skew-Hermitian Splitting Research Articles

Modified Newton-PHSS method for solving nonlinear systems with positive definite Jacobian matrices

By making use of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration as the inner solver for the modified Newton method, we establish the modified Newton-PHSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Moreover, the local convergence theorem is proved under proper conditions. Numerical results are given to show its feasibility and effectiveness. In addition, the advantages of the modified Newton-PHSS method over some other methods are shown by solving two systems of nonlinear equations.

On the Preconditioning Properties of RHSS Preconditioner for Saddle-Point Linear Systems

In this paper, for the regularized Hermitian and skew-Hermitian splitting (RHSS) preconditioner introduced by Bai and Benzi (BIT Numer Math 57: 287–311, 2017) for the solution of saddle-point linear systems, we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters. We accurately describe the numbers of eigenvalues clustered at (0, 0) and (2, 0), if the iteration parameter is close to 0. An estimate about the condition number of the corresponding eigenvector matrix, which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method, is also studied in this work.

Convergence analysis for modified PAHSS-PU method with new parameter setting

Based on the modified preconditioned accelerated Hermitian and skew-Hermitian splitting (MPAHSS) and triangular splitting iterative (TSI) methods, this paper presents a new parameter setting to overcome the drawbacks of the MPAHSS-parameterized Uzawa (MPAHSS-PU) method proposed by Huang et al. (J Comput Appl Math 332:1–12, 2018) for sthe saddle point problems. A sufficient condition is provided to ensure the convergence of MPAHSS-PU method with the new parameter setting, and a selection strategy for its parameters is also given. The new parameter setting not only lessens the parameter limitation of the MPAHSS-PU method, but also improves its performance. The validity of the obtained results and the performance of MPAHSS-PU method with the new parameter setting are demonstrated by numerical examples.

On the regularization matrix of the regularized DPSS preconditioner for non-Hermitian saddle-point problems

In this work, with respect to the regularization matrix of the new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for non-Hermitian saddle-point problems, we provide a class of Hermitian positive semidefinite matrices, which depend on certain parameters, for practical computations. A precise description about the eigenvalue distribution of the corresponding preconditioned matrix is given. The condition number of the eigenvector matrix, which partly determines the convergence rate of the related preconditioned Krylov subspace method, is also discussed in this work. Finally, some numerical experiments are carried out to identify the effectiveness of the presented special choices for the regularization matrix to solve the non-Hermitian saddle-point problems.

Updated preconditioned Hermitian and skew-Hermitian splitting-type iteration methods for solving saddle-point problems

For the saddle-point problems, we discuss and analyze the updated preconditioned Hermitian and skew-Hermitian splitting (UPHSS) iteration method in detail. On this basis, we introduce a two-stage iteration method for the UPHSS iteration method. Theoretical analysis shows that the UPHSS and two-stage UPHSS iteration methods are convergent to the unique solution of the saddle-point linear system when the parameter is suitably chosen. Numerical examples show the correctness of the theory and the effectiveness of these methods.

Regularized DPSS preconditioners for generalized saddle point linear systems

By introducing a regularization matrix and an additional iteration parameter, a new class of regularized deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioners are proposed for generalized saddle point linear systems. Compared with the well-known Hermitian and skew-Hermitian splitting (HSS) preconditioner and the regularized HSS preconditioner (Bai, 2019) studied recently, the new RDPSS preconditioners have much better computing efficiency especially when the (1,1) block matrix is non-Hermitian. It is proved that the corresponding RDPSS stationary iteration method is unconditionally convergent. In addition, clustering property of the eigenvalues of the RDPSS preconditioned matrix is studied in detail. Two numerical experiments arising from the meshfree discretization of a static piezoelectric equation and the finite element discretization of the Navier–Stokes equation show the effectiveness of the new proposed preconditioners.

Regularized DPSS Preconditioners for Singular Saddle Point Problems

Recently, Cao proposed a regularized deteriorated positive and skewHermitian splitting (RDPSS) preconditioner for the non-Hermitian nonsingular saddle point problem. In this paper, we consider applying RDPSS preconditioner to solve the singular saddle point problem. Moreover, we propose a two-parameter accelerated variant of the RDPSS (ARDPSS) preconditioner to further improve its efficiency. Theoretical analysis proves that the RDPSS and ARDPSS methods are semi-convergent unconditionally. Some spectral properties of the corresponding preconditioned matrices are analyzed. Numerical experiments indicate that better performance can be achieved when applying the ARDPSS preconditioner to accelerate the GMRES method for solving the singular saddle point problem.

