal to the last search direction. The residual orthogonality, expressed via Frobenius inner product, actually holds against all previous search directions, making each iterate globally optimal, that is, that minimizes the Frobenius A-norm of the error over the affine Krylov subspace of $A^2$ generated by the initial gradient. The LO(P)MR method is a two-dimensional projection method that enriches iterates produced by the (preconditioned) minimal residual (P)MR method. These approaches differ from existing descent methods and aim to accelerate convergence compared to other global iteration methods, including (P)MR and (preconditioned) steepest descent (P)SD, previously used for SPAI computation. The methods are implemented with practical dropping strategies to control the growth of nonzero components in the approximate inverse. Numerical experiments are performed in which approximate inverses of several sparse SPD matrices are computed. N(P)CG provides a slight improvement over (P)SD, but remains generally less effective than (P)MR. On the other hand, while (P)CG does improve (P)MR, its convergence is more affected by the dropping of nonzero components, ill-conditioning, and small eigenvalues. LO(P)MR is more robust than (P)MR and (P)CG, consistently outperforms other methods, converging faster to better and often sparser approximations.