Method For non-Hermitian Linear Systems Research Articles

Asynchronous iterations of HSS method for non-Hermitian linear systems

A general asynchronous alternating iterative model is designed, for which convergence is theoretically ensured both under classical spectral radius bound and, then, for a classical class of matrix splittings for -matrices. The computational model can be thought of as a two-stage alternating iterative method, which well suits to the well-known Hermitian and skew-Hermitian splitting (HSS) approach, with the particularity here of considering only one inner iteration. Experimental parallel performance comparison is conducted between the generalized minimal residual (GMRES) algorithm, the standard HSS and our asynchronous variant, on both real and complex non-Hermitian linear systems, respectively, arising from convection–diffusion and structural dynamics problems. A significant gain on execution time is observed in both cases.

Convergence conditions for splitting iteration methods for non-Hermitian linear systems

Necessary and sufficient convergence conditions are studied for splitting iteration methods for non-Hermitian system of linear equations when the coefficient matrix is nonsingular. When this theory is specialized to the generalized saddle-point problem, we obtain convergence theorem for a class of modified accelerated overrelaxation iteration methods, which include the Uzawa and the inexact Uzawa methods as special cases. Moreover, we apply this theory to the two-stage iteration methods for non-Hermitian positive definite linear systems, and obtain sufficient conditions for guaranteeing the convergence of these methods.

A finite integration method for conformal, structured-grid, electromagnetic simulation

We describe a numerical scheme for solving Maxwell’s equations in the frequency domain on a conformal, structured, non-orthogonal, multi-block mesh. By considering Maxwell’s equations in a volume parameterized by dimensionless curvilinear coordinates, we obtain a set of tensor equations that are a continuum analogue of common circuit equations, and that separate the metrical and metric-free parts of Maxwell’s equations and the material constitutive relations. We discretize these equations using a new formulation that treats the electric field and magnetic induction using simple basis-function representations to obtain a discrete form of Faraday’s law of induction, but that uses finite integral representations for the displacement current and magnetic field to obtain a discrete form of Ampere’s law, as in the finite integration technique [T. Weiland, A discretization method for the solution of Maxwell’s equations for six-component fields, Electron. Commun. (AE U) 31 (1977) 116; T. Weiland, Time domain electromagnetic field computation with finite difference methods, Int. J. Numer. Model: Electron. Netw. Dev. Field 9 (1996) 295–319]. We thereby derive new projection operators for the discrete tensor material equations and obtain a compact numerical scheme for the discrete differential operators. This scheme is shown to exhibit significantly reduced numerical dispersion when compared to the standard linear finite element method. We take advantage of the mesh structure on a block-by-block basis to implement these numerical operators efficiently, and achieve computational speed with modest memory requirements when compared to explicit sparse matrix storage. Using the Jacobi–Davidson [G.L.G. Sleijpen, H.A. van der Vorst, A Jacobi–Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17 (2) (1996) 401–425; S.J. Cooke, B. Levush, Eigenmode solution of 2-D and 3-D electromagnetic cavities containing absorbing materials using the Jacobi–Davidson algorithm, J. Comput. Phys. 157 (1) (2000) 350–370] and quasi-minimal residual [R.W. Freund, N.M. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math. 60 (1991) 315–339] iterative matrix solution algorithms, we solve the resulting discrete matrix eigenvalue equations and demonstrate the convergence characteristics of the algorithm. We validate the model for three-dimensional electromagnetic problems, both cavity eigenvalue solutions and a waveguide scattering matrix calculation.

Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems

The convergence of the Krylov subspace methods, e.g., Full Orthogonal Method (FOM) and Generalized Minimal Residual Method (GMRES), etc., for solving large non-Hermitian linear systems is studied in a unified and detailed way when the coefficient matrix is defective; in particular, when its spectrum lies in the open right (left) half plane or is on the real axis. Related theoretical error bounds are established, which reveal some intrinsic relationships between the convergence properties and the eigen-characteristics of the coefficient matrix. These results not only generalize all the known ones for the diagonalizable matrices in the literature, but also sharp the corresponding estimates in Jia (Acta Mathematica Sinica (New Series) 14 (1998) 507–518).

Acceleration of Relaxation Methods for Non-Hermitian Linear Systems

Let $A = I - B \in \mathbb{C}^{n,n} $, with diag$(B) = 0$, denote a nonsingular non-Hermitian matrix. To iteratively solve the linear system $A{\bf x} = {\bf b}$, two splittings of A, together with induced relaxation methods, have been recently investigated in [W. Niethammer and R. S. Varga, Results in Math., 16 (1989), pp. 308–320]. The Hermitian splitting of A is defined by $A = M^h - N^h $, where $M^h : = ( A + A^ * )/2$ is the Hermitian part of A. The skew-Hermitian splitting of A is similarly defined by $A = M^s - N^s $ with $M^s : = I + ( A - A^ *)/2$. This paper considers k-step iterative methods to accelerate the relaxation schemes (involving a relaxation factor $\omega $) that are generated by these two splittings. The primary interest is not to determine the optimal relaxation factor $\omega $ that minimizes the spectral radius of the associated iteration operator. Rather, a value of $\omega $ is sought such that the resulting relaxation method can be most efficiently accelerated by a k-step method. For the Hermitian splitting, the choice $\omega = 1$ (together with a suitable Chebyshev acceleration) turns out to be optimal in this sense. For the skew-Hermitian splitting, a hybrid scheme is proposed that is nearly optimal. As another application of this latter hybrid procedure, the block Jacobi method arising from a model equation for a convection-diffusion problem is analyzed.

QMR: a quasi-minimal residual method for non-Hermitian linear systems

The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.

Relaxation Methods for Non-Hermitian Linear Systems

For the matrix equation Ax = b, we consider here two splittings A = M1 − N1 = M2 − N2 of the matrix A, where M1 ≔ (A + A*)/2 is the Hermitian part of A, and M2 ≔ I + (A − A*)/2 is the identity plus the skew-Hermitian part of A. To these two splittings of A, we apply an extrapolation, with extrapolation factor ω, and we find associated regions for ω, in the complex plane, for which these extrapolated splittings yield convergent iterative methods. From this, further applications to semiiterative methods are indicated.