Minimum residual modified HSS iteration method for a class of complex symmetric linear systems

For solving a class of complex symmetric system of linear equations, we apply the minimum residual technique to the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme and propose an iteration method referred to as minimum residual MHSS (MRMHSS) iteration method. Compared with the classical MHSS method, the MRMHSS method has two more iteration parameters, which can be automatically and easily computed. Then, some properties of the MRMHSS iteration method are carefully studied. Finally, we use four examples to test the performance of the MRMHSS iteration method by comparing its numerical results with three other iteration methods.

HSS-like method for solving complex nonlinear Yang–Baxter matrix equation

Many works on solving matrix equations are related to the complex nonlinear Yang–Baxter matrix equation $$ AXA=XAX $$, where $$ A \in {\mathbb {C}}^{n \times n}$$ is a given matrix and X is an unknown matrix. The Yang–Baxter matrix equation has been widely studied by its application in various fields of mathematics and physics. In this paper, we introduce an iterative method based on the Hermitian and skew-Hermitian splitting of coefficient matrix A for solving complex nonlinear Yang–Baxter matrix equation. Then, we prove the convergence of the new scheme subject to some conditions. Finally, an example is solved to discover the applicability of the new method via comparing it with some related previous methods.

Numerical Solution of the Navier–Stokes Equations Using Multigrid Methods with HSS-Based and STS-Based Smoothers

Multigrid methods (MGMs) are used for discretized systems of partial differential equations (PDEs) which arise from finite difference approximation of the incompressible Navier–Stokes equations. After discretization and linearization of the equations, systems of linear algebraic equations (SLAEs) with a strongly non-Hermitian matrix appear. Hermitian/skew-Hermitian splitting (HSS) and skew-Hermitian triangular splitting (STS) methods are considered as smoothers in the MGM for solving the SLAE. Numerical results for an algebraic multigrid (AMG) method with HSS-based smoothers are presented.

HSS-iteration-based iterative interpolation of curves and surfaces with NTP bases

Based on the Hermitian and skew-Hermitian splitting iteration technique, a new progressive iteration approximation technique called HPIA for curves and surfaces interpolation with normalized totally positive (NTP) bases and its weighted version, namely WHPIA, are proposed. We take the previous iteration and the current iteration into account simultaneously, and establish a function based on NTP bases as a perturbation term in the iteration process. Convergence analyses and the approximate optimal weight of WHPIA are given. Theoretical and experimental results show that HPIA and WHPIA are effective.

Extrapolated New Hermitian and Skew-Hermitian Splitting Method for Non-Hermitian Positive Definite Linear System

Extrapolated new Hermitian and skew-Hermitian splitting method is presented for solving non-Hermitian and normal positive definite linear systems. We theoretically prove that this method is convergent in three different ranges of the parameters α and ω. Specially, the convergence results of the NHSS method are also obtained, which generalize the conclusions in [2]. The numerical example further verifies the results.

An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

We extend the positive-definite and skew-Hermitian splitting (PSS) iterative method to EPSS (for extended PSS) for solving non-Hermitian positive-definite systems of linear equations. In this kind of extension, the shift matrix is replaced by a Hermitian positive-definite matrix Σ. It is proved that the method is unconditionally convergent. Then, the EPSS method with a 2 × 2-block diagonal matrix Σ as the shift matrix is applied to generalized saddle point problems and the convergence of the method is verified. Naturally, the induced preconditioner can be applied to generalized saddle point problems. In some special cases, this preconditioner coincides with some existing preconditioners. A special case of the EPSS preconditioner, say SEPSS, is used to accelerate the convergence of GMRES. Effectiveness of this preconditioner is investigated by some numerical experiments.

On single-step HSS iterative method with circulant preconditioner for fractional diffusion equations

By exploiting Toeplitz-like structure and non-Hermitian dense property of the discrete coefficient matrix, a new double-layer iterative method called SHSS-PCG method is employed to solve the linear systems originating from the implicit finite difference discretization of fractional diffusion equations (FDEs). The method is a combination of the single-step Hermitian and skew-Hermitian splitting (SHSS) method with the preconditioned conjugate gradient (PCG) method. Further, the new circulant preconditioners are proposed to improve the efficiency of SHSS-PCG method, and the computation cost is further reduced via using the fast Fourier transform (FFT). Theoretical analysis shows that the SHSS-PCG iterative method with circulant preconditioners is convergent. Numerical experiments are given to show that our SHSS-PCG method with circulant preconditioners preforms very well, and the proposed circulant preconditioners are very efficient in accelerating the convergence rate.

