Scaled norm minimization method for computing the parameters of the HSS and the two‐parameter HSS preconditioners

Abstract

Summary The performance of the Hermitian and skew‐Hermitian splitting (HSS) preconditioner for the non‐Hermitian positive definite system of linear equations is largely dependent on the choice of its parameter value. In this work, an efficient scaled norm minimization (SNM) method is proposed to compute the parameter value of the HSS preconditioner. In addition, by choosing different parameters for the Hermitian and the skew‐Hermitian matrices in the HSS preconditioner, a two‐parameter HSS preconditioner is proposed. Moreover, an efficient and practical formula for computing the parameter values of this new preconditioner is also derived by using the SNM method. Numerical examples are illustrated to verify the performances of the HSS and the two‐parameter HSS preconditioners when their parameters are computed by the SNM method.

A class of block alternating splitting implicit iteration methods for double saddle point linear systems

“…In virtue of the fact that it is constructed from alternating splitting, the BASI preconditioner can be viewed as a generalization of the HSS based preconditioners; see Remark 2. Recently, many readily available and practical strategies have been studied in the literature for choosing the optimal parameters of such methods, such as the local Fourier analysis (LFA) method used for the relaxed dimensional factorization (RDF) preconditioners [22,40], the estimation methods by minimizing the traces of corresponding iteration matrices [40] and the Frobenius norm minimization methods [17,37,41,42] and so on. Due to their similarities in structures, it is not difficult to generalize these mentioned methods to select the optimal parameter of the BASI preconditioner.…”

A two‐step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations

“…Finding quasi‐optimal parameters that minimize an upper bound of one of the above quantities or their appropriate approximations; see, for example, References 7,17,19,29,39,54. Estimated Formula . Finding the parameters by minimizing a cost function that is well defined through the solution error, or the residual vector, or an approximated mixture of them; see, for example, References 55‐61. Reduced‐Model Formula . Finding the parameters in accordance with the above three approaches for smaller‐dimensional linear system that is reasonably reduced and approximated from the original linear system, then using these available parameters to analyze and implement the TMSI paradigm in solving the target linear system; see, for example, References 18,62,63.…”

“…Estimated Formula . Finding the parameters by minimizing a cost function that is well defined through the solution error, or the residual vector, or an approximated mixture of them; see, for example, References 55‐61.…”

Newton-PGSS and Its Improvement Method for Solving Nonlinear Systems with Saddle Point Jacobian Matrices

“…Some of them have not been solved analytically, so we can only explore the method to obtain the numerical simulation at our utmost. In the past, researchers have developed some methods to solve nonlinear function [1][2][3][4][5][6][7][8][9][10]. In these methods, the most typical and popular method for solving the nonlinear system (1) is the Newton method.…”