J. Comp. Math., 33 (2015), pp. 209-226. |
A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians John C. Urschel ^{1}, Xiaozhe Hu ^{1}, Jinchao Xu ^{1}, Ludmil T. Zikatanov ^{2} 1 Department of Mathematics, Tufts University, Medford, MA 021552 Department of Mathematics, Penn State University, Pennsylvania, USA; Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria Received 2014-8-22 Accepted 2014-12-22 Available online 2015-3-13 doi:10.4208/jcm.1412-m2014-0041 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. Key words: Graph Laplacian; Cascadic Multigrid; Fiedler vector; Elliptic eigenvalue problems. AMS subject classifications: 65N55, 65N25. Email: jcurschel@gmail.com, xiaozhe.hu@tufts.edu, xu@math.psu.edu, ludmil@psu.edu |