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Bootstrap Algebraic Multigrid: Status Report, Open Problems, and Outlook

Published online by Cambridge University Press:  03 March 2015

Achi Brandt ,
James Brannick ,
Karsten Kahl  and
Ira Livshits
Achi Brandt
Affiliation:
Department of Applied Mathematics & Computer Science, The Weizmann Institute of Science, 76100 Rehovot, Israel
James Brannick*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
Karsten Kahl
Affiliation:
Department of Mathematics, University of Wuppertal, Wuppertal, Germany
Ira Livshits
Affiliation:
Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA
*
*Email addresses: achi.brandt@weizmann.ac.il (A. Brandt), brannick@psu.edu (J. Brannick), kkahl@math.uni-wuppertal.de (K. Kahl), ilivshits@bsu.edu (I. Livshits)
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Abstract

This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process. We review some recent research in the development, analysis and application of bootstrap algebraic multigrid and point to open problems in these areas. Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.

Keywords

65F1065N2265N55Bootstrap algebraic multigridcompatible relaxationalgebraic distanceleast squares interpolation

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Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 1 , February 2015 , pp. 112 - 135
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Brandt, A.. Multi-level adaptive solutions to boundary value problems. Math. Comp., 31:333–390, 1977.CrossRefGoogle Scholar
[2]Brandt, A.. Algebraic multigrid theory: The symmetric case. Appl. Math. Comput., 19(1–4):23–56, 1986.Google Scholar
[3]Brandt, A.. General highly accurate algebraic coarsening. Elect. Trans. Numer. Anal., 10:1–20, 2000.Google Scholar
[4]Brandt, A.. Multiscale scientific computation: Review 2001. In Multiscale and Multireso-lution Methods, pages 1–96. Springer Verlag, 2001.Google Scholar
[5]Brandt, A.. Principles of systematic upscaling. In Bridging the Scales in Science and Engineering, pages 193–215. Oxford University Press, 2010.Google Scholar
[6]Brandt, A., Brannick, J., Bolten, M., Frommer, A., Kahl, K., and Livshits, I.. Bootstrap AMG for Markov chains. SIAMJ. Sci. Comp., 33:3425–3446, 2011.Google Scholar
[7]Brandt, A., Brannick, J., Kahl, K., and Livshits, I.. Bootstrap AMG. SIAM J. Sci. Comput., 33:612–632, 2011.CrossRefGoogle Scholar
[8]Brandt, A., Brannick, J., Kahl, K., and Livshits, I.. Algebraic distance as a measure ofstrength of connection in AMG. Electronic Transactions in Numerical Analysis, 2013. Submitted: April 2013. Also available as arXiv:1106.5990 [math.NA].Google Scholar
[9]Brandt, A., McCormick, S., and Ruge, J.W.. Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Technical report, Colorado State University, Fort Collins, Colorado, 1983.Google Scholar
[10]Brandt, A., McCormick, S.F., and Ruge, J.W.. Algebraic multigrid (AMG) for sparse matrix equations. In Evans, D.J. editor, Sparsity and Its Applications. Cambridge University Press, Cambridge, 1984.Google Scholar
[11]Brandt, A. and Ron, D.. Renormalization multigrid (RMG): Statistically optimal renor-malization group flow and coarse-to-fine monte carlo acceleration. Journal of Statistical Physics, 102:231–257, 2001. 10.1023/A:1026520927784. First appeared as Renormal-ization Multigrid (RMG): Statistically Optimal Renormalization Group Flow and Coarse to-Fine Monte-Carlo Acceleration, Technical Report GMC-11, Weizmann Institute of Science, 1999.Google Scholar
[12]Brannick, J.. Adaptive algebraic Multigrid coarsening strategies. PhD thesis, University of Colorado at Boulder, 2005.Google Scholar
[13]Brannick, J., Chen, Y., and Zikatanov, L.. An algebraic multilevel method for anisotropic elliptic equations based on subgraph matching. Numer. Linear Algebra Appl., To appear, accepted for publication November 17, 2011.Google Scholar
[14]Brannick, J. and Falgout, R.. Compatible relaxation and coarsening in algebraic multigrid. SIAM J. Sci. Comput., 32(3):1393–1416, 2010.CrossRefGoogle Scholar
