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DOE OSTI · 1240975

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

Ponce, Colin·Vassilevski, Panayot S.
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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.

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97 MATHEMATICS AND COMPUTINGgraph Laplacianrecursive bisectionsupport graph preconditionerstwo-grid methods

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Ponce, Colin, Vassilevski, Panayot S.. 2016-02-18. Solving Graph Laplacian Systems Through Recursive Bisections and Two-Grid Preconditioning. https://doi.org/10.2172/1240975

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