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. A novel BCG like approach is presented called 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

Closer to the solutions: iterative linear solvers

The solution of dense linear systems received much attention after the second world war, and by the end of the sixties, most of the problems associated with it had been solved. For a long time, Wilkinson's \The Algebraic Eigenvalue Problem" [107], other than the title suggests, became also the standard textbook for the solution of linear systems. When it became clear that partial dierential equations could be solved numerically, to a level of accuracy that was of interest for application areas (such as reservoir engineering, and reactor diusion modeling), there was a strong need for the fast solution of the discretized systems, and iterative methods became popular for these problems

Recent advances in Lanczos-based iterative methods for nonsymmetric linear systems

In recent years, there has been a true revival of the nonsymmetric Lanczos method. On the one hand, the possible breakdowns in the classical algorithm are now better understood, and so-called look-ahead variants of the Lanczos process have been developed, which remedy this problem. On the other hand, various new Lanczos-based iterative schemes for solving nonsymmetric linear systems have been proposed. This paper gives a survey of some of these recent developments

Low-rank Linear Fluid-structure Interaction Discretizations

Fluid-structure interaction models involve parameters that describe the solid and the fluid behavior. In simulations, there often is a need to vary these parameters to examine the behavior of a fluid-structure interaction model for different solids and different fluids. For instance, a shipping company wants to know how the material, a ship's hull is made of, interacts with fluids at different Reynolds and Strouhal numbers before the building process takes place. Also, the behavior of such models for solids with different properties is considered before the prototype phase. A parameter-dependent linear fluid-structure interaction discretization provides approximations for a bundle of different parameters at one step. Such a discretization with respect to different material parameters leads to a big block-diagonal system matrix that is equivalent to a matrix equation as discussed in [KressnerTobler 2011]. The unknown is then a matrix which can be approximated using a low-rank approach that represents the iterate by a tensor. This paper discusses a low-rank GMRES variant and a truncated variant of the Chebyshev iteration. Bounds for the error resulting from the truncation operations are derived. Numerical experiments show that such truncated methods applied to parameter-dependent discretizations provide approximations with relative residual norms smaller than $10^{-8}$ within a twentieth of the time used by individual standard approaches.Comment: 30 pages, 7 figure

Absolute value preconditioning for symmetric indefinite linear systems

We introduce a novel strategy for constructing symmetric positive definite (SPD) preconditioners for linear systems with symmetric indefinite matrices. The strategy, called absolute value preconditioning, is motivated by the observation that the preconditioned minimal residual method with the inverse of the absolute value of the matrix as a preconditioner converges to the exact solution of the system in at most two steps. Neither the exact absolute value of the matrix nor its exact inverse are computationally feasible to construct in general. However, we provide a practical example of an SPD preconditioner that is based on the suggested approach. In this example we consider a model problem with a shifted discrete negative Laplacian, and suggest a geometric multigrid (MG) preconditioner, where the inverse of the matrix absolute value appears only on the coarse grid, while operations on finer grids are based on the Laplacian. Our numerical tests demonstrate practical effectiveness of the new MG preconditioner, which leads to a robust iterative scheme with minimalist memory requirements

Parallel Iterative Solution Methods for Linear Systems arising from Discretized PDE's

In these notes we will present an overview of a number of related iterative methods for the solution of linear systems of equations. These methods are so-called Krylov projection type methods and the include popular methods as Conjugate Gradients, Bi-Conjugate Gradients, CGST Bi-CGSTAB, QMR, LSQR and GMRES. We will show how these methods can be derived from simple basic iteration formulas. We will not give convergence proofs, but we will refer for these, as far as available, to litterature. Iterative methods are often used in combination with so-called preconditioning operators (approximations for the inverses of the operator of the system to be solved). Since these preconditions are not essential in the derivation of the iterative methods, we will not give much attention to them in these notes. However, in most of the actual iteration schemes, we have included them in order to facilitate the use of these schemes in actual computations. For the application of the iterative schemes one usually thinks of linear sparse systems, e.g., like those arising in the finite element or finite difference approximatious of (systems of) partial differential equations. However, the structure of the operators plays no explicit role in any of these schemes, and these schemes might also successfully be used to solve certain large dense linear systems. Depending on the situation that might be attractive in terms of numbers of floating point operations. It will turn out that all of the iterative are parallelizable in a straight forward manner. However, especially for computers with a memory hierarchy (i.e. like cache or vector registers), and for distributed memory computers, the performance can often be improved significantly through rescheduling of the operations. We will discuss parallel implementations, and occasionally we will report on experimental findings

A biconjugate gradient type algorithm on massively parallel architectures

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. Recently, Freund and Nachtigal have proposed a novel BCG type approach, the quasi-minimal residual method (QMR), which overcomes the problems of BCG. Here, an implementation is presented of QMR based on an s-step version of the nonsymmetric look-ahead Lanczos algorithm. The main feature of the s-step Lanczos algorithm is that, in general, all inner products, except for one, can be computed in parallel at the end of each block; this is unlike the other standard Lanczos process where inner products are generated sequentially. The resulting implementation of QMR is particularly attractive on massively parallel SIMD architectures, such as the Connection Machine

The Anderson model of localization: a challenge for modern eigenvalue methods

We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large, sparse, real, symmetric, and indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the implicitly restarted Arnoldi method coupled with polynomial and several shift-and-invert convergence accelerators as well as with a sparse hybrid tridiagonalization method. We demonstrate that for our problem the Lanczos implementation is faster and more memory efficient than the other approaches. This seemingly innocuous problem presents a major challenge for all modern eigenvalue algorithms.Comment: 16 LaTeX pages with 3 figures include

BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

In the present paper, we introduce a new extension of the conjugate residual (CR) for solving non-Hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient (BiCG), biconjugate residual (BiCR) and biconjugate A-orthogonal residual (BiCOR) methods. The proposed Krylov subspace method, referred to as the BiCGCR2 method, is based on short-term vector recurrences and is mathematically equivalent to both BiCR and BiCOR. We demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than BiCG, BiCR and BiCOR. Hence, it may be exploited for the development of new variants of non-optimal Krylov subspace methods