Ben Polman

On approximate factorization methods for block matrices suitable for vector and parallel processors

Abstract Some existence results for methods based on the approximate factorization of block matrices are proven. These methods are based on recursive computations of diagonal block matrices and the approximation of their inverses to preserve sparsity. We also discuss a recently proposed [1] inverse free factorization method and present some numerical tests for it.

Incomplete blockwise factorizations of (block) H-matrices

The use of incomplete blockwise factorizations as preconditioners in conjugate gradient like methods has become more and more popular in recent years. Most of the theory concerning existence and applicability of these factorizations has been limited to M-matrices so far. Here we introduce a more general definition of block H-matrices (Robert [8]) and we extend the theory to this class of matrices.

Decay rates of inverses of banded M-matrices that are near to Toeplitz matrices

The decaying behavior of inverses of positive definite band matrices is analysed for M-matrices that are in some sense close to Toeplitz matrices. Estimates based on the factorization are derived that are better than existing ones, in particular for nonsymmetric matrices. Some examples are given.

Solving the Stokes Problem on a Massively Parallel Computer

Abstract We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distri...

Incomplete block-matrix factorization iterative methods for convection-diffusion problems

Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems.This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter.In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter ε=1.

On incomplete block factorization methods of generalized SSOR type for H-matrices

Abstract Two types of (modified) incomplete block factorization methods are considered, and the existence and convergence of the related splittings are established for H -matrices. This is done by studying in parallel the similar preconditioner for an M -matrix A which satisfies A ⩽ M ( A ). We show also for the M -matrix case somewhat more general results for the type of preconditioners considered than previously found in the literature.

Experimental comparison of three-dimensional point and line modified incomplete factorizations

We examine how the variations of the coefficients of 3-dimensional (3D) partial differential equations (PDEs) influence the convergence of the conjugate gradient method, preconditioned by standard pointwise and linewise modified incomplete factorizations. General analytical spectral bounds obtained previously are applied, which displays the conditions under which good performances could be expected. The arguments also reveal that, if the total number of unknowns is very large or the number of unknowns in one direction is much larger than in both other ones, or if there are strong jumps in the variation of the PDE coefficients or fewer Dirichlet boundary conditions, then linewise preconditionings could be significantly more efficient than the corresponding pointwise ones. We also discuss reasons to explain why in the case of constant PDE coefficients, the advantage of preferring linewise methods to pointwise ones is not as pronounced as in 2D problems. Results of numerical experiments are reported.

Efficient Planewise-like Preconditioners for Solving 3D Problems

We deal with the numerical solution of large linear systems resulting from discretizations of three-dimensional boundary value problems. It has been shown recently that, if the use of presently available planewise preconditionings is as pathological as thought by many people, except for some trivial anisotropic problems, linewise preconditionings could fairly outperform pointwise methods of approximately the same computational complexity. We propose here a zebra (or line red-black) like numbering strategy of the grid points that leads to a rate of convergence comparable to the one predicted for ideal planewise preconditionings. The keys to the success of this strategy are threefold. On the one hand, one gets rid of the, time and memory consuming, task of computing some accurate approximation to the inverse of each pivot plane matrix. On the other hand, at each PCG iteration, there is no longer a need to solve linear systems whose matrices have the same structure as a two-dimensional boundary value problem matrix. Finally, it is well suited to parallel computations. Copyright

On the eigenvalues of the structure matrix of matrices of zeros and ones

Abstract We give answers to questions raised by R. A. Brualdi and by G. Sierksma and E. Sterken concerning the eigenvalues of the structure matrix T of (0,1) matrices and of the corresponding matrix T ∗ as introduced by Ryser. We completely solve the question in case rank T is 1 or 2. In case rank T is 3 we give the characteristic polynomial for T (as well as for T ∗ ), from which the eigenvalues can be computed. Furthermore we prove that in all cases the eigenvalues are real, and we give estimates for the eigenvalues in terms of the dimension and the sparsity of the matrices.