Quasi-minimal Residual Research Articles

Software for simplified Lanczos and QMR algorithms

The nonsymmetric Lanczos process simplifies when applied to J-symmetric and J-Hermitian matrices, and work and storage requirements are roughly halved compared to the general case. In this paper, we describe FORTRAN-77 implementations of simplified versions of the look-ahead Lanczos algorithm and of the quasiminimal residual (QMR) method, which is a Lanczos-based iterative procedure for the solution of linear systems. These implementations of the simplified algorithms complete our software package QMRPACK, which so far contained only codes for Lanczos and QMR algorithms for general matrices. We describe in some detail the use of two routines, one for the solution of linear systems, and the other for eigenvalue computations. We present examples that lead to J-symmetric and J-Hermitian matrices. Results of numerical experiments are reported.

Frequency‐domain modelling of airborne electromagnetic responses using staggered finite differences1

AbstractA 3D frequency‐domain EM modelling code has been implemented for helicopter electromagnetic (HEM) simulations. A vector Helmholtz equation for the electric fields is employed to avoid convergence problems associated with the first‐order Maxwell's equations when air is present. Additional stability is introduced by formulating the problem in terms of the scattered electric fields. With this formulation the impressed dipole source is replaced with an equivalent source, which for the airborne configuration possesses a smoother spatial dependence and is easier to model. In order to compute this equivalent source, a primary field arising from dipole sources of either a whole space or a layered half‐space must be calculated at locations where the conductivity is different from that of the background.The Helmholtz equation is approximated using finite differences on a staggered grid. After finite‐differencing, a complex‐symmetric matrix system of equations is assembled and preconditioned using Jacobi scaling before it is solved using the quasi‐minimum residual (QMR) method. The modelling code has been compared with other 1D and 3D numerical models and is found to produce results in good agreement.We have used the solution to simulate novel HEM responses that are computationally intractable using integral equation (IE) solutions. These simulations include a 2D conductor residing at a fault contact with and without topography. Our simulations show that the quadrature response is a very good indicator of the faulted background, while the in‐phase response indicates the presence of the conductor. However when interpreting the in‐phase response, it is possible erroneously to infer a dipping conductor due to the contribution of the faulted background.

MANY MASSES ON ONE STROKE: ECONOMIC COMPUTATION OF QUARK PROPAGATORS

The computational effort in the calculation of Wilson fermion quark propagators in Lattice Quantum Chromodynamics can be considerably reduced by exploiting the Wilson fermion matrix structure in inversion algorithms based on the non-symmetric Lanczos process. We consider two such methods: QMR (quasi minimal residual) and BCG (biconjugate gradients). Based on the decomposition M/κ = 1/κ−D of the Wilson mass matrix, using QMR, one can carry out inversions on a whole trajectory of masses simultaneously, merely at the computational expense of a single propagator computation. In other words, one has to compute the propagator corresponding to the lightest mass only, while all the heavier masses are given for free, at the price of extra storage. Moreover, the symmetry γ5M = M†γ5 can be used to cut the computational effort in QMR and BCG by a factor of two. We show that both methods then become — in the critical regime of small quark masses — competitive to BiCGStab and significantly better than the standard MR method, with optimal relaxation factor, and CG as applied to the normal equations.

The quasi-minimal residual algorithm applied to complex symmetric linear systems in quantum reactive scattering

The solution of systems of linear equations Ax=b with complex symmetric coefficient matrix A of size N, typically appearing in quantum-reactive scattering problems, is discussed. The quasiminimal residual (QMR) method is introduced to solve the complex symmetric linear system and is compared to the generalized minimal residual (GMRES) method. The methods are applied to two different chemical problems: the initial state-selected reaction probability for the H2+OH→H +H2O reaction, and the cumulative reaction probability for the isomerization of ketene, both with N≳104. It is shown that the QMR method behaves more favorably, i.e., converges faster, than the GMRES for large N, especially when high accuracy is needed.

