## Section: Scientific Foundations

### High-Performance Computing

Participants : Amine Abdelmoula, Jenny Al Khoury, Guy Antoine Atenekeng Kahou, Étienne Bresciani, Édouard Canot, Caroline de Dieuleveult, Jocelyne Erhel, Frédéric Guyomarc'h, Laura Grigori, Noha Makhoul, Bernard Philippe, Damien Tromeur-Dervout, Petko Yanev, Samih Zein, Mohammed Ziani.

The focus of this topic is the development of parallel algorithms and software. The objectives are to solve large scale equations in linear algebra ( 3.1 ) and to use high performance computing for dealing with problems arising from hydrogeology and geophysics ( 4.1 ).

#### Parallel sparse linear algebra

Algorithms have been described above (
3.1 ).
The team works on the development of parallel software for sparse direct solvers (LU factorization), iterative solvers (PCG, GMRES, subdomain method), least-squares solvers (QR factorization). The target is Giga-systems with billions (10^{9} ) of unknowns.

#### Parallel spatial discretization

Our applications in hydrogeology and geophysics ( 4.1 ) are in the framework of Partial Differential Algebraic Equations (PDAE). We usually discretize time by a classical one-step or multi-step scheme and space by a Finite Element Method or a similar method. To get a fully parallel implementation, it is necessary to parallelize the matrix computation and generation. A common approach is to divide the computational domain into subdomains. Once the matrix is computed, it is used in linear solvers. The challenge is to reduce communication between the two phases. Recently, we have also investigated particle methods. Parallel particle trackers are still an area of research.

#### Software components for coupled problems

Our applications are quite often multi-physics models, where nonlinear coupling occurs. Our objective is to design software components, which provide a great modularity and flexibility for using the models in different contexts. The main numerical difficulty is to design a coupling algorithm with parallel potentiality. We also investigate the implementation on grid architectures, in collaboration with Paris Inria-team. The challenge is to develop and use a middleware for high-level applications.

#### Grid computing for stochastic simulations

In our applications, we use stochastic modelling in order to take into account geophysical variability. From a numerical point of view, it amounts to run multiparametric simulations. The objective is to use the power of grid computing. The target architecture is a heterogeneous collection of parallel clusters, with high-speed networks in clusters and slower networks interconnecting the clusters.