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MiS Preprint

Parallel Black Box Domain Decomposition Based ${\cal H}$-LU Preconditioning

Lars Grasedyck, Ronald Kriemann and Sabine Le Borne


Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an ${\cal H}$-matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition. This new approach is based on a domain decomposition technique and yields a block structure in which large subblocks of the finite element stiffness matrix are zero and remain zero in a subsequent LU factorization, thus leading to, rigorously proven and numerically verified, improved storage and computational complexity requirements compared to ${\cal H}$-matrices constructed by a standard geometric bisection process. Furthermore, we introduce a black box clustering technique which no longer requires geometric grid information. The new algorithms have been implemented in parallel, and we provide numerical results in which an ${\cal H}$-LU factorization based on black box domain decomposition clustering is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.

MSC Codes:
65F05, 65F30, 65F50, 65N55
hierarchical matrices, domain decomposition, nested dissection, lu, parallel

Related publications

2009 Journal Open Access
Lars Grasedyck, Ronald Kriemann and Sabine LeBorne

Domain decomposition based \(\mathscr {H}\)-LU preconditioning

In: Numerische Mathematik, 112 (2009) 4, pp. 565-600