Numerical linear algebra lies at the core of virtually all scientific computing. It provides the mathematical basis for operating on all geometric and image data, it is the foundation for the numerical solution of ordinary and partial differential equations, and it has become the central toolbox for the current hype-theme in computing: big data analytics.
The deep relationship between numerical linear algebra and differential as well as integral equations remains important to engineering and science. Unless explicit methods are being used, the discretization of PDEs leads to systems of linear or nonlinear equations. The nonlinear case requires the repeated solution of linear equations. For most equations and discretization methods, the linear matrix systems are sparse. CERFACS has thirty years of expertise in solving such systems and is internationally acknowledged as one of the leading research centres in this field. During the last decade, work has centred on developing direct methods with a high degree of parallelism. More recently, the focus at Cerfacs lies on highly-parallel iterative solvers and on using these in hybrid methods to extend the capability of direct methods. Furthermore, the Algo team concentrates on multigrid and multilevel methods which are among the primary candidates to achieve the scalability and performance needed for future extreme scale computing. To show the applicability to real-world engineering problems we have been testing a range of all these solution methods for large scale problems in collaboration with our academic and industrial partners.
Some examples of our recent research work:
Sparse direct methods, hybrid direct-iterative and iterative methods
We focus on core problems in numerical linear algebra that are relevant for both CERFACS, its shareholders and are of academic interest. For example, we worked on solution methods for linear systems arising in the structural analysis of a reactor containment building with our industrial partner EDF. The developed iterative solver showed excellent convergence in only a few iterations for this problem [2]. It has been made available to EDF's finite-element code code_aster through the implementation in the open source linear algebra library PETSc. An example of our academic research work is the recently finished PhD thesis of Philippe Leleux on Hybrid direct and iterative solvers for sparse indefinite and overdetermined systems on future exascale architectures, in which attention is given to iterative and hybrid methods.
Multilevel and multigrid methods
Multigrid methods belong to the few solvers that have scalable numerical complexity and are thus among the primary candidates to exploit future extreme scale computing to its full potential. A current research topic is to develop such hierarchical algorithms and the software techniques to implement them on massively parallel and heterogeneous computer systems. The algo team takes part in the H2020 Energy oriented Centre of Excellence EoCoE 2. Here, a tailored multigrid solver for the gyrokinetic Poisson equation on disk-like geometries described by curvilinear coordinates has been developed [1]. This algorithm has the potential for highly parallel scalability and, since it is a matrix free algorithm, it has a low memory footprint.
[1] Kuehn, M., Kruse, C., Ruede, U., Energy-minimizing, symmetric finite differences for anisotropic meshes and energy functional extrapolation, 2020, https://hal.archives-ouvertes.fr/hal-02941899/document
[2] Kruse, C., Darrigrand, V., Tardieu, N. et al. Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints. Adv. Model. and Simul. in Eng. Sci. 7, 45 (2020). https://doi.org/10.1186/s40323-020-00181-2
