An adaptive multigrid method based on path cover

Xiaozhe Hu, Junyuan Lin, Ludmil T. Zikatanov

Research output: Contribution to journalArticle

Abstract

We propose a path cover adaptive algebraic multigrid (PC-αAMG) method for solving linear systems with weighted graph Laplacians that can also be applied to discretized second order elliptic partial differential equations. The PC-αAMG is based on unsmoothed aggregation AMG (UA-AMG). To preserve the structure of smooth error down to the coarse levels, we approximate the level sets of the smooth error by first forming a vertex-disjoint path cover with paths following the level sets. The aggregations are then formed by matching along the paths in the path cover. In such a manner, we are able to build a multilevel structure at a low computational cost. The proposed PC-αAMG provides a mechanism to efficiently rebuild the multilevel hierarchy during the iterations and leads to a fast nonlinear multilevel algorithm. Traditionally, UA-AMG requires more sophisticated cycling techniques, such as AMLI-cycle or K-cycle, but as our numerical results show, the PC-αAMG proposed here leads to a nearly optimal standard V-cycle algorithm for solving linear systems with weighted graph Laplacians. Numerical experiments for some real-world graph problems also demonstrate PC-αAMG’s effectiveness and robustness, especially for ill-conditioned graphs.

Original languageEnglish (US)
Pages (from-to)S220-S241
JournalSIAM Journal on Scientific Computing
Volume41
Issue number5
DOIs
StatePublished - Jan 1 2019

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Multigrid Method
Adaptive Method
Agglomeration
Cover
Algebraic multigrid
Path
Linear systems
Graph Laplacian
Aggregation
Weighted Graph
Cycle
Level Set
Partial differential equations
Linear Systems
Algebraic multigrid Method
Path Following
Disjoint Paths
Elliptic Partial Differential Equations
Cycling
Graph in graph theory

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

Hu, Xiaozhe ; Lin, Junyuan ; Zikatanov, Ludmil T. / An adaptive multigrid method based on path cover. In: SIAM Journal on Scientific Computing. 2019 ; Vol. 41, No. 5. pp. S220-S241.
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An adaptive multigrid method based on path cover. / Hu, Xiaozhe; Lin, Junyuan; Zikatanov, Ludmil T.

In: SIAM Journal on Scientific Computing, Vol. 41, No. 5, 01.01.2019, p. S220-S241.

Research output: Contribution to journalArticle

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