A Peaceman–Rachford Splitting Method with Monotone Plus Skew-Symmetric Splitting for Nonlinear Saddle Point Problems

This paper is devoted to solving the linearly constrained convex optimization problems by Peaceman–Rachford splitting method with monotone plus skew-symmetric splitting on KKT operators. This approach generalizes the Hermitian and skew-Hermitian splitting method, an unconditionally convergent algorithm for non-Hermitian positive definite linear systems, to the nonlinear scenario. The convergence of the proposed algorithm is guaranteed under some mild assumptions, e.g., the strict convexity on objective functions and the consistency on constraints, even though the Lions–Mercier property is not fulfilled. In addition, we explore an inexact version of the proposed algorithm, which allows solving the subproblems approximately with some inexactness criteria. Numerical simulations on an image restoration problem demonstrate the compelling performance of the proposed algorithm.

A parameterized deteriorated PSS preconditioner and its optimization for nonsymmetric saddle point problems

Based on the deteriorated positive definite and skew-Hermitian splitting (DPSS) preconditioner, we develop a parameterized DPSS (PDPSS) preconditioner for the nonsymmetric saddle point linear systems. The spectral properties of the PDPSS preconditioned coefficient matrix are analyzed. Moreover, an upper bound of the degree of the minimal polynomial of the PDPSS preconditioned matrix is also obtained. Inasmuch as the efficiency of the PDPSS preconditioner depends on the values of its parameters, we further derive a fast and effective method to compute the optimal parameter values of the PDPSS preconditioner. Finally, numerical examples are employed to illustrate the feasibility and the efficiency of the PDPSS preconditioner.

Preconditioned modified Hermitian and skew-Hermitian splitting iteration methods for fractional nonlinear Schrödinger equations

A variant of preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method is proposed for a class of Toeplitz-like complex linear equations arising from the space fractional coupled nonlinear Schrödinger equations. Theoretical analysis show that the new method is convergent unconditionally. Moreover, the choice of the theoretical optimal parameter is further studied in detail, which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the Toeplitz matrix and the extreme absolute values of the diagonal matrix. In addition, the splitting can be used as a preconditioner for Krylov subspaces method. Numerical experiments show that the proposed method is efficient, and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem.

An Efficient Iterative Method Based on Two-Stage Splitting Methods to Solve Weakly Nonlinear Systems

In this paper, an iterative method for solving large, sparse systems of weakly nonlinear equations is presented. This method is based on Hermitian/skew-Hermitian splitting (HSS) scheme. Under suitable assumptions, we establish the convergence theorem for this method. In addition, it is shown that any faster and less time-consuming two-stage splitting method that satisfies the convergence theorem can be replaced instead of the HSS inner iterations. Numerical results, such as CPU time, show the robustness of our new method. This method is easy, fast and convenient with an accurate solution.

ERHSS iteration method for PDE optimal control problem

An extended regularized Hermitian and skew-Hermitian splitting iteration method, for solving the double saddle point linear system arising from PDE optimal control problem, is proposed. The unconditional convergence of the method is proved. Furthermore, some spectral properties of the preconditioned matrix are investigated. Numerical experiments show that the iteration method is feasible and efficient.

How to compute the minimum norm least squares solution of singular linear system by using the preconditioned HSS method?

For the singular, non-Hermitian, and positive semi-definite system of linear equations Ax = b, we introduce a kind of preconditioners for the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method. Then, the iteration sequence generated by the corresponding PHSS iteration method converges to the minimum norm least squares solution A‡b for any initial guess no matter the singular system of linear equations is consistent or inconsistent. In addition, a kind of PHSS preconditioners are derived from the PHSS iteration method. The PHSS preconditioned generalized minimum residual (PGMRES) methods also determine the minimum norm least squares solution of the consistent singular linear system at breakdown. Numerical experiments are used to verify the effectiveness and the robustness of the PHSS iteration method and the PHSS preconditioner.