Hybrid direct and iterative solvers for the sparse indefinite and overdetermined systems on future exascale architectures

Philippe Leleux (CERFACS)

11 juin 2021

In scientific computing, the numerical simulation of systems is crucial to get a deep understanding of the physics underlying real world applications. The models used in simulation are often based on partial differential equations (PDE) which, after fine discretisation, give rise to huge sparse systems of equations to solve. Historically, 2 classes of methods were designed for the solution of such systems: direct methods, robust but expensive in both computations and memory; and iterative methods, cheap but with a very problem-dependent convergence. In the context of high performance computing, hybrid direct-iterative methods were then introduced in order to combine the advantages of both methods, while using efficiently the increasingly large and fast supercomputing facilities. In this thesis, we focus on the latter type of methods with two complementary research axes. In a first research track, we detail the mechanisms behind the efficient implementation of multigrid methods. The latter makes use of several levels of increasingly refined grids to solve linear systems with a combination of fine grid smoothing and coarse grid corrections. The efficient parallel implementation of such a scheme is a difficult task. We then focus on the improvement of the parallel efficiency of a multigrid scheme and in particular the scalability of the solver used on the coarsest grid. At extreme scale, this study is carried in the HHG framework (Hierarchical Hybrid Grids) for the solution of a Stokes problem with jumping coefficients, inspired from Earth’s mantle convection simulation.In the following chapters, we study some hybrid methods derived from the classical row-projection method block Cimmino, and interpreted as domain decomposition methods. These methods are based on the partitioning of the matrix into blocks of rows on which projections are computed to iteratively approximate the solution of a linear system. Both methods are implemented in the parallel solver ABCD-Solver (Augmented Block Cimmino Distributed solver). Finally, for the solution of discretized PDE problems, we propose a new approach using a coarse representation of the space to obtain an iterative method with fast linear convergence, demonstrated on Helmholtz and Convection-Diffusion problems.;