Interconnected Hierarchical Structures for Fast Direct Elliptic Solution

We propose an interconnected hierarchical rank structure and use it to design a fast direct elliptic solver that can significantly reduce the amount of low-rank compression operations used in usual structured direct solvers. Interconnected structures within two hierarchical layers are exploited: the...

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Published in:	Journal of scientific computing Vol. 91; no. 1; p. 15
Main Authors:	Liu, Xiao, Xia, Jianlin, V. de Hoop, Maarten, Ou, Xiaofeng
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.04.2022 Springer Nature B.V
Subjects:	Algorithms Approximation Computational Mathematics and Numerical Analysis Design Factorization Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Solvers Theoretical Interconnected hierarchical structure Elliptic equation Basis reuse Fast sparse direct solver Neighbor tree Schur complement update
ISSN:	0885-7474, 1573-7691
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Abstract	We propose an interconnected hierarchical rank structure and use it to design a fast direct elliptic solver that can significantly reduce the amount of low-rank compression operations used in usual structured direct solvers. Interconnected structures within two hierarchical layers are exploited: the hierarchical partitioning of a large problem into subproblems that are local Schur complements on smaller subdomains, and the interconnected hierarchical structured approximations of the subproblems. The interconnected structures make it feasible to extensively reuse off-diagonal basis matrices produced in the rank-structured approximation of smaller local Schur complements. Such basis matrices are produced only once and then reused across multiple hierarchical levels of the sparse factorization. Unlike many existing rank-structured direct solvers where explicit low-rank compression is often the major computation, our new solver can then avoid most of the compression operations. This helps to both conveniently preserve the rank structures and reduce the cost. Under moderate conditions, the total factorization cost is O ( rn ), where r is an appropriate off-diagonal numerical rank bound. The interconnected structures are further extended to accelerate a factorization update problem where many local coefficient updates are involved. Numerical tests on some PDE problems are used to demonstrate the efficiency and speedup. In particular, some reuse factors in our tests indicate dramatic reduction in the number of low-rank compression operations.
AbstractList	We propose an interconnected hierarchical rank structure and use it to design a fast direct elliptic solver that can significantly reduce the amount of low-rank compression operations used in usual structured direct solvers. Interconnected structures within two hierarchical layers are exploited: the hierarchical partitioning of a large problem into subproblems that are local Schur complements on smaller subdomains, and the interconnected hierarchical structured approximations of the subproblems. The interconnected structures make it feasible to extensively reuse off-diagonal basis matrices produced in the rank-structured approximation of smaller local Schur complements. Such basis matrices are produced only once and then reused across multiple hierarchical levels of the sparse factorization. Unlike many existing rank-structured direct solvers where explicit low-rank compression is often the major computation, our new solver can then avoid most of the compression operations. This helps to both conveniently preserve the rank structures and reduce the cost. Under moderate conditions, the total factorization cost is O ( rn ), where r is an appropriate off-diagonal numerical rank bound. The interconnected structures are further extended to accelerate a factorization update problem where many local coefficient updates are involved. Numerical tests on some PDE problems are used to demonstrate the efficiency and speedup. In particular, some reuse factors in our tests indicate dramatic reduction in the number of low-rank compression operations. We propose an interconnected hierarchical rank structure and use it to design a fast direct elliptic solver that can significantly reduce the amount of low-rank compression operations used in usual structured direct solvers. Interconnected structures within two hierarchical layers are exploited: the hierarchical partitioning of a large problem into subproblems that are local Schur complements on smaller subdomains, and the interconnected hierarchical structured approximations of the subproblems. The interconnected structures make it feasible to extensively reuse off-diagonal basis matrices produced in the rank-structured approximation of smaller local Schur complements. Such basis matrices are produced only once and then reused across multiple hierarchical levels of the sparse factorization. Unlike many existing rank-structured direct solvers where explicit low-rank compression is often the major computation, our new solver can then avoid most of the compression operations. This helps to both conveniently preserve the rank structures and reduce the cost. Under moderate conditions, the total factorization cost is O(rn), where r is an appropriate off-diagonal numerical rank bound. The interconnected structures are further extended to accelerate a factorization update problem where many local coefficient updates are involved. Numerical tests on some PDE problems are used to demonstrate the efficiency and speedup. In particular, some reuse factors in our tests indicate dramatic reduction in the number of low-rank compression operations.
ArticleNumber	15
Author	Liu, Xiao Ou, Xiaofeng Xia, Jianlin V. de Hoop, Maarten
Author_xml	– sequence: 1 givenname: Xiao surname: Liu fullname: Liu, Xiao organization: Department of Computational and Applied Mathematics, Rice University – sequence: 2 givenname: Jianlin surname: Xia fullname: Xia, Jianlin email: xiaj@math.purdue.edu organization: Department of Mathematics, Purdue University – sequence: 3 givenname: Maarten surname: V. de Hoop fullname: V. de Hoop, Maarten organization: Department of Computational and Applied Mathematics, Rice University – sequence: 4 givenname: Xiaofeng surname: Ou fullname: Ou, Xiaofeng organization: Department of Mathematics, Purdue University
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SubjectTerms	Algorithms Approximation Computational Mathematics and Numerical Analysis Design Factorization Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Solvers Theoretical
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