The Bramble-Pasciak preconditioner for saddle point problems

The Bramble-Pasciak Conjugate Gradient method is a well known tool to solve linear systems in saddle point form. A drawback of this method in order to ensure applicability of Conjugate Gradients is the need for scaling the preconditioner which typically involves the solution of an eigenvalue problem. Here, we introduce a modified preconditioner and inner product which without scaling enable the use of a MINRES variant and can be used for the simplified Lanczos process. Furthermore, the modified preconditioner and inner product can be combined with the original Bramble-Pasciak setup to give new preconditioners and inner products. We undermine the new methods by showing numerical experiments for Stokes problems

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

USRA/RIACS

The Research Institute for Advanced Computer Science (RIACS) was established by the Universities Space Research Association (USRA) at the NASA Ames Research Center (ARC) on 6 June 1983. RIACS is privately operated by USRA, a consortium of universities with research programs in the aerospace sciences, under a cooperative agreement with NASA. The primary mission of RIACS is to provide research and expertise in computer science and scientific computing to support the scientific missions of NASA ARC. The research carried out at RIACS must change its emphasis from year to year in response to NASA ARC's changing needs and technological opportunities. A flexible scientific staff is provided through a university faculty visitor program, a post doctoral program, and a student visitor program. Not only does this provide appropriate expertise but it also introduces scientists outside of NASA to NASA problems. A small group of core RIACS staff provides continuity and interacts with an ARC technical monitor and scientific advisory group to determine the RIACS mission. RIACS activities are reviewed and monitored by a USRA advisory council and ARC technical monitor. Research at RIACS is currently being done in the following areas: Parallel Computing; Advanced Methods for Scientific Computing; Learning Systems; High Performance Networks and Technology; Graphics, Visualization, and Virtual Environments

Lanczos-type solvers for nonsymmetric linear systems of equations

Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by look-ahead are also discusse

A parallel algorithm of ICSYM for complex symmetric linear systems in quantum chemistry

Computational effort is a common issue for solving large-scale complex symmetric linear systems, particularly in quantum chemistry applications. In order to alleviate this problem, we propose a parallel algorithm of improved conjugate gradient-type iterative (CSYM). Using three-term recurrence relation and orthogonal properties of residual vectors to replace the tridiagonalization process of classical CSYM, which allows to decrease the degree of the reduce-operator from two to one communication at each iteration and to reduce the amount of vector updates and vector multiplications. Several numerical examples are implemented to show that high performance of proposed improved version is obtained both in convergent rate and in parallel efficiency