Abstract:
We establish two types of block triangular preconditioners applied to the linear saddle point problems with the singular (1,1) block. These preconditioners are based on the results presented in the paper of Rees and Greif (2007). We study the spectral characteristics of the preconditioners and show that all eigenvalues of the preconditioned matrices are strongly clustered. The choice of the parameter is involved. Furthermore, we give the optimal parameter in practical. Finally, numerical experiments are also reported for illustrating the efficiency of the presented preconditioners.

Abstract:
For the singular saddle-point problems with nonsymmetric positive definite $(1,1)$ block, we present a general constraint preconditioning (GCP) iteration method based on a singular constraint preconditioner. Using the properties of the Moore-Penrose inverse, the convergence properties of the GCP iteration method are studied. In particular, for each of the two different choices of the $(1,1)$ block of the singular constraint preconditioner, a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix. Numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the GCP iteration method. Moreover, the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual (GMRES) and quasi-minimal residual (QMR) methods are also tested.

Abstract:
In this paper, the generalized shift-splitting preconditioner is implemented for saddle point problems with symmetric positive definite (1,1)-block and symmetric positive semidefinite (2,2)-block. The proposed preconditioner is extracted form a stationary iterative method which is unconditionally convergent. Moreover, a relaxed version of the proposed preconditioner is presented and some properties of the eigenvalues distribution of the corresponding preconditioned matrix are studied. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioners.

Abstract:
In this paper, we propose a preconditioner based on the shift-splitting method for generalized saddle point problems with nonsymmetric positive definite (1,1)-block and symmetric positive semidefinite $(2,2)$-block. The proposed preconditioner is obtained from an basic iterative method which is unconditionally convergent. We also present a relaxed version of the proposed method. Some numerical experiments are presented to show the effectiveness of the method.

Abstract:
In this paper, we present a new alternating local Hermitian and skew-Hermitian splitting preconditioner for solving saddle point problems. The spectral property of the preconditioned matrices is studies in detail. Theoretical results show all eigenvalues of the preconditioned matrices will generate two tight clusters, one is near (0, 0) and the other is near (2, 0) as the iteration parameter tends to zero from positive. Numerical experiments are given to validate the performances of the preconditioner. Mathematical suject classification: Primary: 65F10; Secondary: 65F50.

Abstract:
A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node bifurcation point in the weak-noise limit, while the final value of the deterministic solution changes discontinuously at the point. A systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method. In this formulation, the critical nature turns out to originate from the neutrality of exiting time from a saddle-point. The theoretical calculation explains results of numerical simulations.

Abstract:
We study the Langevin equation for a single harmonic saddle as an elementary model for the beta-relaxation in supercooled liquids close to Tc. The input of the theory is the spectrum of the eigenvalues of the dominant stationary points at a given temperature. We prove in general the existence of a time-scale t_eps, which is uniquely determined by the spectrum, but is not simply related to the fraction of negative eigenvalues. The mean square displacement develops a plateau of length t_eps, such that a two-step relaxation is obtained if t_eps diverges at Tc. We analyze the specific case of a spectrum with bounded left tail, and show that in this case the mean square displacement has a scaling dependence on time identical to the beta-relaxation regime of Mode Coupling Theory, with power law approach to the plateau and power law divergence of t_eps at Tc.

Abstract:
We investigate the solution of large linear systems of saddle point type with singular block by preconditioned iterative methods and consider two parameterized block triangular preconditioners used with Krylov subspace methods which have the attractive property of improved eigenvalue clustering with increased ill-conditioning of the block of the saddle point matrix, including the choice of the parameter. Meanwhile, we analyze the spectral characteristics of two preconditioners and give the optimal parameter in practice. Numerical experiments that validate the analysis are presented. 1. Introduction We study preconditioners for general nonsingular linear systems of the type Such systems arise in a large number of applications, for example, the (linearized) Navier-Stokes equations and other physical problems with conservation laws as well as constrained optimization problems [1–5]. As such systems are typically large and sparse, solution by iterative methods has been studied extensively [1, 5–16]. Much attention has focused on the Navier-Stokes problem; see, for example, [1, 2, 5, 17, 18]. The techniques for solving systems like (1) are so numerous that it is almost impossible to give an overview. In addition to the methods developed specifically for Navier-Stokes problems, existing techniques also include splitting schemes [2, 6, 12, 19, 20], constraint preconditioning [2, 20, 21], Uzawa-type algorithms [2, 12, 22], and (preconditioned) Krylov subspace methods based on (approximations to) the Schur complement [2, 16, 23]. We start with augmentation block triangular preconditioners for the general system (1); see Section 2 for our assumptions. When is nonsingular, results for the general system have been obtained before; for example, Murphy et al. [23, 24] propose the block diagonal Schur complement preconditioner and the block triangular Schur complement preconditioner as follows: If defined, it has been shown that the preconditioned matrices (cf. [24]) are diagonalizable and have only three distinct eigenvalues , and two distinct eigenvalues , , respectively. However, when is singular, it cannot be inverted and the Schur complement does not exist. For symmetric saddle point systems, that is, and is symmetric, one possible way of dealing with the systems is by augmentation, that is, by replacing with , where is an symmetric positive definite weight matrix [4, 7, 9, 14, 17, 18, 25–27]. Recently, for symmetric saddle point systems with block that has a high nullity, Greif and Sch？tzau [25, 26] studied the application of the following block diagonal

Abstract:
A relaxed splitting preconditioner based on matrix splitting is introduced in this paper for linear systems of saddle point problem arising from numerical solution of the incompressible Navier-Stokes equations. Spectral analysis of the preconditioned matrix is presented, and numerical experiments are carried out to illustrate the convergence behavior of the preconditioner for solving both steady and unsteady incompressible flow problems.

Abstract:
in this paper, we explore the block triangular preconditioning techniques applied to the iterative solution of the saddle point linear systems arising from the discretized maxwell equations. theoretical analysis shows that all the eigenvalues of the preconditioned matrix arestrongly clustered. numerical experiments are given to demonstrate the efficiency of the presented preconditioner. mathematical subject classification: 65f10.