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On ApproximateInverse Preconditioners
, 1995
"... We investigate the use of sparse approximateinverse preconditioners for the iterative solution of unsymmetric linear systems of equations. Such methods are of particular interest because of the considerable scope for parallelization. We propose a number of enhancements which may improve their perfo ..."
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We investigate the use of sparse approximateinverse preconditioners for the iterative solution of unsymmetric linear systems of equations. Such methods are of particular interest because of the considerable scope for parallelization. We propose a number of enhancements which may improve their performance. When run in a sequential environment, these methods can perform unfavourably when compared with other techniques. However, they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment. 1 Current reports available by anonymous ftp from joyousgard.cc.rl.ac.uk (internet 130.246.9.91) in the directory "pub/reports". Computing and Information Systems Department, Atlas Centre, Rutherford Appleton Laboratory, Oxfordshire OX11 0QX, England. June 23, 1995. 1 INTRODUCTION 1 1 Introduction Suppose that A is a real n by n unsymmetric matrix, whose columns are a j , 1 j n. We are principally concerned wit...
On an augmented Lagrangianbased preconditioning of Oseen type problems
"... The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized NavierStokes equations (Oseen’s problem). We utilize the socalled augmented Lagrangian approach, where the original linear system of equations is ..."
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The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized NavierStokes equations (Oseen’s problem). We utilize the socalled augmented Lagrangian approach, where the original linear system of equations is first transformed to an equivalent one, which latter is then solved by a preconditioned iterative solution method. The matrices in the linear systems, arising after the discretization of Oseen’s problem, are of twobytwo block form as are the best known preconditioners for these. In the augmented Lagrangian formulation, a scalar regularization parameter is involved, which strongly influences the quality of the blockpreconditioners for the system matrix (referred to as outer), as well as the conditioning and the solution of systems with the resulting pivot block (referred to as inner) which, in the case of large scale numerical simulations has also to be solved using an iterative method. We analyse the impact of the value of the regularization parameter on the convergence of both outer and inner solution methods. The particular preconditioner used in this work exploits the inverse of the pressure mass matrix. We study the effect of various approximations of that inverse on the performance of the preconditioners, in particular that of a sparse approximate inverse, computed in an elementbyelement fashion. We analyse and compare the spectra of the preconditioned matrices for the different approximations and show that the resulting preconditioner is independent of problem, discretization and method parameters, namely, viscosity, mesh size, mesh anisotropy. We also discuss possible approaches to solve the modified pivot matrix block. Keywords: NavierStokes equations, saddle point systems, augmented Lagrangian, finite elements, approximation of mass matrixiterative methods, preconditioning 1
IMPROVED BALANCED INCOMPLETE FACTORIZATION
"... In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix, and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Marín, J. Mas and M. Tůma, SIAM J. Sci. Comput., 30 (2008), pp. 2302– 2318]. The improvements ..."
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Cited by 2 (1 self)
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In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix, and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Marín, J. Mas and M. Tůma, SIAM J. Sci. Comput., 30 (2008), pp. 2302– 2318]. The improvements are based on a deeper understanding of the Inverse ShermanMorrison (ISM) decomposition and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors influence numerically each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systems.
Robust preconditioning methods for algebraic problems, arising in multiphase flow models
, 2011
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SEE PROFILE
, 2011
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
ON INCREMENTAL CONDITION ESTIMATORS IN THE 2NORM
"... Abstract. The paper deals with estimating the condition number of triangular matrices in the Euclidean norm. The two main incremental methods, based on the work of Bischof and on the later work of Duff and Vömel, are compared. The paper presents new theoretical results revealing their similarities ..."
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Abstract. The paper deals with estimating the condition number of triangular matrices in the Euclidean norm. The two main incremental methods, based on the work of Bischof and on the later work of Duff and Vömel, are compared. The paper presents new theoretical results revealing their similarities and differences. As typical in condition number estimation, there is no universal alwayswinning strategy, but theoretical and experimental arguments show that the clearly preferable approach is the algorithm of Duff and Vömel when appropriately applied to both the triangular matrix itself and its inverse. This leads to a highly accurate incremental condition number estimator. Key words. condition number estimation, matrix inverses, incremental condition estimator, incremental norm estimator AMS subject classifications. 65F05,65F08,65F10,65A35 1. Introduction. The