TY - JOUR
T1 - A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians
AU - Urschel , John C.
AU - Xu , Jinchao
AU - Hu , Xiaozhe
AU - Zikatanov , Ludmil T.
JO - Journal of Computational Mathematics
VL - 2
SP - 209
EP - 226
PY - 2015
DA - 2015/04
SN - 33
DO - http://doi.org/10.4208/jcm.1412-m2014-0041
UR - https://global-sci.org/intro/article_detail/jcm/9837.html
KW - Graph Laplacian, Cascadic multigrid, Fiedler vector, Elliptic eigenvalue problems.
AB - <p style="text-align: justify;">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.</p>