Section: New Results

Direct solvers for sparse linear systems

This work is closely related to the Mumps solver (see Section  5.1 ) and was performed in close collaboration with INPT (Toulouse). First, we have pursued the study of low-rank representations to speed-up sparse direct solvers using the so called BLR (Block Low Rank) format [44] . This work was done in collaboration with LSTC (Livermore Software Technology Corp., USA) and in the context of a contract with EDF which funded the PhD thesis of Clément Weisbecker at INPT. We also worked on shared-memory parallelism [61] in the context of the PhD thesis of Wissam M. Sid-Lakhdar. Concerning low-rank approximations, they were experimented on geophysics applications [38] (Helmholtz equations) in the context of a collaboration with members of the ISTerre and Geoazur laboratories. The impact of both low-rank compression and shared-memory parallelism was also studied on electromagnetism problems [17] , in collaboration with University of Padova (Italy) and CEDRAT.

We have started the design and implementation of a distributed-memory low-rank multifrontal solver. When computations are faster (thanks to low-rank compression or multithreading within each node), we observed that communications become critical; we are therefore currently studying the limits of the communication schemes from the Mumps approach and their possible improvements.

On numerical and industrial aspects, we worked on rank detection and null space basis computations (in collaboration with CERFACS and Total/Hutchinson) as well as on improved parallel pivoting strategies for symmetric indefinite systems, in collaboration with ESI-Group (see Section  7.1 ).