BootCMatch: A Software Package for Bootstrap AMG Based on Graph Weighted Matching

Abstract

This article has two main objectives: one is to describe some extensions of an adaptive Algebraic Multigrid (AMG) method of the form previously proposed by the first and third authors, and a second one is to present a new software framework, named BootCMatch, which implements all the components needed to build and apply the described adaptive AMG both as a stand-alone solver and as a preconditioner in a Krylov method. The adaptive AMG presented is meant to handle general symmetric and positive definite (SPD) sparse linear systems, without assuming any a priori information of the problem and its origin; the goal of adaptivity is to achieve a method with a prescribed convergence rate. The presented method exploits a general coarsening process based on aggregation of unknowns, obtained by a maximum weight matching in the adjacency graph of the system matrix. More specifically, a maximum product matching is employed to define an effective smoother subspace (complementary to the coarse space), a process referred to as compatible relaxation, at every level of the recursive two-level hierarchical AMG process.

Results on a large variety of test cases and comparisons with related work demonstrate the reliability and efficiency of the method and of the software.

References

1. A. Abdullahi, P. D’Ambra, D. di Serafino, and S. Filippone. 2018. Parallel aggregation based on compatible weighted matching for AMG. In Large Scale Scientific Computing. LSSC 2017. (Lecture Notes in Computer Science), I. Lirkov and S. Margenov (Eds.), Vol. 10665. Springer, Cham, Switzerland, 563--571.
2. P. R. Amestoy, I. S. Duff, J. Koster, and J. Y. L’Excellent. 2001. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23 (2001), 15--41.
3. M. Avila, A. Folch, G. Houzeaux, B. Eguzkitza, L. Prieto, and D. Cabezón. 2013. A parallel CFD model for wind farms. Procedia Computer Science 18 (2013), 2157--2166. International Conference on Computational Science, 2013.
4. A. H. Baker, Tz. V. Kolev, and U. M. Yang. 2010. Improving algebraic multigrid interpolation operators for linear elasticity. Numer. Linear Algebra Appl. 17 (2010), 495--517.
5. D. P. Bertsekas. 1988. The auction algorithm: A distributed relaxation method for the assignment problem. Annals of Operations Research 14 (1988), 105--123.
6. D. P. Bertsekas. 1992. Auction algorithms for network flow problems: A tutorial introduction. Computational Optimization and Applications 1 (1992), 7--66.
7. BootCMatch. 2017. Bootstrap algebraic multigrid based on compatible weighted matching. Retrieved from https://github.com/bootcmatch/BootCMatch.
8. A. Brandt. 2000. General highly accurate algebraic coarsening. Electronic Transactions on Numerical Analysis 10 (2000), 1--20.
9. A. Brandt, J. Brannick, K. Kahl, and I. Livshits. 2015. Algebraic distance for anisotropic diffusion problems: Multilevel results. Electronic Transactions on Numerical Analysis 44 (2015), 472--496.
10. A. Brandt, J. Brannick, K. Kahl, and I. Livshits. 2015. Bootstrap algebraic multigrid: Status report, open problems, and outlook. Numerical Mathematics: Theory, Methods and Applications 8 (2015), 112--135.
11. A. Brandt, J. Brannick, K. Kahl, and I. Livshitz. 2011. Bootstrap AMG. SIAM J. Sci. Comput. 33 (2011), 612--632.
12. A. Brandt, S. McCormick, and J. Ruge. 1985. Algebraic multigrid (AMG) for sparse matrix equations. In Sparsity and its Applications. Cambridge Univ. Press, Cambridge, 257--284.
13. J. Brannick, Y. Chen, J. Kraus, and L. Zikatanov. 2013. Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs. SIAM J. Numer. Anal. 51 (2013), 1805--1827.
14. J. Brannick, Y. Chen, and L. Zikatanov. 2012. An algebraic multilevel method for anisotropic elliptic equations based on subgraph matching. Numer. Linear Algebra Appl. 19 (2012), 279--295.
15. J. Brannick and R. D. Falgout. 2010. Compatible relaxation and coarsening in algebraic multigrid. SIAM J. Sci. Comput. 32 (2010), 1393--1416.
16. M. Brezina, R. D. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick, and J. Ruge. 2005. Adaptive smoothed aggregation SA multigrid. SIAM Rev. 47 (2005), 317--346.
17. M. Brezina, R. D. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick, and J. Ruge. 2006. Adaptive algebraic multigrid. SIAM J. Sci. Comput. 27 (2006), 1261--1286.
18. P. D’Ambra and P. S. Vassilevski. 2013. Adaptive AMG with coarsening based on compatible weighted matching. Comput. Visual Sci. 16 (2013), 59--76.
