Multi-layer Hierarchical Structures and Factorizations

نویسنده

  • JIANLIN XIA
چکیده

We propose multi-layer hierarchically semiseparable (MHS) structures for the fast factorizations of dense matrices arising from multi-dimensional discretized problems such as certain integral operators. The MHS framework extends hierarchically semiseparable (HSS) forms (which are essentially one dimensional) to higher dimensions via the integration of multiple layers of structures, i.e., structures within the dense generators of HSS forms. In the 2D case, we lay theoretical foundations for MHS structures and justify the feasibility of MHS approximations based on the fast multipole method (FMM) and algebraic techniques such as structure-preserving rank-revealing factorizations. Rigorous rank bounds and conditions for the structures are given. Representative subsets of mesh points and a multi-layer tree are used to intuitively illustrate the structure. The MHS framework makes it convenient to explore FMM structures and perform direct factorizations. We can naturally design and analyze MHS algorithms by taking advantage of existing methods and analysis for simple HSS methods. In particular, we can design fully stable and scalable multi-layer ULV (MULV) factorizations that can preserve the inner structures and have nearly linear complexity under certain conditions. An idea of reduced matrices is used to show the structured factorizations and the recursive sparsification of the mesh. We also establish intrinsic connections between dense MULV factorizations and sparse structured multifrontal factorizations, which bridges the gaps among different types of hierarchical solvers, and facilitates the sharing of ideas and the study and design of new algorithms. The new structures and algorithms can be used for the direct solution of some multi-dimensional discretized problems with nearly linear complexity and storage.

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تاریخ انتشار 2016