Serge Gratton, Habilitation à diriger les recherches : December 13, 2005

Fast and robust solvers in Scientific Computing Applications in Geosciences.

Jury

• G. Balmino : Research Director CNES-CNRS
• C. Brezinski : Professor at "Université des sciences et technologies" of Lille
• F. Chaitin-Chatelin : Professor at the University Toulouse I and Groupe leader at CERFACS
• I.S. Duff : Groupe leader at RAL (UK) and Project leader at CERFACS
• A Frommer : Professor at the University of Wuppertal (Germany)
• L. Giraud : Professor at EENSEEIHT
• Y. Saad : Professor at the University of Minnesota (USA)
• P. Spiteri : Professor at EENSEEIHT
• Ph.L. Toint : Professor at the University of Namur (Belgium)

Abstract

The availability of very powerful computers enables the solution of larger and more complex problems in various applications. This has favoured the use of very sophisticated models, but has also stimulated many questions in the domain of numerical analysis. Most of the questions are related to the design of algorithms that are fast and accurate. Simulations tend to replace real-life experiments, because they are often cheaper, but it is of primary interest to be sure that a result obtained using a fast parallel computer is also close to the phenomenon that real-life experiment would have produced.

The interplay of these two questions (computational efficiency and accuracy) is ubiquitous in the domain of linear algebra too. The domain of the solution of linear systems of equations provides numerous examples of this duality. For example, a perfect iterative method would be a method able to use efficiently a parallel machine to drive a stopping criterion down to a target value as quickly as possible.

In this work, we explore tools that enable us to obtain fast and accurate solutions on modern parallel architectures. We present theoretical tools for the analysis of the sensitivity and estimators for the reliability of the solution.These theoretical tools enable the analysis of complex solution methods without performing a complicated step-by-step study of round-off error propagation. A careful use of these theoretical tools is also a key ingredient in relaxed solution methods to introduce better parallelism and accelerate the computations. This will be clearly demonstrated on inexact Krylov methods and on preconditioning techniques for the solution of linear systems.

These ideas are being integrated in a set of CERFACS routines for the solution of linear systems that is very popular, as measured by the two thousand downloads of this software.

Most of the ideas presented here will be supported by numerical expriments on real-life applications in Data Assimilation for Geosciences and Electromagnetics.