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...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of scientific computing Jg. 91; H. 1; S. 15
Hauptverfasser: Liu, Xiao, Xia, Jianlin, V. de Hoop, Maarten, Ou, Xiaofeng
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.04.2022
Springer Nature B.V
Schlagworte:
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
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-022-01761-7