Skip to main content Accessibility help

Algebraic Theory of Two-Grid Methods

  • