Matrix Splitting Preconditioner Research Articles

Rotated block diagonal preconditioners for Navier-Stokes control problems

Fluid flow control problems play a crucial role in various industrial applications. The optimal control of the Navier-Stokes equations poses a significant challenge. By employing Oseen's approximation to linearize the problem, we encounter a series of large sparse structured linear systems. These coefficient matrices exhibit a two-by-two block structure with square blocks. Leveraging this block structure, we exploit matrix splitting preconditioners. Theoretical analysis indicates that the real parts of all eigenvalues of preconditioned matrix are equal to 1/2. A large number of eigenvalues are clustered in (1±i)/2. The imaginary part of the eigenvalues is bounded explicitly. The eigenvalue distribution predicts the fast convergence of Krylov subspace methods. To avoid solving the saddle point subsystems in the preconditioning procedure, we propose a practical variant of the preconditioner. The theoretical analysis demonstrates that if the approximation is sufficiently close to Schur complement, the eigenvalues of the modified preconditioned matrix will lie within a circle centered at the original eigenvalues with a radius less than 1. This implies that the modified preconditioner exhibits excellent performance in terms of preconditioning. The numerical experiments demonstrate that the generalized minimal residual method, when combined with the proposed preconditioners, proves to be an efficient and effective solution for Oseen's control optimization with a variable viscosity coefficient. The number of iterations required by the preconditioned methods remains unaffected by the mesh size used in finite element discretization and only marginally depends on the regularization parameter.

A modified new matrix splitting preconditioner for double saddle point problems

A modified new matrix splitting preconditioner for double saddle point problems

Fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations

The discretizations of two- and three-dimensional spatial fractional diffusion equations with the shifted finite-difference formulas of the Grünwald-Letnikov type can result in discrete linear systems whose coefficient matrices are of the form D+T, where D is a nonnegative diagonal matrix and T is a block-Toeplitz with Toeplitz-block matrix or a block-Toeplitz with each block being block-Toeplitz with Toeplitz-block matrix. For these discrete spatial fractional diffusion matrices, we construct diagonal and block-circulant with circulant-block splitting preconditioner for the two-dimensional case, and diagonal and block-circulant with each block being block-circulant with circulant-block splitting preconditioner for the three-dimensional case, to further accelerate the convergence rates of Krylov subspace iteration methods, and we analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1, and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the Krylov subspace iteration methods. Moreover, this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant-plus-diagonal preconditioners in both iteration counts and computing times.

Spectral analysis of the matrix splitting preconditioner for block two-by-two linear systems

This paper further generalizes the matrix splitting preconditioner proposed by Huang et al. (J Comput Appl Math 332:1–12, 2018) to solve the general block two-by-two linear systems. All eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 by analyzing its spectral properties in detail. Numerical results are provided to support the validity of the theoretical results and demonstrate the effectiveness of the preconditioner by making a comparison of it with the classical block-diagonal and block-triangular preconditioners applied to the preconditioned generalized minimal residual method (GMRES).

Quasi‐Toeplitz splitting iteration methods for unsteady space‐fractional diffusion equations

We construct a class of quasi‐Toeplitz splitting iteration methods to solve the two‐sided unsteady space‐fractional diffusion equations with variable coefficients. By making full use of the structural characteristics of the coefficient matrix, the method only requires computational costs of O(n log n) with n denoting the number of degrees of freedom. We develop an appropriate circulant matrix to replace the Toeplitz matrix as a preconditioner. We discuss the spectral properties of the quasi‐circulant splitting preconditioned matrix. Numerical comparisons with existing approaches show that the present method is both effective and efficient when being used as matrix splitting preconditioners for Krylov subspace iteration methods.

A note on the block alternating splitting implicit iteration method for complex saddle‐point problems

SummaryRecently, Bai has proposed a class of block alternating splitting implicit (BASI) iteration methods [Numer. Linear Algebra Appl. 19(6):914‐936] for the complex generalized saddle‐point problems. The author demonstrates the unconditional convergence of the methods when the (1,1) block of the saddle‐point coefficient matrix is positive definite. In this paper, we display that the BASI iteration method is convergent for wider problems, for instance, the (1,1) block is positive semidefinite, provided the splittings meet certain requirements. The theoretical result encourages us to explore more efficient iteration schemes for more general problems. We present new BASI iteration schemes for the algebraic linear systems from time‐harmonic eddy current models. The corresponding matrix splitting preconditioners are presented for the generalized minimal residual method. Experimental results show the feasibility and effectiveness of our new iteration methods and the corresponding preconditioned generalized minimal residual methods.

Regularized HSS iteration methods for stabilized saddle-point problems

Abstract We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval $(0, \, 2)$ when the iteration parameter is close to $0$ and, furthermore, they can be clustered near $0$ and $2$ when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

Respectively scaled HSS iteration methods for solving discretized spatial fractional diffusion equations

SummaryFor the discrete linear systems resulted from the discretization of the one‐dimensional anisotropic spatial fractional diffusion equations of variable coefficients with the shifted finite‐difference formulas of the Grünwald–Letnikov type, we propose a class of respectively scaled Hermitian and skew‐Hermitian splitting iteration method and establish its asymptotic convergence theory. The corresponding induced matrix splitting preconditioner, through further replacements of the involved Toeplitz matrices with certain circulant matrices, leads to an economic variant that can be executed by fast Fourier transforms. Both theoretical analysis and numerical implementations show that this fast respectively scaled Hermitian and skew‐Hermitian splitting preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods employed as effective linear solvers for the target discrete linear systems.

Modified PHSS iterative methods for solving nonsingular and singular saddle point problems

For large sparse saddle point problems, we establish a new version of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method, called the modified PHSS (MPHSS) method in this paper. Then, we theoretically study its convergence and semi-convergence properties and determine its optimal iteration parameter and corresponding optimal convergence factor. Furthermore, the spectral properties of the MPHSS preconditioned matrix are discussed in detail. Numerical experiments show that the MPHSS iteration method is effective and robust when it is used either as a solver or as a matrix splitting preconditioner for the generalized minimal residual (GMRES) method.

An efficient two-step iterative method for solving a class of complex symmetric linear systems

In this paper, a new two-step iterative method called the two-step parameterized (TSP) iteration method for a class of complex symmetric linear systems is developed. We investigate its convergence conditions and derive the quasi-optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the TSP iteration method. Meanwhile, some more practical ways to choose iteration parameters for the TSP iteration method are proposed. Furthermore, comparisons of the TSP iteration method with some existing ones are given, which show that the upper bound of the spectral radius of the TSP iteration method is smaller than those of the modified Hermitian and skew-Hermitian splitting (MHSS), the preconditioned MHSS (PMHSS), the combination method of real part and imaginary part (CRI) and the parameterized variant of the fixed-point iteration adding the asymmetric error (PFPAE) iteration methods proposed recently. Inexact version of the TSP iteration (ITSP) method and its convergence properties are also presented. Numerical experiments demonstrate that both TSP and ITSP are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems.

On a type of matrix splitting preconditioners for a class of block two-by-two linear systems

Recently, Bai et al. (2013) proposed an effective and efficient matrix splitting iterative method, called preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method, for two-by-two block linear systems of equations. The eigenvalue distribution of the iterative matrix suggests that the splitting matrix could be advantageously used as a preconditioner. In this study, the CGNR method is utilized for solving the PMHSS preconditioned linear systems, and the performance of the method is considered by estimating the condition number of the normal equations. Furthermore, the proposed method is compared with other PMHSS preconditioned Krylov subspace methods by solving linear systems arising in complex partial differential equations and a distributed control problem. The numerical results demonstrate the difference in the performance of the methods under consideration.

Spectral properties of the matrix splitting preconditioners for generalized saddle point problems

In this paper, we study the spectral properties of two different kinds of preconditioners for generalized saddle point problems. One is based on the two-parameter matrix splitting preconditioner for saddle point problems by Wang et al., we generalize this preconditioner to generalized saddle point problems and analyze the spectral properties of the corresponding preconditioned matrix. The other is based on the Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems by Huang et al., we study the spectral properties of the HSS preconditioner with two different parameters α and β (the generalized HSS preconditioner) for generalized saddle point problems. In addition, some numerical tests are given to verify the validity of the presented theoretical results.

Quasi‐HSS iteration methods for non‐Hermitian positive definite linear systems of strong skew‐Hermitian parts

SummaryFor large sparse non‐Hermitian positive definite linear systems, we establish exact and inexact quasi‐HSS iteration methods and discuss their convergence properties. Numerical experiments show that both iteration methods are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods. In addition, these two iteration methods are, respectively, much more powerful than the exact and inexact HSS iteration methods, especially when the linear systems have nearly singular Hermitian parts or strongly dominant skew‐Hermitian parts, and they can be employed to solve non‐Hermitian indefinite linear systems with only mild indefiniteness.

A new matrix splitting preconditioner for generalized saddle point problems

For generalized saddle point problems, we establish a new matrix splitting preconditioner and give the implementing process in detail. The new preconditioner is much easier to be implemented than the modified dimensional split (MDS) preconditioner. The convergence properties of the new splitting iteration method are analyzed. The eigenvalue distribution of the new preconditioned matrix is discussed and an upper bound for the degree of its minimal polynomial is derived. Finally, some numerical examples are carried out to verify the effectiveness and robustness of our preconditioner on generalized saddle point problems discretizing the incompressible Navier–Stokes equations.

A new iterative method for solving a class of complex symmetric system of linear equations

We present a new stationary iterative method, called Scale-Splitting (SCSP) method, and investigate its convergence properties. The SCSP method naturally results in a simple matrix splitting preconditioner, called SCSP-preconditioner, for the original linear system. Some numerical comparisons are presented between the SCSP-preconditioner and several available block preconditioners, such as PGSOR (Hezari et al. Numer. Linear Algebra Appl. 22, 761---776, 2015) and rotate block triangular preconditioners (Bai Sci. China Math. 56, 2523---2538, 2013), when they are applied to expedite the convergence rate of Krylov subspace iteration methods for solving the original complex system and its block real formulation, respectively. Numerical experiments show that the SCSP-preconditioner can compete with PGSOR-preconditioner and even more effective than the rotate block triangular preconditioners.

Spectral properties of a class of matrix splitting preconditioners for saddle point problems

Based on the accelerated Hermitian and skew-Hermitian splitting iteration scheme (Bai and Golub, 2007), we propose a new two-parameter matrix splitting preconditioner in this paper. Spectral properties of the preconditioned matrix are analyzed in detail. Furthermore, based on this preconditioner, an improved version of matrix splitting preconditioner is presented and analyzed. Finally, performance of the preconditioners is compared by using GMRES(m) as an iterative solver on linear systems arising from the discretization of Stokes and Navier–Stokes equations.