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Title: Solving Graph Laplacian Systems Through Recursive Bisections and Two-Grid Preconditioning

Abstract

We present a parallelizable direct method for computing the solution to graph Laplacian-based linear systems derived from graphs that can be hierarchically bipartitioned with small edge cuts. For a graph of size n with constant-size edge cuts, our method decomposes a graph Laplacian in time O(n log n), and then uses that decomposition to perform a linear solve in time O(n log n). We then use the developed technique to design a preconditioner for graph Laplacians that do not have this property. Finally, we augment this preconditioner with a two-grid method that accounts for much of the preconditioner's weaknesses. We present an analysis of this method, as well as a general theorem for the condition number of a general class of two-grid support graph-based preconditioners. Numerical experiments illustrate the performance of the studied methods.

Authors:
 [1];  [2]
  1. Cornell Univ., Ithaca, NY (United States)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1240975
Report Number(s):
LLNL-TR-683260
DOE Contract Number:  
AC52-07NA27344
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; graph Laplacian; recursive bisection; support graph preconditioners; two-grid methods

Citation Formats

Ponce, Colin, and Vassilevski, Panayot S. Solving Graph Laplacian Systems Through Recursive Bisections and Two-Grid Preconditioning. United States: N. p., 2016. Web. doi:10.2172/1240975.
Ponce, Colin, & Vassilevski, Panayot S. Solving Graph Laplacian Systems Through Recursive Bisections and Two-Grid Preconditioning. United States. https://doi.org/10.2172/1240975
Ponce, Colin, and Vassilevski, Panayot S. 2016. "Solving Graph Laplacian Systems Through Recursive Bisections and Two-Grid Preconditioning". United States. https://doi.org/10.2172/1240975. https://www.osti.gov/servlets/purl/1240975.
@article{osti_1240975,
title = {Solving Graph Laplacian Systems Through Recursive Bisections and Two-Grid Preconditioning},
author = {Ponce, Colin and Vassilevski, Panayot S.},
abstractNote = {We present a parallelizable direct method for computing the solution to graph Laplacian-based linear systems derived from graphs that can be hierarchically bipartitioned with small edge cuts. For a graph of size n with constant-size edge cuts, our method decomposes a graph Laplacian in time O(n log n), and then uses that decomposition to perform a linear solve in time O(n log n). We then use the developed technique to design a preconditioner for graph Laplacians that do not have this property. Finally, we augment this preconditioner with a two-grid method that accounts for much of the preconditioner's weaknesses. We present an analysis of this method, as well as a general theorem for the condition number of a general class of two-grid support graph-based preconditioners. Numerical experiments illustrate the performance of the studied methods.},
doi = {10.2172/1240975},
url = {https://www.osti.gov/biblio/1240975}, journal = {},
number = ,
volume = ,
place = {United States},
year = {Thu Feb 18 00:00:00 EST 2016},
month = {Thu Feb 18 00:00:00 EST 2016}
}