Generalized approximate inverse preconditioners for least squares problems

Abstract

This paper is concerned with a new approach for preconditioning large sparse least squares problems. Based on the idea of the approximate inverse preconditioner, which was originally developed for square matrices, we construct a generalized approximate inverse (GAINV) M which approximately minimizes III -MAIIF or III -AMIIF• Then, we also discuss the theoretical issues such as the equivalence between the original least squares problem and the preconditioned problem. Finally, numerical experiments on problems from Matrix Market collection and random matrices show that although the preconditioning is expensive, it pays off in certain cases.

A Class of Generalized Approximate Inverse Solvers for Unsymmetric Linear Systems of Irregular Structure Based on Adaptive Algorithmic Modelling for Solving Complex Computational Problems in Three Space Dimensions

“…During the last decades extensive research work has been focused in the preconditioned approach and preconditioned iterative methods for solving large linear and nonlinear problems in sequential and parallel environments [5] [58]. A predominant role in the usage of the preconditioned iterative schemes possess the explicit preconditioned Conjugate Gradient (EPCG) method and its variants using the sparse approximate inverse M * due to its superior convergence rate for solving very large complex computational problems [47].…”

Preconditioners for Krylov subspace methods: An overview

“…We also highlight a number of other developments regarding the iterative solution of least‐squares problems, including the limited memory preconditioner for the normal equations developed in [2], preconditioners based on approximate inverses [323,324], the solution of such problems based on augmented matrix methods [325, chapter 7], and preconditioners based on LU factorizations for both the normal equations [326] and the augmented system [327]. Specialized preconditioners can also be devised for least‐squares problems with more specific properties, such as weighted Toeplitz least‐squares problems [328] (cf Section 3.3).…”

A New High-Order Stable Numerical Method for Matrix Inversion

“…We should note that choosing the initial value as given above can satisfy the necessary condition of arriving to the convergence phase. Some ways for updating the initial matrix for sparse matrices are brought forward by [ 21 ].…”