## Section: New Results

### Linear algebra

#### Parallel Preconditioning of Krylov methods using Domain Decomposition

Participants : Guy Antoine Atenekeng Kahou, Laura Grigori, Bernard Philippe.

We have pursued the work on a parallel version of the GMRES method preconditionned by Multiplicative Schwarz by testing the parallel codes on large problems. It appears that the method is very robust since convergence occured almost always at the desired accuracy. In collaboration with M. Sosonkina, from the University of Iowa (USA), we compared our method to the approaches developed in the software pARMS http://www-users.cs.umn.edu/~saad/software/pARMS/ . Only a Schur complement approach was potentially performing better than our approach. Further experiments are going on [47] , [46] , [50] , [42] , [11] , [41] .

A graph partitioning algorithm has been designed. It aims at partitioning a sparse matrix into a
block-diagonal form, such that any two consecutive blocks overlap. We denote this form of the
matrix as the overlapped block-diagonal matrix. The partitioned matrix is suitable for applying
the explicit formulation of Multiplicative Schwarz preconditioner (EFMS) described in a previous work (see the 2005 activity report).
The graph partitioning algorithm partitions the graph of the input matrix into K partitions, such
that every partition _{i} has at most two neighbors _{i + 1} and _{i-1} . First, an ordering algorithm,
such as the reverse Cuthill-McKee algorithm, that reduces the matrix profile is performed. An
initial overlapped block-diagonal partition is obtained from the profile of the matrix. An iterative
strategy is then used to further refine the partitioning by allowing nodes to be transfered between
neighboring partitions.
Some experiments have been performed on matrices arising from real-world applications
to show the feasibility and usefulness of this approach [19] , [40] .
Further work is going on to improve the efficiency of the block partitionning.

In a common work with Nabil Gmati in Tunis, we have studied the convergence of the GMRES method when it is efficiently preconditionned. It is already well-known that, when the eigenvalues of the transformed system are all included in the disk centered at 1 and of unit radius, convergence occurs. We have proved that, when p eigenvalues lie out of that disk, they may delay the convergence by p steps at most. That result can be applied to the Schwarz Alternate preconditionner and more generally to the Multiplicative Schwarz preconditionner [48] , [49] .

#### Aitken-Schwarz Domain Decomposition for flow transport on Grid architecture

Participants : Jocelyne Erhel, Damien Tromeur-Dervout.

The Aitken-Schwarz domain decomposition method makes use of the convergence property of Schwarz type domain decomposition methods in order to accelerate the solution convergence at the artificial interfaces using the Aitken convergence acceleration technique [70] . The generalized Schwarz alternating method (GSAM) was introduced in [68] . Its purely linear convergence in the case of linear operators, as shown in [14] , suggests that the convergent sequence of traces solution at the artificial interfaces can be accelerated by the well-known process of Aitken convergence acceleration.

In [29] , we faced the problem of extending Aitken
acceleration method to nonuniform meshes. For this purpose, we
developed a new original method to compute the Non Uniform
Discrete Fourier Transform (NUDFT) based on the function values at
the nonuniform points. Moreover, this technique creates a robust
framework for the adaptive acceleration of the Schwarz method, by
using an approximation of the error operator at artificial
interfaces based on *a posteriori* estimates of the Fourier
mode behavior (preprint [61] , submitted).

The target application is flow in heterogeneous porous media; in collaboration with J.-R. De Dreuzy, the structure of the code was modified in order to handle Aitken-Schwarz Domain Decomposition.

#### Comparison of direct and iterative linear solvers

Participants : Jocelyne Erhel, Damien Tromeur-Dervout.

This work was done in the context of Grid'5000 project, in collaboration with A. Beaudoin, from the University of le Havre and J.-R. de Dreuzy, from the department of Geosciences at the University of Rennes 1. We have compared a direct linear solver (PSPASES from the University of Minnesota and IBM) and two iterative multigrid linear solvers (SMG/HYPRE and Boomer-AMG/HYPRE from Lawrence Livermore National Laboratory) to compute the flow in a heterogeneous 2D porous medium. Computations were done on a cluster of PC of the grid at Irisa. Our results show that we are able to deal with very large 2D highly heterogeneous porous media [20] , [21] , [25] . The direct solver is highly parallel and scalable, but the complexity and the memory requirements are prohibitive for very large computational domains. On the other hand, multigrid solvers are not so scalable, but the complexity and the memory requirements are kept low. Moreover, Boomer-AMG is more efficient than SMG when the heterogeneity increases (paper in preparation).

#### Rank-Revealing QR factorization

Participants : Laura Grigori, Frédéric Guyomarc'h, Bernard Philippe, Petko Yanev.

We have presented an algorithm to compute a rank revealing sparse QR factorization. First, a sparse QR factorization with no pivoting is performed, that allows us to obtain efficiently a sparse upper triangular factor R. Second, an incremental condition number estimator is used iteratively on the factor R to identify redundant columns. These columns are moved to the end of the matrix, and R is kept in an upper triangular form by means of Givens rotations and its sparsity is preserved as most as possible. Numerical results have shown the effectiveness of our algorithm [38] (papers in preparation).

#### Eigenvalue solvers using Domain Decomposition

Participant : Bernard Philippe.

Eigenvalue solvers are described in [12] . By considering the eigenvalue problem as a system of nonlinear equations, it is possible to develop a number of solution schemes which are related to the Newton iteration. For example, to compute eigenvalues and eigenvectors of an n×n matrix A, the Davidson and the Jacobi-Davidson techniques, construct `good' basis vectors by approximately solving a ``correction equation'' which provides a correction to be added to the current approximation of the sought eigenvector. That equation is a linear system with the residual r of the approximated eigenvector as right-hand side.

In cooperation with Yousef Saad, we have extended this general technique to the ``block'' situation, i.e., the case where a set of p approximate eigenpairs is available, in which case the residual r becomes an n×p matrix. The paper [13] defines two algorithms based on this approach. For symmetric real matrices, the first algorithm converges quadratically and the second cubically. In a second part of the same paper is considered the class of substructuring methods such as the Component Mode Synthesis (CMS) and the Automatic Multi-Level Substructuring (AMLS) methods, and to view them from the angle of the block correction equation. In particular this viewpoint allows us to define an iterative version of well-known one-level substructuring algorithms (CMS or one-level AMLS).