[15]Brannick, J. and Kahl, K.. Bootstrap algebraic multigrid for the 2d wilson-dirac system. SIAM Journal of Scientific Computing. Submitted (August 28, 2013). Also available on aRxiV:1308.5992 [math.NA].Google Scholar
[16]Brannick, J., Kahl, K., and Sokolovic, S.. Journal of Applied Numerical Mathematics, Submitted January 19, 2014.Google Scholar
[17]Brannick, J. and Zikatanov, L.. Algebraic multigrid methods based on compatible relaxation and energy minimization. In Widlund, O.B. and Keyes, D.E.. editors, Lecture Notes in Computational Science and Engineering, volume 55, pages 15–26. Springer, 2007.Google Scholar
[18]Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., and Ruge, J.. Adaptive smoothed aggregation (αSA). SIAM J. Sci. Comput., 25(6):1896–1920, 2004.CrossRefGoogle Scholar
[19]Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T.McCormick, S. and Ruge, J.Adaptive amg (aAMG). SIAMJ. Sci. Comput., 26:1261–1286, 2005.Google Scholar
[20]Brezina, M., Ketelsen, C., Manteuffel, T., McCormick, S., Park, M., and Ruge, J.. Relaxation-corrected bootstrap algebraic multigrid (rBAMG). Journal of Numerical Linear Algebra with Applications, 19(2):178–193, March 2012.CrossRefGoogle Scholar
[21]Brezina, M., Manteuffel, T., McCormick, S., Ruge, J., Sanders, G., and Vassilevski, P.. A generalized eigensolver based on smoothed aggregation (ges-sa) for initializing smoothed aggregation (sa) multigrid. Numer. Linear Algebra Appl., 15:249–269, 2008.CrossRefGoogle Scholar
[22]Chartier, T., Falgout, R.D., Henson, V.E., Jones, J.E., and Manteuffel, T., McCormick, S.F., Ruge, J.W., and Vassilevski, P.S.. Spectral AMGe (ρAMGe). SIAM J. Sci. Comput., 25:1–26, 2003.CrossRefGoogle Scholar
[23]Falgout, R. and Vassilevski, P.. On generalizing the amg framework. SIAM J. Numer. Anal., 42(4):1669–1693, 2004.CrossRefGoogle Scholar
[24]Halko, N., Martinsson, P., and Tropp, J.. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53:217–288, 2011.CrossRefGoogle Scholar
[25]Livne, O.E.. Coarsening by compatible relaxation. Num. Lin. Alg. Appl., 11:205–227, 2004.CrossRefGoogle Scholar
[26]Livne, O.E. and Brandt, A.. Lean algebraic multigrid (LAMG): Fast graph Laplacian linear solver. SIAM Journal of Scientific Computing, 34:B499–B522, 2012.CrossRefGoogle Scholar
[27]Livshits, I.. One-dimensional algorithm for finding eigenbasis of the schrödinger operator. SIAM J. Sci. Comput., 30(1):416–440, 2008.CrossRefGoogle Scholar
[28]Livshits, I.. The least squares amg solver for the one-dimensional helmholtz operator. Computing and Visualization in Science, (14):17–25, 2011.CrossRefGoogle Scholar
[29]MacLachlan, S.Manteuffel, T., and McCormick, S.. Adaptive reduction-based amg. Numer. Linear Algebra Appl., 13:599–620, 2006.CrossRefGoogle Scholar
[30]MacLachlan, S. and Saad, Y.. A greedy strategy for coarse-grid selection. SIAM J. Sci. Comput., 29(5):1825–1853, 2007.CrossRefGoogle Scholar
[31]Manteuffel, T., McCormick, S., Park, M., and Ruge, J.. Operator-based interpolation for bootstrap algebraic multigrid. Numer. Linear Algebra Appl., 17(2–3):519–537, 2010.CrossRefGoogle Scholar
[32]Bobby, Philip and Chartier, Timothy P.. Adaptive algebraic smoothers. J. Computational Applied Mathematics, 236:2277–2297, 2012.Google Scholar
[33]Ron, D., Safro, I., and Brandt, A.. Relaxation-based coarsening and multiscale graph organization. Multiscale Modeling and Simulation, 9:407–423, 2011.CrossRefGoogle Scholar
[34]Ruge, J.W. and Stuoben, K.. Algebraic multigrid (AMG). In McCormick, S.F., editor, Multigrid Methods, volume 3 of Frontiers in Applied Mathematics, pages 73–130. SIAM, Philadelphia, PA, 1987.CrossRefGoogle Scholar
[35]Safro, I., Ron, D., and Brandt, A.. Multilevel algorithms for linear ordering problems. J. Exp. Algorithmics, 13:1.4–1.20, 2009.CrossRefGoogle Scholar
[36]Schroder, J.B.. Smoothed aggregation solvers for anisotropic diffusion. Numer. Linear Algebra Appl., 19:296–312, 2012.CrossRefGoogle Scholar
[37]Wilkinson, J.. The Algebraic eigenvalue problem. Clarendon Press, Oxford, 1965.Google Scholar