ACCELERATION OF ONE-PARAMETER RELAXATION METHODS FOR SINGULAR SADDLE POINT PROBLEMS

A generalization of parameterized inexact Uzawa method for singular saddle point problems

Recently, Guo et al. proposed a modified SOR-like (MSOR-like) iteration method for solving the nonsingular saddle point problem. In this paper, we further prove the semi-convergence of this method when it is applied to solve the singular saddle point problems under suitable conditions on the involved iteration parameters. Moreover, the optimal iteration parameters and the corresponding optimal semi-convergence factor for the MSOR-like method are determined. In addition, numerical experiments are used to show the feasibility and effectiveness of the MSOR-like method for solving singular saddle point problems, arising from the incompressible flow problems.

A new SOR-Like method for the saddle point problems

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.

Recently, there have been many studies on solving different kinds of fuzzy equations. In this paper, the solution of a trapezoidal fully fuzzy linear system (FFLS) is studied. Uzawa approach, which is a popular iterative technique for saddle point problems, is considered for solving such FFLSs. In our Uzawa approach, it is possible to compute the solution of a fuzzy system using various relaxation iterative methods such as Richardson, Jacobi, Gauss-Seidel, SOR, SSOR as well as Krylov subspace methods such as GMRES, QMR and BiCGSTAB. Krylov subspace iterative methods are known to converge for a larger class of matrices than relaxation iterative methods and they exhibit higher convergence rates. Thus, they are more widely used in practical problems. Numerical experiments are to illustrate the performance of our suggested methods.

In this paper, we propose GSTS–Uzawa method for solving a class of complex singular saddle point problems based on generalized skew-Hermitian triangular splitting (GSTS) iteration method and classical Uzawa method. We research on its semi-convergence properties and the eigenvalues distributions of its preconditioned matrix. The resulting GSTS–Uzawa preconditioner is used to precondition Krylov subspace methods such as the restarted generalized minimal residual (GMRES) method for solving the equivalent formulation of the complex singular saddle point problems. The theoretical results and effectiveness of the GSTS–Uzawa method are supported by a numerical example.

In this paper, we consider the regularized deteriorated positive and skew-Hermitian splitting (RDPSS) iteration method to solve singular non-Hermitian saddle point problems arising from the steady Navier–Stokes equation, which was first presented by Cao (Appl. Math. Lett., 2018). The unconditional semi-convergence property of the RDPSS iteration method and some spectral properties of the corresponding preconditioned matrix are analysed detailedly. Meanwhile, the strategies to estimate the parameters of the RDPSS preconditioner are discussed. In the end, some numerical experiments are performed to verify the feasibility and effectiveness of the proposed method and outperform several other commonly used ones.

The Uzawa-PPS iteration methods for nonsingular and singular non-Hermitian saddle point problems

The relaxed nonlinear PHSS-like iteration method for absolute value equations

General constraint preconditioning iteration method for singular saddle-point problems

<abstract><p>The strong <italic>P</italic>-regular splitting is put forward and defined for iterative methods of non-Hermitian linear systems in the paper. The strong <italic>P</italic>-regular splitting combining SOR iterative methods and relaxed SOR iterative methods are established, and conditions guaranteeing the convergence are presented. Furthermore, two numerical experiments are done to illustrate the convergence and effectiveness of our iterative methods.</p></abstract>

High-resolution methods are generally defined to be high-performance methods for estimating and/or detecting the desired and/or undesired signal components present in a given set of data. The term “high-resolution” also implies a good ability to resolve very “similar” signal components. One of the most common problems in signal processing is known as frequency estimation. In frequency estimation, “high-resolution” often refers to a good ability to resolve two or more closely located frequencies in the given data. There are two groups of high-resolution methods. One is parametric methods, and the other non-parametric methods. The parametric high-resolution methods result from ingenious exploitations of known data structures. The non-parametric high-resolution methods maximize the output of some desired information with little knowledge of the data structure. The choice between parametric methods and non-parametric methods largely depends on one’s confidence in the assumed data model. In this chapter, we expose the readers to a range of existing parametric high-resolution methods. In Section 1.2, we present several frequency estimation techniques using algebraic principles. They are linear prediction, matrix pencil, and iterative quadratic maximum likelihood. The linear prediction method and the matrix pencil method can achieve near-optimal accuracy of estimation without the local convergence issues associated with the optimal methods. The computational complexities of the two methods are among the most efficient. The iterative quadratic maximum likelihood method is an approximation of the (exact) maximum likelihood method. Under some condition (e.g., high SNR), this approximation achieves the optimal accuracy. Concepts like forwardand-backward averaging, total least square, and (joint) singular value decomposition of orthonormal matrices will also be discussed. In Section 1.3, we present methods that exploit large sample theorems in statistics. In particular, we focus on data of multiple independent measurements. The key data structure is captured by the dominant (principal) subspace of the data matrix or the dominant eigenvectors of the data covariance matrix. The principal subspace is referred

This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski–Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski–Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relaxation parameter sequence since the optimal relaxation parameter sequence involves fixed points of the operators and can't be used in actual computing. Thirdly, we apply our results to the operators splitting method, such as forward–backward splitting methods and Douglas–Rachford splitting method. Finally, numerical experiments are provided to illustrate that Krasnosel'ski–Mann iteration and the relaxed projected method with our proposed relaxation parameter sequence behave better than others.

On the semi-convergence of preconditioned GLHSS iteration method for non-Hermitian singular saddle point problem

In this paper we study the piecewise collocation method for a class of functional integral equations with state-dependent delays that is, where the delays depend on the solution. It is well known that these equations typically have discontinuity in the solution or its derivatives at the initial point of integration domain. This discontinuity propagated along the integration interval giving rise to subsequent points, called "singular points", which can not be determined priori and the solution derivatives in these points are smoothed out along the interval. Most of the known numerical methods for this type of equations are generally very sensitive to the singular points and therefore must have a process that detects these points and insert them into the mesh to guarantee the required accuracy. Here, we present a numerical algorithm based on the piecewise collocation method and an approach for tracking the singular points relying on the recent strategy for implicit delay differential equations which has been proposed by Guglielmi and Hairer in 2008. The convergence analysis of the method is investigated and some numerical experiments are presented for clarifying the robustness of the method.