19. P. D’Ambra and P. S. Vassilevski. 2016. Adaptive AMG based on weighted matching for systems of elliptic PDEs arising from displacement and mixed methods. In Progress in Industrial Mathematics at ECMI 2014 (Mathematics in Industry), Russo G. et al. (Eds.), Vol. 22. Springer-Verlag, Berlin, Germany, 1013--1020.
20. T. A. Davis and Y. Hu. 2011. The University of Florida sparse matrix collection. ACM Trans. Math. Software 38, 1 (2011), 1:1--1:25.
21. R. Diestel. 2010. Graph Theory, 4th ed.Springer, Heidelberg, GTM 173.
22. I. S. Duff and J. Koster. 2001. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Anal. Appl. 22 (2001), 973--996.
23. I. S. Duff and S. Pralet. 2005. Strategies for scaling and pivoting for sparse symmetric indefinite problems. SIAM J. Matrix Anal. Appl. 27 (2005), 313--340.
24. R. D. Falgout, J. E. Jones, and U. Meyer Yang. 2006. The design and implementation of Hypre, a library of parallel high-performance preconditioners. In Numerical Solutions of Partial Differential Equations on Parallel Computers (Lecture Notes in Computational Science and Engineering), A. M. Bruaset and A. Tveito (Eds.), Vol. 15. Springer-Verlag, Berlin, Germany, 267--294.
25. R. D. Falgout and P. S. Vassilevski. 2004. On generalizing the algebraic multigrid framework. SIAM J. Numer. Anal. 42 (2004), 1669--1693.
26. R. D. Falgout, P. S. Vassilevski, and L. T. Zikatanov. 2005. On two-grid convergence estimates. Numer. Linear Algebra Appl. 12 (2005), 471--494.
27. M. Hagemann and O. Schenk. 2006. Weighted matchings for preconditioning symmetric indefinite linear systems. SIAM J. Sci. Comput. 28 (2006), 403--420.
28. M. Halappanavar, J. Feo, O. Villa, A. Tumeo, and A. Pothen. 2012. Approximate weighted matching on emerging manycore and multithreaded architectures. Int. J. High Perform. Comput. Appl. 26 (2012), 413--430.
29. J. Hogg. 2016. Sparse Parallel Robust Algorithms Library (SPRAL). Retrieved from https://github.com/ralna/spral.
30. J. Hogg and J. Scott. 2013. Optimal weighted matchings for rank-deficient sparse matrices. SIAM J. Matrix Anal. Appl. 34 (2013), 1431--1447.
31. J. Hogg and J. Scott. 2015. On the use of suboptimal matchings for scaling and ordering sparse symmetric matrices. Numer. Linear Algebra Appl. 22 (2015), 648--663.
32. H. Kuhn. 1955. The Hungarian method for the assignment problem. Naval Research Logistics Quarterly 2 (1955), 83--97.
33. X. S. Li. 2005. An overview of SuperLU: Algorithms, implementation, and user interface. ACM Trans. Math. Software 31, 3 (September 2005), 302--325.
34. O. E. Livne. 2004. Coarsening by compatible relaxation. Num. Linear Alg. Appl. 11 (2004), 205--227.
35. M. Metcalf, J. Reid, and M. Cohen. 2011. Modern Fortran Explained (4th ed.). Oxford University Press, Inc., New York, NY.
36. MFEM. 2015. Modular finite element methods. mfem.org. (2015).
37. A. Napov and Y. Notay. 2012. An algebraic multigrid method with guaranteed convergence rate. SIAM J. Sci. Comput. 34 (2012), A1079--A1109.
38. Y. Notay. 2010. An aggregation-based algebraic multigrid method. Electronic Transactions on Numerical Analysis 37 (2010), 123--146.
39. Y. Notay and P. S. Vassilevski. 2008. Recursive Krylov-based multigrid cycles. Numer. Linear Algebra Appl. 15 (2008), 473--487.
40. A. Pothen, F. Dobrian, and M. Halappanavar. 2013. Matchbox, a library of graph matching algorithms. (2013). http://www.cs.odu.edu/ mhalappa/matching/.
41. R. Preis. 1999. Linear time 1/2-approximation algorithm for maximum weighted matching in general graphs. In STACS’99 (Lecture Notes in Computer Science), J. Dongarra, K. Madsen, and J. Wasniewski (Eds.), Vol. 1563. Springer-Verlag, Berlin, Germany, 259--269.
42. J. W. Ruge. 1986. AMG for problems of elasticity. Appl. Math. Comput. 19 (1986), 293--309.
43. J. W. Ruge and K. Stüben. 1987. Algebraic multigrid (AMG). In Multigrid Methods, S. F. McCormick (Ed.). SIAM, Philadelphia, 73--130.
44. M. Sathe, O. Schenk, and H. Burkhart. 2012. An auction-based weighted matching implementation on massively parallel architectures. Parallel Comput. 38 (2012), 595--614.
45. The Numerical Analysis Group. 2011. HSL (2011). A collection of Fortran codes for large scale scientific computation. (2011). http://www.hsl.rl.ac.uk.
46. P. Vaněk, J. Mandel, and M. Brezina. 1996. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems.Computing 56, 3 (1996), 179--196.
47. P. S. Vassilevski. 2008. Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations. Springer, New York.

Index Terms

• Mathematics of computing

  • Mathematical software

    • Solvers