Direct methods for the solution of sparse systems of linear equations are used in a wide range of numerical simulation applications. Such methods are based on the decomposition of the matrix into the product of triangular factors, followed by triangular solves. In comparison to iterative methods, they are known for their numerical accuracy and robustness. However, they are also characterized by a high memory consumption (especially for 3D problems) and a large amount of computations. The quality of the computed solution, the numerical functionalities and the computation time are essential parameters, while the use of material resources (number of processors and memory usage) must be carefully optimized. In this habilitation thesis, we describe some work to pursue these objectives in the context of the sparse direct solver MUMPS, developed in Toulouse, Lyon-Grenoble and Bordeaux. The approach relies on an original parallelization of the multifrontal method for distributed-memory machines, in which an asynchronous management of parallelism associated with distributed scheduling algorithms allows for dynamic datastructures and numerical pivoting. We consider task scheduling, optimization of the memory usage, and various numerical functionalities. On- going and future work aim at efficiently solving problems that are always bigger, while maintaining numerical stability and adapting our approaches to the quick evolutions of computer platforms: increase of the number of computing nodes, increase of the number of cores per node, but decrease of memory per core. In this context, software engineering and technology transfer aspects become critical in order to maintain in the long term a software package like MUMPS. This software is both necessary to our research and widely used in industry, maximizing feedback that validates our work and provides future work directions.
