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Parallelism and robustness in hybrid solvers for large linear systems : Application to design optimization in fluid dynamics

Désiré Nuentsa Wakam 1
1 SAGE - Simulations and Algorithms on Grids for Environment
Inria Rennes – Bretagne Atlantique , IRISA-D1 - SYSTÈMES LARGE ÉCHELLE
Abstract : This thesis presents a set of numerical schemes that aim at solving large linear systems on parallel computers. The proposed approaches are part of a hybrid scheme where the direct and iterative methods are combined through domain decomposition techniques. The initial problem is first divided into subproblems by partitioning the coefficient graph of the system. The Schwarz-based methods are then used as preconditioners for GMRES-based Krylov methods. We first consider a hybrid scheme using an explicit formulation of the multiplicative Schwarz preconditioner. We define two levels of data parallelism : the first level has been used to parallelize the GMRES method at the global level; we introduce the second level to solve the subsystems induced by the Schwarz preconditioner through a parallel direct method. We show that this splitting guarantee a certain robustness in the global hybrid approach by reducing the total number of partitions. Moreover, this approach enables a better usage of CPU resources allocated inside a compute node. Then we study the convergence and the parallelism in the GMRES method which is used as the global accelerator in the hybrid method. The general observation is that the number of iterations in that method increases very fast with the number of partitions in the hybrid solver, and so the total CPU time. To limit this effect, we propose several implementations of the GMRES method with the deflation methods. These approaches formulate a deflation process either as an adaptive preconditioner or an augmented subspace basis technique. We show the usefulness of these approaches in their ability to limit the influence of the right choice of the Krylov basis size, and thus to avoid the stagnation of the global hybrid solver. Moreover, these approaches are very efficient to reduce the memory usage as well as the global CPU time and also the exchanged MPI messages between the working processors. The benefits are given throughout this thesis on moderate to large linear systems arising from several applications fields, and mainly from design optimisation in computational fluid dynamics.
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Submitted on : Wednesday, April 25, 2012 - 12:37:18 AM
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  • HAL Id : tel-00690965, version 1


Désiré Nuentsa Wakam. Parallelism and robustness in hybrid solvers for large linear systems : Application to design optimization in fluid dynamics. Distributed, Parallel, and Cluster Computing [cs.DC]. Université Rennes 1, 2011. English. ⟨tel-00690965⟩



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