On September 11--12, 2000, CERFACS (Toulouse, France) hosted two Industrial Days on Inner-Outer Iterations, a workshop focusing on numerical quality and software coupling. This workshop was organized at the request of three out of the four industrial shareholders of CERFACS, namely AEDS (the new European consortium for Defence and Aerospace), CNES (the French NASA) and EDF (French Electricity). More and more often in their simulations, these industries are making heavy use of either embedded iterative processes (at two or more levels) or coupled software that exchanges information successively between the subunits. Preliminary discussions lead to the acknowledgement that the state-of-the-art in the understanding of the numerical behaviour of embedded or coupled processes was not advanced enough to fulfill the needs of the many users.
 
A workshop was therefore planned to bring together engineers and researchers to allow the first to better specify their needs and thel atter to present up-to-date tools and existing research results on the topic. This resulted in a two-day workshop supported by SMAI and organised as follows. The first morning of the workshop consisted of two tutorials. After the welcome address given by Iain Duff (CERFACS and RAL), the first speaker, Sven Hammarling from Nag Ltd, gave a lively and clear introduction to error analysis, particularly concentrating on backward error analysis, and discussed its implementation in modern software such as LAPACK. Then Andreas Griewank from the Technical University in Dresden presented the basics of automatic differentiation, a powerful tool to perform forward error analysis and a sensitivity analysis on complex codes. He also described its use for embedded processes arising in optimal design. In the afternoon, industrial users were invited to report on  their experience in using embedded processes. Jean-Louis Vaudescal, from EDF, described various applications in mechanics and neutronics particularly focusing on a generalised eigenvalue problem arising in the modelling of a nuclear  core.  Such a problem is solved by successive  iterations with Chebyshev acceleration  but each step requires the solution of a linear system which has to be computed with  an iterative solver due to the characteristics of the problem. Jean-Louis Vaudescal insisted on the extreme sensitivity of  the convergence of the outer process with respect to the accuracy of theinner one.  Philippe Homsi from EADS concluded the first day by abrief description of the major issues encountered  in software coupling for multidisciplinary physics such as the coupling of acoustics and structure for aircraft design.
 
The second day of the workshop gave the opportunity for expert researchers to present the state-of-the-art in the control of embedded iterative processes.  Even when there are only two levels of iterations corresponding to solving a linear problem, many questions concerning the optimal tuning of the inner process  remain open.  A pioneer in the field of the control of inner-outer iterations and inexact methods for Linear Algebra, Gene Golub from Stanford University opened this session with an overview of his many contributions with his young colleagues to the topic including inexact variants of the Chebyshev method, an inexact Uzawa method, inexact conjugate gradient methods or the Lanczos process with inner-outer  iterations. When the outer process is the Chebyshev method, Golub has  shown that it is possible in some cases to define an optimal strategy for monitoring the inner process while minimising the global cost.
 
Whereas it is known that an outer Newton-like method requires increasingly inner accuracy as convergence proceeds, the picture is completely different when the outer scheme is a  Krylov-type method. In her talk,  Francoise Chaitin-Chatelin from CERFACS and University Toulouse I focused on this fundamental difference and illustrated it by recent numerical experiments carried out by her group at CERFACS. These experiments show that Krylov-type methods need an accurate inner process only at the beginning of  the convergence. This led Amina Bouras and Val'erie Frayss'e (CERFACS) to propose and study a relaxation strategy for Krylov-type methods which allows  the inner accuracy to deteriorate as convergence proceeds. This counter-intuitive strategy can be shown to provide a substantial reduction of the overall computer time required to obtain the solution with a prescribed backward error.
 
Many important industrial domains can be successfully tackled by this approach.  As an example, Val'erie Frayss'e presented promising results on domain decomposition methods for heterogeneous andanisotropic problems, obtained jointly with Amina Bouras and Luc Giraud at CERFACS. Returning to inexact Newton methods, the last speaker Andreas Frommer from the University of Wuppertal discussed their implementation and performance on parallel computers.
 
The workshop was ended by a lively round-table. Everyone acknowledged that the industrial needs presented during the workshop were particularly challenging and the engineers were strongly urged to provide the community with detailed examples and data. The workshop gathered 6 engineers and 24 researchers from 5 countries(although this would increase to 13 if the multi-ethnic distribution of CERFACS researchers is included). Its moderate size and the nice and informal atmosphere really favoured fruitful discussions andindeed fostered the birth of new collaborations.  Abstracts,  slides and pictures are available from the URL: http://www.cerfacs.fr/algor/PastWorkshops/IndustrialDays2000/index.html
 
 Val'erie Frayss'e and Elisabeth Traviesas
 CERFACS