Accelerated multigrid for graph Laplacian operators

Pietro Dell'Acqua, Antonio Frangioni, Stefano Serra-Capizzano
2015 Applied Mathematics and Computation  
We consider multigrid type techniques for the numerical solution of large linear systems, whose coefficient matrices show the structure of (weighted) graph Laplacian operators. We combine ad hoc coarser-grid operators with iterative techniques used as smoothers. Empirical tests suggest that the most effective smoothers have to be of Krylov type with subgraph preconditioners, while the projectors, which define the coarser-grid operators, have to be designed for maintaining as much as possible
more » ... graph structure of the projected matrix at the inner levels. The main theoretical contribution of the paper is the characterization of necessary and sufficient conditions for preserving the graph structure. In this framework it is possible to explain why the classical projectors inherited from differential equations are good in the differential context and why they may behave unsatisfactorily for unstructured graphs. Furthermore, we report and discuss several numerical experiments, showing that our approach is effective even in very difficult cases where the known approaches are rather slow. As a conclusion, the main advantage of the proposed approach is the robustness, since our multigrid type technique behaves uniformly well in all cases, without requiring either the setting or the knowledge of critical parameters, as it happens when using the best known preconditioned Krylov methods. Keywords: graph matrices, multigrid, conditioning and preconditioning 2000 MSC: 05C50, 15A12, 65F10 [46] to general Markov chains [20], from consensus algorithms [53] to optimization problems such as the Minimum Cost Flow (MCF) in networks [1, 4] . In all these cases the dimension of the involved matrices is either very large or extremely large. Among these applications, the central one in the development of this article, at least as far as the computational part is concerned, involves linear systems arising in Interior Point (IP) techniques for MCF problems [1, 4] . In this setting, a linear system involving the weighted Laplacian of the underlying network has to be solved at each iteration, with varying vector of arc weights Θ (cf. relation (2) ). Not only the networks can be very large (e.g., with up to 2 22 arcs [54]), but these approaches allow a more or less direct extension [13, 14] to multicommodity flow problems [30, 15, 16] , that have a huge range of practical applications from telecommunication [18] to transportation [31] and beyond. In the latter setting, the size of the matrix is further multiplied by the number of commodities (different types of flows in the network), that can be easily run into the thousands (e.g. being quadratic in the number of nodes in some applications). Furthermore, the graph structure is somewhat "muddled" by the rows corresponding to mutual capacity constraints. In summary, the considered class of problems exhibits three main issues: • the large size of the considered linear systems; • the sparsity and the graph structure of the involved matrices;
doi:10.1016/j.amc.2015.08.033 fatcat:a7z6buplnnhlfcw4xtrwwcr2wq