Julien Langou : May 27, 2003

Contribution of the numerical linear algebra techniques for solving the monostatic radar cross section calculation, an electromagnetism problem where a large linear system with multiple right-hand sides is involved.

Abstract

The starting point of this study is a problem posed by the electromagnetism group at EADS--CCR: How to solve several linear systems with the same coefficient matrix but various right-hand sides? For the targetted application the matrices are complex, dense and huge (of order of a few millions). Because such matrices cannot be computed nor stored in numerical simulation involved in a design process, the use of an iterative scheme with an approximate matrix-vector product is the only alternative. The matrix-vector product is performed using the Fast Multipole Method developped by EADS. In this talk, we show how to adapt some Krylov solvers so that they can handle efficiently multiple right--hand sides. We mainly focused on GMRES's variants.

In the context of the monostatic radar cross section calculation, it is possible to show that the number of independent right-hand sides is finite. We give a sketch of the proof and compare the dimension of the space of right-hand sides given by our theory and the dimension numerically observed. This properties enables to reduce considerably the number of right-hand sides to solve. In particular, an implementation of the block-GMRES method is given.

Finally more prospective results are given. The Fast Multipole Method is an inexact matrix-vector product for which the accuracy can be tuned. The less accurate the matrix vector product is, the fastest the computation. We show how to take advantage of this by using a relaxed schemes (inexact Krylov methods). Also we study the relevance of the normwise backward error as a stopping criterion for the iterative solvers.

This talk uses results and codes provided from EADS, Guillaume Sylvand and Bruno Carpentieri. Part I is joint work with Luc Giraud and Emeric Martin and Part II is joint work with Luc Giraud and Francis Collino.