Algebraic multigrid for directed graph Laplacian linear systems (NS‐LAMG)

Fox, Alyson; Manteuffel, Thomas

doi:10.1002/nla.2152

Title: Algebraic multigrid for directed graph Laplacian linear systems (NS‐LAMG)

Journal Article · 31 January 2018 · Numerical Linear Algebra with Applications

DOI: https://doi.org/10.1002/nla.2152 · OSTI ID:1418882

^[1]; Manteuffel, Thomas ^[2]

Center for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore CA 94551 USA
Department of Applied Mathematics University of Colorado Boulder Boulder CO 80303 USA

Summary We propose nonsymmetric lean algebraic multigrid (NS‐LAMG), a new algebraic multigrid algorithm for directed graph Laplacian systems that combines ideas from undirected graph Laplacian multigrid solvers and multigrid algorithms for Markov chain stationary distribution systems. Low‐degree elimination, proposed in LAMG for undirected graphs, is generalized to directed graphs and is a key component of NS‐LAMG. In the setup phase, we propose a simple stationary‐aggregation multigrid algorithms for Markov chain stationary distribution systems solver enhanced by low‐degree elimination to find the right null‐space vector that is used for the intergrid transfer operators. Numerical results show that low‐degree elimination improves performance and that NS‐LAMG outperforms generalized minimal residual method with restart and stable bi‐conjugate gradient method for real‐world, directed graph Laplacian linear systems.

View Accepted Manuscript (Publisher)

Cite

Export

Save

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1418882

Journal Information:: Numerical Linear Algebra with Applications, Journal Name: Numerical Linear Algebra with Applications Journal Issue: 3 Vol. 25; ISSN 1070-5325

Publisher:: Wiley Blackwell (John Wiley & Sons)Copyright Statement

Country of Publication:: United Kingdom

Language:: English

References (30)

