My recent works have been focussing on solving two-dimensional and three-dimensional boundary value problems by Krylov subspace type methods, preconditioned by means of incomplete factorization based techniques and/or algebraic multigrid methods, on serial as well as on parallel computers. I am also involved in Monte-Carlo simulation techniques for the risk assessment of radioactive waste repositories.

Self-adjoint two-dimensional problems

In collaboration with Y. Notay, a new dynamically relaxed block preconditioning has been introduced and favourably compared with previous block versions whose conditioning analysis has been recently refined. The new method both generalizes and improves the well known relaxation method RILU of O. Axelsson and Lindskog which interpolates, through a relaxation parameter, between the popular incomplete factorization and its modified variant that preserves rowsums. The generalization consists in that, now, the relaxation parameter is variable and dynamically computed during the incomplete factorization phase. Like its precursor RILU and unlike standard ILU methods, the new version does not suffer a lot from existence problem and its performance is stable with respect to both discontinuities and (strong) anisotropies. On top of that and contrary to RILU, its behavior does not critically depends on the choice of the relaxation parameter. A rigorous theoretical support which generalizes O. Axelsson's theory is provided.

Self-adjoint three-dimensional problems

Together with B. Polman, analytical spectral bounds obtained previously, that can help in understanding the convergence behavior of various modified incomplete factorization preconditionings, have been applied to three-dimensional problems. The three classical partitionings (point, line and plane) are compared, showing that (and how) the variations of the PDE coefficients and the boundary conditions influence the convergence rate of the methods involved. In harmony with previous works, planewise methods, though potentially the most efficient, are found, at least in their present forms, to hardly compete with pointwise methods, except for some trivial anisotropic PDEs. As concers linewise methods, both theoretical and experimental arguments are provided, that display that, in either of the following situations :   there are strong jumps in the variation of the PDE coefficients,   there are less Dirichlet boundary conditions,   the total number of unknowns is very large,   the number of unknowns in one direction is much larger than in both other ones, pointwise methods of (approximately) the same computational complexity could be fairly defeated, in contrat to the popular belief according to which block methods are not efficient for solving three-dimensional problems. As in the two-dimensional case, we expect even better performances with optimized relaxation methods and dynamically modified methods. Both aspects are receiving more attention in a collaboration with Y. Notay. B. Polman and I have proposed a line red-black like numbering strategy of the grid points that leads to rate of convergence comparable to the one predicted for idealized planewise (block) preconditionings. We have at the same time avoided the main inconveniences of the latter preconditionings, say, the - time and memory consuming - task of computing accurate approximate inverse to each pivot plane matrix and, the problem of solving, at each iteration, linear systems whose matrices have the same structure as a two-dimensional PDE matrix. To prevent instability inherent to standard incomplete factorization preconditioners, and for efficiency purposes, we have been led to combine our ordering with dynamically relaxed methods. The method turns out to be suitable for parallel computations. Unlike all other parallel orderings, the number of iterations does not increase, neither with the number of processors, nor with respect to standard methods, without using any coarse grid correction.

Parallel implementations

As is well known, incomplete factorization based preconditionings are not easy to parallelize without seriously affecting the convergence. Several attempts have been made in the literature, ranging from reordering strategies to domain decomposition methods, which reflects the difficulty of the task. Jointly with H.A. van der Vorst, I have designed and implemented a new family of parallel pointwise incomplete factorization. It is inspired from an improved version of a parallel block method that I have previously explored. The new technique combines the idea of domain decomposition method with a sort of hierarchical data exchange between adjacent subdomains. A judiciously fill-in strategy around interface nodes allows to cope with the dramatic decrease of the convergence rate inherent in both domain decomposition methods and most of reordering techniques. As opposed to most of previous approaches, communications between adjacent subdomains are required during both the construction and the application of the preconditioner. In doing so, we obtain a simple generalization of H.A. van der Vorst's 4-processors technique. Preliminary results are encouraging. Properly handled, the method leads to a decrease of the number of iterations for large number of processors.

Parallel iterative solvers for (vibro-) acoustic problems

Within the framework of the European ERPRIT project DOMINOS, R. Beauwens, G. Warzee and I are investigating parallel iterative solvers for solving large-scale complex-symmetric indefinite linear systems that arise from finite element discretizations of (vibro-) acoustic simulation problems. We aim at applying domain decomposition techniques combined with algebraic multigrid methods as preconditioners in Krylov subspace type methods like, a.o.,   the generalized minimal residual (GMRES), preferably implicitly restarted,   the bi-conjugate gradient stabilized of degree l (BiCGstab(l)),   the quasi-minimal residual method (QMR), possibly with look-ahead. I have designed a powerfull and robust code for solving, on parallel computers, large and sparse complex-symmetric (highly indefinite) linear systems that arise from acoustic wave propagation simulation industrial problems.

    Updated : Wed Feb 21 12:14:20 MET 2001           magolu@ulb.ac.be