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Parallel space-time multilevel methods with application to electrophysiology : theory and implementation

Benedusi, Pietro ; Krause, Rolf (Dir.)

Thèse de doctorat : Università della Svizzera italiana, 2020 ; 2020INFO005.

The goal of this thesis is to design and study an efficient strategy to solve possibly non-linear parabolic partial differential equations on massively parallel machines. Traditionally, when solving time-dependent problems, time stepping methods are used to advance the solution in time. These techniques are inherently sequential and therefore they introduce a bottleneck in the overall... More

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    Summary
    The goal of this thesis is to design and study an efficient strategy to solve possibly non-linear parabolic partial differential equations on massively parallel machines. Traditionally, when solving time-dependent problems, time stepping methods are used to advance the solution in time. These techniques are inherently sequential and therefore they introduce a bottleneck in the overall computational scalability. To overcome this limitation, we focus on the design of time parallel solvers. To achieve parallel efficiency in both space and time, we employ a multilevel space-time finite element discretization, coupled with parallel block preconditioners. We use continuous finite elements to discretize in space and, for stability reasons, we adopt discontinuous finite elements in the time dimension. In space, in particular, we consider the generic finite element framework of isogeometric analysis. We consider a space-time multilevel method, based on a hierarchy of non-nested meshes, created using a semi-geometric approach. With this technique, we can automatically generate space-time coarse spaces, starting from a single fine spatial mesh, in any dimension and in the presence of complex geometries. Through a detailed spectral analysis, we can design convenient preconditioners for space-time operators and give estimates of their conditioning, with respect to problem, discretization and multigrid parameters. We numerically investigate how different iterative solution strategies, coarsening strategies and spectral based preconditioners, can affect the overall convergence and robustness of our multilevel approach. Finally, we run strong and weak scalability experiments, mostly focusing on time parallelism. In this analysis, we consider two model problems: the heat equation, possibly anisotropic or with jumping coefficients, and the monodomain equation, a non-linear reaction-diffusion model arising from the study of the electrical activation of the human heart.