Preconditioning Techniques for Large Linear Systems: A Survey

Abstract

This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper. c 2002 Elsevier Science (USA)

Application of a Fast Boundary Element Algorithm for Open Floating Potential Electrostatic Problem

“…The bi-conjugate gradient method, which approaches great advantage in smaller storage requirement by replacing the orthogonal sequence of residuals by two mutually orthogonal sequences, is an effective method for equation (7). The implement steps can be described below.…”

“…a preconditioner is well suited for fast multipole BEM to the analysis of 3D crack problems [6]. The preconditioning of fast multipole BEM has been solved successfully by a new technique based on the inverse of the block diagonal matrix corresponding to element or node neighbors with satisfactory rate of convergence in modeling electron guns [7]. Remarkable results have been obtained from the 3-D BEM analysis of electron guns problem solved by sparse inverse preconditioner based on the leaves [8].…”

“…The computation of M T is reduced to solutions of N independent linear least-squares problems. The algorithm to calculate every column m j of M is the solution of a linear system in Ref [7].…”

Geometry Based Preconditioner for Radiation Problems Involving Wire and Surface Basis Functions

“…The former have more natural parallelism than the latter, which also present well-known instability problems that sometimes can be overcome by pivoting [15]. Recent works have proposed the preconditioning of the ordinary equation with the transpose complex conjugate of the impedance matrix [16], a multiplicative preconditioner using Calderon identities [17] or a localized preconditioner in the vicinity of a radiation problem antenna [9].…”

Quantitative susceptibility mapping‐based cerebral metabolic rate of oxygen mapping with minimum local variance

“…To reduce error propagation and improve convergence speed, a right preconditioning technique is applied (33). The new system is defined as A P x P ¼ b, in which A P ¼ AP and x P ¼ P À1 x.…”