Preconditioning Linear Systems Arising from Graph Laplacians of Complex Networks
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1 Preconditioning Linear Systems Arising from Graph Laplacians of Complex Networks Kevin Deweese 1 Erik Boman 2 1 Department of Computer Science University of California, Santa Barbara 2 Scalable Algorithms Department Sandia National Laboratories Copper Mountain Conference, 2014 Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy s National Nuclear Security Administration under contract DE-AC04-94AL85000.
2 Problem Description Our focus: Solve the the system of equations Lx = b where L is a combinatorial Laplacian matrix Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
3 Network Science Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
4 Best Techniques Direct Solver: Sparse Cholesky Iterative Solver: Conjugate Gradient Use a preconditioner to accelerate convergence. B 1 Ax = B 1 b A key challenge is to find a good preconditioner. Traditional preconditioners include Jacobi, Incomplete Cholesky, and Multigrid methods. Are different techniques needed for network science problems? Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
5 Combinatorial Preconditioning Solving linear equations with SDD matrices by constructing good preconditioners (Vaidya, 1991) Support theory for preconditioning (Boman and Hendrickson, 2003) Vaidya s preconditioners: Implementation and experimental Study (Chen and Toledo, 2003) Nearly linear time algorithms for preconditioning and solving SDD linear systems (Spielman and Teng, 2006) A nearly m log n time solver for SDD linear systems (Koutis, Miller, and Peng, 2011) mostly theory, few experiments Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
6 Support Graphs Use a graph approximation as a preconditioner. - Complete factorization of an incomplete matrix Specifically we use max-weight spanning trees. Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
7 Experimental Setup We use the Trilinos software framework (Heroux et al. 2005) for our experiments. We implemented MST as an Ifpack2 preconditioner. We use - Ifpack2 s Jacobi and RILUK preconditioners. - Tpetra for matrix/vector operations. - Belos for iterative methods (PCG). - Zoltan for graph partitioning. Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
8 Serial Results Flickr graph, 800k Rows (Vertices), 13M NNZ (2xEdges) Preconditioner Iterations Solve Time (s) None Jacobi ILUT MST Normalized Residual Iteration MST Jacobi ILUT Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
9 Parallel Setup We use the Additive Schwarz domain decomposition scheme to split the problem over subdomains and do inexact subdomain solves. - No overlap (Block Jacobi) We use Zoltan s interface to the Scotch graph partitioning library to - Load balance. - Reduce communication cost. - Improve subdomain preconditioner quality. Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
10 Effect of Partitioning as-skitter, 1.7M Rows (Vertices), 22M NNZ (2xEdges), 24 cores Preconditioner Iterations Iterations Solve Time (s) Solve Time (s) No Partitioning Scotch No Partitioning Scotch MST Jacobi RILUK Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
11 Experiments Strong Scaling (24, 48, 96 cores of hopper) - as-skitter ( 11M edges) - as-skitter with random edge weights (0-100) Weak Scaling (12, 24, 48 cores of hopper) - Used the BTER graph generator (Kolda et al. 2014) to generate graphs of 250k, 500k, and 1M nodes. - BTER (average degree 20, global clustering coefficient.3) - BTER with random edge weights (0-100) - 2.6M, 5M, 10M edges respectively Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
12 Strong Scaling as-skitter, 1.7M Rows (Vertices), 22M NNZ (2xEdges) Iterations Sove Time (s) Jacobi RILUK MST Processors Processors Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
13 Strong Scaling as-skitter with edge weights, 1.7M Rows (Vertices), 22M NNZ (2xEdges) Iterations Sove Time (s) Jacobi RILUK MST Processors Processors Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
14 Weak Scaling BTER Iterations Sove Time (s) Jacobi RILUK MST Processors Processors Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
15 Weak Scaling BTER with edge weights Iterations Sove Time (s) Jacobi RILUK MST Processors Processors Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
16 Summary Support graph preconditioners with domain decomposition scale well and are a promising alternative for solving linear problems of Laplacians. Other support graphs should be examined. We plan to target low-stretch trees next. Edge weights matter. Can we generate graphs with more realistic edge weights? Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
17 Thank You John Gilbert, Siva Rajamanickam, Karen Devine Kevin Deweese, Erik Boman (UCSB,SNL) Preconditioning Graph Laplacians CM / 17
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