## Section: New Results

### Large Scale Peer to Peer Performance Evaluations

**Peer to Peer Large Scale Linear Algebra, programming and experimentations**[36]

We discuss the deployment of large scale numerical algorithms on a Grid. We minimize the communications needs by using persistent storage of data and we introduce out-of-core programming for the task farming paradigm. We discuss the performances of the bisection method to compute the eigenvalues of a real symmetric tridiagonal matrix and a block-based matrix-vector product. As experimental middleware, we use the XtremWeb system on two geographic sites: the university of Lille 1 and Paris-XI university at Orsay.

**Matrix Peer-to-Peer Computing With Very Large Heterogeneous Plateforms**[35]

After a short overview of global computing, also known as peer-to-peer computing, we study the deployment of linear algebra problems on such distributed environments. Some applications are very easy to adapt by means of parametric parallelism. We propose several techniques such that the persistence of data and out-of-core programming which aim to decrease communications and to deal with limited quantity of memory on peers. The experimentations use an XtremWeb platform deployed on two geographic sites in Lille, France, and Tsukuba, Japan.

**Large Scale Linear System Global Computing**[38]

We present a typical parallel method GMRES to solve large sparse linear systems by the use of a lightweight GRID system XtremWeb. We discuss the performances of this implementation deployed on two XtremWeb networks: a local network with 128 nondedicated PCs in Polytech-Lille of University of Lille I in France, a remote network with 3 clusters (91 CPUs) at the HPCS laboratory of Tsukuba in Japan. We do the tests as well on the platform of supercomputer IBM SP4 and in a LAN MPI computing environment LAM-MPI. We present the advantages and drawbacks of our implementations on the three computing systems.

**GMRES Method on Lightweight GRID System**[28]

We have implemented an important algorithm GMRES which is one of the key methods to resolve large, nonsymmetric, linear problems. We discuss the performances of this algorithm deployed on two XtremWeb networks: a local network with 128 non-dedicated PCs in Polytech-Lille of University of Lille I, a remote network with 3 clusters (91 CPUs) in the High Performance Computing Center of University of Tsukuba. We compare these performances with those of a MPI implementation of GMRES on the same platform.

**Toward global and grid computing for large scale linear algebra problems**[21]

In this paper, we gather resources of global and grid computing platforms in order to solve a linear algebra problem. We fit the algorithm of bisection on the platform of global computing, XtremWeb, and on the platform of RPC programming, OmniRPC. Those software are deployed on two different geographic sites at the engineer school of Polytech'Lille, France, and at the HPCS laboratory of Tsukuba, Japan. The combination of two different software and two geographic sites allows to do and analyse a wide range of tests.

**Cluster and Grid Matrix Computation with Persistent Storage and Out-of-core Programming**[18]

We present a performance evaluation of a large-scale numerical application on a cluster and a global Grid/Cluster platform. The computational resources are a cluster of clusters (34 nodes, 84 processors) and a local area network Grid (128 nodes), distributed on two geographic sites: Tsukuba university (Japan) and university of Lille I (France). As experimental Grid middleware we use the XtremWeb. We compare a classical MPI version with global Grid/Cluster versions. We also present and test some techniques based on out-of-core programming and an efficient data placement. We discuss the performances of a block-based Gauss-Jordan method for large matrix inversion.

**Towards a scheduling policy for hybrid methods on computational grids**[31]

We propose a cost model for running particular component based applications on a computational Grid. This cost is evaluated by a metascheduler and negotiated with the user by a broker. A specific set of applications is considered: hybrid methods, where components have to be launched simultaneously.

**A Hybrid GMRES-LS-Arnoldi method to accelerate the parallel solution of linear systems**[17]

We present a parallel hybrid asynchronous method to solve large sparse linear systems by the use of a large parallel machine. This method combines a parallel GMRES (m) algorithm with the Least Squares method that needs some eigenvalues obtained from a parallel Arnoldi's algorithm. All of the algorithms run on the different processors of an IBM SP3 or IBM SP4 computer simultaneously. This implementation of this hybrid method allows to take advantage of the parallelism available and to accelerate the convergence by decreasing considerably the number of iterations.

**Multiple Explicitly Restarted Arnoldi Method for Solving Large Eigenproblems**[15]

We propose a new approach for calculating some eigenpairs of large sparse non-Hermitian matrices. This method, called Multiple Explicitly Restarted Arnoldi (MERAM), is well suited for environments that combine different parallel programming paradigms. This technique is based on a multiple use of the Explicitly Restarted Arnoldi method (ERAM) and improves its convergence.

This technique is implemented and tested on a distributed environment consisting of two interconnected parallel machines. The MERAM technique is compared with ERAM, and one can notice that the convergence is improved. In some cases, more than a twofold improvement can be seen in MERAM results. We also implemented MERAM on a cluster of workstations. According to our experiments, MERAM converges better than the Explicitly Restarted Block Arnoldi method and, for some matrices, more quickly than the PARPACK package, which implements the Implicitly Restarted Arnoldi method.