A robust algebraic multilevel preconditioner for non-symmetricM-matrices Notay, Y. van Numerical Linear Algebra with Applications, Vol. 7, Issue 5 https://doi.org/10.1002/1099-1506(200007/08)7:5<243::AID-NLA195>3.0.CO;2-Y	journal	January 2000
Recursive Krylov-based multigrid cycles Notay, Yvan; Vassilevski, Panayot S. Numerical Linear Algebra with Applications, Vol. 15, Issue 5 https://doi.org/10.1002/nla.542	journal	January 2008
Square and stretch multigrid for stochastic matrix eigenproblems: SQUARE AND STRETCH MULTIGRID Treister, Eran; Yavneh, Irad Numerical Linear Algebra with Applications, Vol. 17, Issue 2-3 https://doi.org/10.1002/nla.708	journal	February 2010
Fast multilevel methods for Markov chains: FAST MULTILEVEL METHODS FOR MARKOV CHAINS Sterck, Hans De; Miller, Killian; Treister, Eran Numerical Linear Algebra with Applications, Vol. 18, Issue 6 https://doi.org/10.1002/nla.800	journal	October 2011
Top-level acceleration of adaptive algebraic multilevel methods for steady-state solution to Markov chains De Sterck, H.; Miller, K.; Manteuffel, T. Advances in Computational Mathematics, Vol. 35, Issue 2-4 https://doi.org/10.1007/s10444-010-9168-x	journal	June 2010
On a two-level multigrid solution method for finite Markov chains Krieger, Udo R. Linear Algebra and its Applications, Vol. 223-224 https://doi.org/10.1016/0024-3795(95)00166-O	journal	July 1995
Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing Koutis, Ioannis; Miller, Gary L.; Tolliver, David Computer Vision and Image Understanding, Vol. 115, Issue 12 https://doi.org/10.1016/j.cviu.2011.05.013	journal	December 2011
Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications Li, Lun; Alderson, David; Doyle, John C. Internet Mathematics, Vol. 2, Issue 4 https://doi.org/10.1080/15427951.2005.10129111	journal	January 2005
The Computer Realization of the QR Decomposition on Matrices with Full Column Rank Feng, Tianxiang; Liu, Hongxia 2009 International Conference on Computational Intelligence and Security https://doi.org/10.1109/CIS.2009.54	conference	December 2009
Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid Brezina, M.; Falgout, R.; MacLachlan, S. SIAM Review, Vol. 47, Issue 2 https://doi.org/10.1137/050626272	journal	January 2005
Multilevel Adaptive Aggregation for Markov Chains, with Application to Web Ranking De Sterck, H.; Manteuffel, Thomas A.; McCormick, Stephen F. SIAM Journal on Scientific Computing, Vol. 30, Issue 5 https://doi.org/10.1137/070685142	journal	January 2008
Smoothed Aggregation Multigrid for Markov Chains De Sterck, H.; Manteuffel, T. A.; McCormick, S. F. SIAM Journal on Scientific Computing, Vol. 32, Issue 1 https://doi.org/10.1137/080719157	journal	January 2010
Towards Adaptive Smoothed Aggregation ($\alpha$SA) for Nonsymmetric Problems Brezina, M.; Manteuffel, T.; MCormick, S. SIAM Journal on Scientific Computing, Vol. 32, Issue 1 https://doi.org/10.1137/080727336	journal	January 2010
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems Saad, Youcef; Schultz, Martin H. SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3 https://doi.org/10.1137/0907058	journal	July 1986
Algebraic Multigrid for Markov Chains De Sterck, H.; Manteuffel, T. A.; McCormick, S. F. SIAM Journal on Scientific Computing, Vol. 32, Issue 2 https://doi.org/10.1137/090753589	journal	January 2010
Recursively Accelerated Multilevel Aggregation for Markov Chains De Sterck, H.; Miller, K.; Sanders, G. SIAM Journal on Scientific Computing, Vol. 32, Issue 3 https://doi.org/10.1137/090770114	journal	January 2010
Nearly Linear Time Algorithms for Preconditioning and Solving Symmetric, Diagonally Dominant Linear Systems Spielman, Daniel A.; Teng, Shang-Hua SIAM Journal on Matrix Analysis and Applications, Vol. 35, Issue 3 https://doi.org/10.1137/090771430	journal	January 2014
Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems van der Vorst, H. A. SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 2 https://doi.org/10.1137/0913035	journal	March 1992
A Spectral Clustering Approach To Finding Communities in Graphs White, Scott; Smyth, Padhraic Proceedings of the 2005 SIAM International Conference on Data Mining https://doi.org/10.1137/1.9781611972757.25	conference	January 2014
A Bootstrap Algebraic Multilevel Method for Markov Chains Bolten, Matthias; Brandt, Achi; Brannick, James SIAM Journal on Scientific Computing, Vol. 33, Issue 6 https://doi.org/10.1137/100791816	journal	January 2011
On-the-Fly Adaptive Smoothed Aggregation Multigrid for Markov Chains Treister, Eran; Yavneh, Irad SIAM Journal on Scientific Computing, Vol. 33, Issue 5 https://doi.org/10.1137/100799034	journal	January 2011
Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver Livne, Oren E.; Brandt, Achi SIAM Journal on Scientific Computing, Vol. 34, Issue 4 https://doi.org/10.1137/110843563	journal	January 2012
Approaching Optimality for Solving SDD Linear Systems Koutis, Ioannis; Miller, Gary L.; Peng, Richard SIAM Journal on Computing, Vol. 43, Issue 1 https://doi.org/10.1137/110845914	journal	January 2014
Julia: A Fresh Approach to Numerical Computing Bezanson, Jeff; Edelman, Alan; Karpinski, Stefan SIAM Review, Vol. 59, Issue 1 https://doi.org/10.1137/141000671	journal	January 2017
Center-piece subgraphs: problem definition and fast solutions Tong, Hanghang; Faloutsos, Christos Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '06 https://doi.org/10.1145/1150402.1150448	conference	January 2006
Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs Christiano, Paul; Kelner, Jonathan A.; Madry, Aleksander Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11 https://doi.org/10.1145/1993636.1993674	conference	January 2011
A simple, combinatorial algorithm for solving SDD systems in nearly-linear time Kelner, Jonathan A.; Orecchia, Lorenzo; Sidford, Aaron Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13 https://doi.org/10.1145/2488608.2488724	conference	January 2013
Almost-linear-time algorithms for Markov chains and new spectral primitives for directed graphs Cohen, Michael B.; Kelner, Jonathan; Peebles, John STOC '17: Symposium on Theory of Computing, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing https://doi.org/10.1145/3055399.3055463	conference	June 2017
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares Paige, Christopher C.; Saunders, Michael A. ACM Transactions on Mathematical Software, Vol. 8, Issue 1 https://doi.org/10.1145/355984.355989	journal	March 1982
Algebraic connectivity of graphs [Algebraic connectivity of graphs] Fiedler, Miroslav Czechoslovak Mathematical Journal, Vol. 23, Issue 2 https://doi.org/10.21136/CMJ.1973.101168	journal	January 1973

Similar Records

A Parallel Solver for Graph Laplacians

Conference · 2018 · OSTI ID:1544266

Konolige, Tristan; Brown, Jed

Final Report:B595949 - Fast Solvers for Discrete Hodge Laplacians

Technical Report · 2013 · OSTI ID:1062227

Zikatanov, Ludmil

Two-Level Adaptive Algebraic Multigrid for a Sequence of Problems with Slowly Varying Random Coefficients [Adaptive Algebraic Multigrid for Sequence of Problems with Slowly Varying Random Coefficients]

Journal Article · 2013 · SIAM Journal on Scientific Computing · OSTI ID:1226208

Kalchev, D.; Ketelsen, C.; Vassilevski, P. S.

Title: Algebraic multigrid for directed graph Laplacian linear systems (NS‐LAMG)

Citation Formats

References (30)

Similar Records

Related Subjects