Year: 2015
Author: John C. Urschel, Jinchao Xu, Xiaozhe Hu, Ludmil T. Zikatanov
Journal of Computational Mathematics, Vol. 33 (2015), Iss. 2 : pp. 209–226
Abstract
In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1412-m2014-0041
Journal of Computational Mathematics, Vol. 33 (2015), Iss. 2 : pp. 209–226
Published online: 2015-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18
Keywords: Graph Laplacian Cascadic multigrid Fiedler vector Elliptic eigenvalue problems.