Preconditioned Krylov Subspace Methods for Lyapunov Matrix Equations

The authors study the iterative solution of Lyapunov matrix equations \[ AX + XA^T = - D^T D \] by reconditioned Krylov subspace methods. These solution techniques are of interest for problems leading to large and sparse matrices A as those arising from certain applications in large space structure control theory. We show how conjugate gradient (CG)-type methods for nonsymmetric linear systems can be applied to this type of equation utilizing the special structure when computing matrix-vector and inner products. In contrast to recently developed methods for such matrix equations based on Krylov subspaces associated with A, the authors implicitly work with the equivalent system of linear equations involving the Kronecker sum $M = A \otimes I_N + I_N \otimes A$. Motivation for this new approach comes from the observation that it allows the straightforward incorporation of preconditioners. Several preconditioners for such problems are presented and analyzed. In particular, since the solution matrix X is known to be symmetric, it is of interest to know which of the methods produces symmetric iterates in each step. It is proven that this is the case for alternating direction implicit (ADI)-type and (point) symmetric successive overrelaxation (SSOR) preconditioning in association with the quasiminimal residual (QMR) method. As numerical results show, it is essential to use reconditioning in association with Krylov subspace methods. Several preconditioners for such problems are presented and analyzed. Finally numerical examples are presented where the different preconditioners are compared.

Residual Smoothing Techniques for Iterative Methods

An iterative method for solving a linear system $Ax = b$ produces iterates $\{ x_k \} $ with associated residual norms that, in general, need not decrease “smoothly” to zero. “Residual smoothing” techniques are considered that generate a second sequence $\{ y_k \} $ via a simple relation $y_k = (1 - \eta _k )y_{k - 1} + \eta _k x_k $. The authors first review and comment on a technique of this form introduced by Schonauer and Weiss that results in $\{ y_k \} $ with monotone decreasing residual norms; this is referred to as minimal residual smoothing Certain relationships between the residuals and residual norms of the biconjugate gradient (BCG) and quasi-minimal residual (QMR) methods are then noted, from which it follows that QMR can be obtained from BCG by a technique of this form; this technique is extended to generally applicable quasi-minimal residual smoothing. The practical performance of these techniques is illustrated in a number of numerical experiments.

An Implementation of the QMR Method Based on Coupled Two-Term Recurrences

Recently, the authors proposed a new Krylov subspace iteration, the quasi-minimal residual (QMR) algorithm, for solving non-Hermitian linear systems. In the original implementation of the QMR method, the Lanczos process with look-ahead is used to generate basis vectors for the underlying Krylov subspaces. In the Lanczos algorithm, these basis vectors are computed by means of three-term recurrences. It has been observed that, in finite-precision arithmetic, vector iterations based on three-term recursions are usually less robust than mathematically equivalent coupled two-term vector recurrences. This paper presents a look-ahead algorithm that constructs the Lanczos basis vectors by means of coupled two-term recursions. Some implementation details are given, and the look-ahead strategy is described. A new implementation of the QMR method, based on this coupled two-term algorithm, is proposed. A simplified version of the QMR algorithm without look-ahead is also presented, and the special case of QMR for complex symmetric linear systems is considered. Results of numerical experiments comparing the original and the new implementations of the QMR method are reported.

A Quasi-Minimal Residual Variant of the Bi-CGSTAB Algorithm for Nonsymmetric Systems

Motivated by a recent method of Freund [SIAM J. Sci. Comput., 14 (1993), pp. 470–482], who introduced a quasi-minimal residual (QMR) version of the conjugate gradients squared (CGS) algorithm, a QMR variant of the biconjugate gradient stabilized (Bi-CGSTAB) algorithm of van der Vorst that is called QMRCGSTAB, is proposed for solving nonsymmetric linear systems. The motivation for both QMR variants is to obtain smoother convergence behavior of the underlying method. The authors illustrate this by numerical experiments that also show that for problems on which Bi-CGSTAB performs better than CGS, the same advantage carries over to QMRCGSTAB.

A Family of Quasi-Minimal Residual Methods for Nonsymmetric Linear Systems

The quasi-minimal residual (QMR) algorithm by Freund, Gutknecht, and Nachtigal [Research Institute forAdvanced Computer Science Tech. Report 91.09, NASA Ames Research Center, Moffett Field, CA; Numer. Math., 60 (1991), pp. 315–339], in addition to its ability to avoid breakdowns, also improves the irregular convergence behavior encountered by the biconjugate gradient (BiCG) method through the incorporation of quasi minimization. This paper studies this quasi-minimal residual approach applied, for simplicity, to the nonlook-ahead version of the QMR method. In particular, a family of quasi-minimal residual methods is defined where on both ends lie the original QMR method and a minimal residual method while lying in between are the variants derived here. These variants are shown to consume a little more computational work per iterations than the original QMR method, but generally give an even better convergence behavior as well as lower iteration counts. The same idea was also applied to a recently proposed transpose-free quasi-minimal residual method. In particular, a few quasi-minimal residual variants of the Van der Vorst BiCGSTAB method are derived. Numerical results are presented comparing the quasi-minimal residual and transpose-free quasi-minimal residual variants derived in this paper.

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.