Skip to main content Accessibility help

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

  • Achi Brandt (a1), James Brannick (a2), Karsten Kahl (a3) and Ira Livshits (a4)


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.


Corresponding author

*Email addresses: (A. Brandt), (J. Brannick), (K. Kahl), (I. Livshits)


Hide All
[1]Brandt, A.. Multi-level adaptive solutions to boundary value problems. Math. Comp., 31:333390, 1977.
[2]Brandt, A.. Algebraic multigrid theory: The symmetric case. Appl. Math. Comput., 19(1–4):2356, 1986.
[3]Brandt, A.. General highly accurate algebraic coarsening. Elect. Trans. Numer. Anal., 10:120, 2000.
[4]Brandt, A.. Multiscale scientific computation: Review 2001. In Multiscale and Multireso-lution Methods, pages 196. Springer Verlag, 2001.
[5]Brandt, A.. Principles of systematic upscaling. In Bridging the Scales in Science and Engineering, pages 193215. Oxford University Press, 2010.
[6]Brandt, A., Brannick, J., Bolten, M., Frommer, A., Kahl, K., and Livshits, I.. Bootstrap AMG for Markov chains. SIAMJ. Sci. Comp., 33:34253446, 2011.
[7]Brandt, A., Brannick, J., Kahl, K., and Livshits, I.. Bootstrap AMG. SIAM J. Sci. Comput., 33:612632, 2011.
[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].
[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.
[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.
[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.
[12]Brannick, J.. Adaptive algebraic Multigrid coarsening strategies. PhD thesis, University of Colorado at Boulder, 2005.
[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.
[14]Brannick, J. and Falgout, R.. Compatible relaxation and coarsening in algebraic multigrid. SIAM J. Sci. Comput., 32(3):13931416, 2010.
[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].
[16]Brannick, J., Kahl, K., and Sokolovic, S.. Journal of Applied Numerical Mathematics, Submitted January 19, 2014.
[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 1526. Springer, 2007.
[18]Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., and Ruge, J.. Adaptive smoothed aggregation (αSA). SIAM J. Sci. Comput., 25(6):18961920, 2004.
[19]Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T.McCormick, S. and Ruge, J.Adaptive amg (aAMG). SIAMJ. Sci. Comput., 26:12611286, 2005.
[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):178193, March 2012.
[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., 15:249269, 2008.
[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., 25:126, 2003.
[23]Falgout, R. and Vassilevski, P.. On generalizing the amg framework. SIAM J. Numer. Anal., 42(4):16691693, 2004.
[24]Halko, N., Martinsson, P., and Tropp, J.. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53:217288, 2011.
[25]Livne, O.E.. Coarsening by compatible relaxation. Num. Lin. Alg. Appl., 11:205227, 2004.
[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.
[27]Livshits, I.. One-dimensional algorithm for finding eigenbasis of the schrödinger operator. SIAM J. Sci. Comput., 30(1):416440, 2008.
[28]Livshits, I.. The least squares amg solver for the one-dimensional helmholtz operator. Computing and Visualization in Science, (14):1725, 2011.
[29]MacLachlan, S.Manteuffel, T., and McCormick, S.. Adaptive reduction-based amg. Numer. Linear Algebra Appl., 13:599620, 2006.
[30]MacLachlan, S. and Saad, Y.. A greedy strategy for coarse-grid selection. SIAM J. Sci. Comput., 29(5):18251853, 2007.
[31]Manteuffel, T., McCormick, S., Park, M., and Ruge, J.. Operator-based interpolation for bootstrap algebraic multigrid. Numer. Linear Algebra Appl., 17(2–3):519537, 2010.
[32]Bobby, Philip and Chartier, Timothy P.. Adaptive algebraic smoothers. J. Computational Applied Mathematics, 236:22772297, 2012.
[33]Ron, D., Safro, I., and Brandt, A.. Relaxation-based coarsening and multiscale graph organization. Multiscale Modeling and Simulation, 9:407423, 2011.
[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 73130. SIAM, Philadelphia, PA, 1987.
[35]Safro, I., Ron, D., and Brandt, A.. Multilevel algorithms for linear ordering problems. J. Exp. Algorithmics, 13:1.4–1.20, 2009.
[36]Schroder, J.B.. Smoothed aggregation solvers for anisotropic diffusion. Numer. Linear Algebra Appl., 19:296312, 2012.
[37]Wilkinson, J.. The Algebraic eigenvalue problem. Clarendon Press, Oxford, 1965.


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

  • Achi Brandt (a1), James Brannick (a2), Karsten Kahl (a3) and Ira Livshits (a4)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed