In this "Habilitation à Diriger des Recherches" (HDR), we present our research in high-performance scientific computing over the recent years. Our work has been mainly related to parallel algorithms for numerical linear algebra solvers and their parallel implementation in public domain software libraries. We illustrate in this manuscript how these calculations can be accelerated using innovative algorithms and be made reliable using specific quantities in error analysis. First we explain how numerical linear algebra solvers can be designed to exploit the capabilities of current heterogeneous multicore+GPU systems. We consider dense factorization algorithms for which we describe the work splitting between the architectural components and its influence in terms of communication cost. These computations can also be significantly enhanced thanks to mixed precision algorithms that use lower precision for the most expensive tasks while achieving higher precision accuracy for the results. Then we present our research in developing fast linear algebra solvers using randomized algorithms. Randomization represents a very promising approach to accelerate linear algebra computations and the class of algorithms that we developed has the advantage of reducing the amount of communication in dense factorizations by removing completely the pivoting phase in linear system solutions. The resulting software has been developed for multicore machines possibly accelerated by GPUs. Finally we propose numerical tools that enable us to assess the quality of the computed solution of overdetermined linear least squares, including the total least squares approach. Our method is based on deriving exact values or estimates for the condition number of these problems. We describe algorithms and software to compute these quantities using standard parallel libraries. Research tracks for the coming years are given in a concluding chapter.
