Combler l’écart entre H-Matrices et méthodes directes creuses pour la résolution de systèmes linéaires de grandes tailles

Abstract : Many physical phenomena may be studied through modeling and numerical simulations, commonplace in scientific applications. To be tractable on a computer, appropriated discretization techniques must be considered, which often lead to a set of linear equations whose features depend on the discretization techniques. Among them, the Finite Element Method usually leads to sparse linear systems whereas the Boundary Element Method leads to dense linear systems. The size of the resulting linear systems depends on the domain where the studied physical phenomenon develops and tends to become larger and larger as the performance of the computer facilities increases. For the sake of numerical robustness, the solution techniques based on the factorization of the matrix associated with the linear system are the methods of choice when affordable. In that respect, hierarchical methods based on low-rank compression have allowed a drastic reduction of the computational requirements for the solution of dense linear systems over the last two decades. For sparse linear systems, their application remains a challenge which has been studied by both the community of hierarchical matrices and the community of sparse matrices. On the one hand, the first step taken by the community of hierarchical matrices most often takes advantage of the sparsity of the problem through the use of nested dissection. While this approach benefits from the hierarchical structure, it is not, however, as efficient as sparse solvers regarding the exploitation of zeros and the structural separation of zeros from non-zeros. On the other hand, sparse factorization is organized so as to lead to a sequence of smaller dense operations, enticing sparse solvers to use this property and exploit compression techniques from hierarchical methods in order to reduce the computational cost of these elementary operations. Nonetheless, the globally hierarchical structure may be lost if the compression of hierarchical methods is used only locally on dense submatrices. We here review the main techniques that have been employed by both those communities, trying to highlight their common properties and their respective limits with a special emphasis on studies that have aimed to bridge the gap between them. With these observations in mind, we propose a class of hierarchical algorithms based on the symbolic analysis of the structure of the factors of a sparse matrix. These algorithms rely on a symbolic information to cluster and construct a hierarchical structure coherent with the non-zero pattern of the matrix. Moreover, the resulting hierarchical matrix relies on low-rank compression for the reduction of the memory consumption of large submatrices as well as the time to solution of the solver. We also compare multiple ordering techniques based on geometrical or topological properties. Finally, we open the discussion to a coupling between the Finite Element Method and the Boundary Element Method in a unified computational framework.
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Aurélien Falco. Combler l’écart entre H-Matrices et méthodes directes creuses pour la résolution de systèmes linéaires de grandes tailles. Other [cs.OH]. Université de Bordeaux, 2019. English. ⟨NNT : 2019BORD0090⟩. ⟨tel-02183902⟩

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