Miroslav Tůma

A comparative study of sparse approximate inverse preconditioners

A number of recently proposed preconditioning techniques based on sparse approximate inverses are considered. A description of the preconditioners is given, and the results of an experimental comparison performed on one processor of a Cray C98 vector computer using sparse matrices from a variety of applications are presented. A comparison with more standard preconditioning techniques, such as incomplete factorizations, is also included. Robustness, convergence rates, and implementation issues are discussed.

Stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics

The solution of linear systems arising in the finite element analysis of shells and solids by the preconditioned conjugate gradient method is considered. Stabilized and block versions of the AINV factorized approximate inverse preconditioner are presented and tested on a variety of difficult problems. Comparisons with other preconditioning methods are also included.

A parallel solver for large-scale Markov chains

We consider the parallel computation of the stationary probability distribution vector of ergodic Markov chains with large state spaces by preconditioned Krylov subspace methods. The parallel preconditioner is obtained as an explicit approximation, in factorized form, of a particular generalized inverse of the generator matrix of the Markov process. Graph partitioning is used to parallelize the whole algorithm, resulting in a two-level method.Conditions that guarantee the existence of the preconditioner are given, and the results of a parallel implementation are presented. Our results indicate that this method is well suited for problems in which the generator matrix can be explicitly formed and stored.

Numerical experiments with two approximate inverse preconditioners

We present the results of numerical experiments aimed at comparing two recently proposed sparse approximate inverse preconditioners from the point of view of robustness, cost, and effectiveness. Results for a standard ILU preconditioner are also included. The numerical experiments were carried out on a Cray C98 vector processor.

Mixed-hybrid finite element approximation of the potential fluid flow problem

Abstract In the paper a mixed-hybrid approximation of the potential fluid flow problem based on prismatic discretization of the domain is presented. Trilateral prismatic elements with vertical faces and nonparallel bases suitable for the modelling of real geological circumstances are considered. The set of linearly independent vector basis functions is defined and existence and uniqueness of the approximate solution from the resulting symmetric indefinite system are examined. Possible approaches to the solution of the discretized system are discussed.

AN ASSESSMENT OF SOME PRECONDITIONING TECHNIQUES IN SHELL PROBLEMS

SUMMARY Preconditioned Krylov subspace methods have proved to be eAcient in solving large, sparse linear systems in many areas of scientific computing. The success of these methods in many cases is due to the existence of good preconditioning techniques. In problems of structural mechanics, like the analysis of heat transfer and deformation of solid bodies, iterative solution of the linear equation system can result in a significant reduction of computing time. Also many preconditioning techniques can be applied to these problems, thus facilitating the choice of an optimal preconditioning on the particular computer architecture available. However, in the analysis of thin shells the situation is not so transparent. It is well known that the stiAness matrices generated by the FE discretization of thin shells are very ill-conditioned. Thus, many preconditioning techniques fail to converge or they converge too slowly to be competitivewith direct solvers. In this study, the performance of some general preconditioning techniques on shell problems is examined. #1998 John Wiley & Sons, Ltd.

Preconditioner updates for solving sequences of linear systems in matrix-free environment

SUMMARY We present two new ways of preconditioning sequences of nonsymmetric linear systems in the special case where the implementation is matrix free. Both approaches are fully algebraic, they are based on the general updates of incomplete LU decompositions recently introduced in (SIAM J. Sci. Comput. 2007; 29(5):1918–1941), and they may be directly embedded into nonlinear algebraic solvers. The first of the approaches uses a new model of partial matrix estimation to compute the updates. The second approach exploits separability of function components to apply the updated factorized preconditioner via function evaluations with the discretized operator. Experiments with matrix-free implementations of test problems show that both new techniques offer useful, robust and black-box solution strategies. In addition, they show that the new techniques are often more efficient in matrix-free environment than either recomputing the preconditioner from scratch for every linear system of the sequence or than freezing the preconditioner throughout the whole sequence. Copyright q 2010 John Wiley & Sons, Ltd. Received 11 December 2008; Revised 16 November 2009; Accepted 4 December 2009

On Signed Incomplete Cholesky Factorization Preconditioners for Saddle-Point Systems

Limited-memory incomplete Cholesky factorizations can provide robust preconditioners for sparse symmetric positive-definite linear systems. In this paper, the focus is on extending the approach to sparse symmetric indefinite systems in saddle-point form. A limited-memory signed incomplete Cholesky factorization of the form

Preconditioned Iterative Methods for Solving Linear Least Squares Problems

LDL^T

HSL_MI28: An Efficient and Robust Limited-Memory Incomplete Cholesky Factorization Code

is proposed, where the diagonal matrix