A class of nonsymmetric preconditioners for saddle point problems

For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric

Some Preconditioning Techniques for Saddle Point Problems

Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud \ud The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336

A fast normal splitting preconditioner for attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian

A linearly implicit conservative difference scheme is applied to discretize the attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian. Complex symmetric linear systems can be obtained, and the system matrices are indefinite and Toeplitz-plus-diagonal. Neither efficient preconditioned iteration method nor fast direct method is available to deal with these systems. In this paper, we propose a novel matrix splitting iteration method based on a normal splitting of an equivalent real block form of the complex linear systems. This new iteration method converges unconditionally, and the quasi-optimal iteration parameter is deducted. The corresponding new preconditioner is obtained naturally, which can be constructed easily and implemented efficiently by fast Fourier transform. Theoretical analysis indicates that the eigenvalues of the preconditioned system matrix are tightly clustered. Numerical experiments show that the new preconditioner can significantly accelerate the convergence rate of the Krylov subspace iteration methods. Specifically, the convergence behavior of the related preconditioned GMRES iteration method is spacial mesh-size-independent, and almost fractional order insensitive. Moreover, the linearly implicit conservative difference scheme in conjunction with the preconditioned GMRES iteration method conserves the discrete mass and energy in terms of a given precision

Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods