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

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.
 [1] ;  [2]
  1. Cornell Univ., Ithaca, NY (United States)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
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Resource Type:
Technical Report
Research Org:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
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Country of Publication:
United States
97 MATHEMATICS AND COMPUTING graph Laplacian; recursive bisection; support graph preconditioners; two-grid methods