|
 |
|
|
The PINEAPL project (Parallel NumErical Applications and Portable Libraries, Fourth Framework Project:20018) was a coordinated effort to produce a general purpose library of parallel numerical software suitable for a wide range of computationally intensive industrial applications and to port several application codes which use this library to parallel computers. PRECISE has been translated into Fortran to allow larger problems to be handled to match industrial needs. It i was used to test each item of numerical software produced during the project and distributed by NAG. NAG is the leader of the consortium which includes British Aerospace, CERFACS, Thomson-CSF, CPS (Napoli), the Danish Hydraulic Institute, IBM SEMEA, the University of Manchester, Math-Tech, and PIAGGIO.
The GPS phase mesurement can be used for precise localisation of beacons, satellites,... Unfortunately, this phase measurement is only known within an additive term called real ambiguity. Some experiments show that, in order to obtain a good accuracy on the localisation, one can take advantage of the fact that we know certain linear combinations of real ambiguities which yield integer numbers. This leads to a mixed least-squares problem with real and integer variables which is difficult and expensive to solve in practice. CERFACS role was to help CNES in making a sensible algorithmic choice for estimation of integer ambiguities, in the context of the search for optimal orbit parameters for a low satellite with GPS phase measurements, in order to achieve centimeter-level accuracy.
We studied the LAMDA method which provides the least-squares estimates for ambiguities which are the best unbiased estimates. To overcome its unpractical slow convergence, an exponential speed-up was achieved for the search algorithm. With such a speed up, the LAMDA method is fitted for data processing from regional network of GPS stations, as well as from low Earth orbiting satellites.
Embedded linear solvers are increasingly used in large scale scientific computing. An important issue is to understand how to tune the level of accuracy of the inner solver, in order to guarantee the convergence of the outer solver at the optimal overall cost. When the outer solver is a Newton-like method, then, as one could expect, the accuracy of the inner solver needs to be increased as the global convergence proceeds. On the contrary, and quite surprisingly, when the outer iteration is a Krylov-type method, the first iterates require to be computed with full accuracy. And this accuracy can be relaxed as the convergence proceeds. This astonishing robustness of Krylov methods to inaccuracy of the matrix-vector products is potentially of great interest for large scale computations for industry (domain decomposition, multipole etc...). A Workshop on this theme was organized with CNES, EDF, EADS at CERFACS on 11-12 September 2000.
We compare, from the point of view of computing efficiency, the computing power of the four real division algebras of numbers, i.e. the real and the complex numbers, the quaternions and the octonions. Our experiments support the conjecture that computation with octonions is to Biology what computation with quaternions is to Physics.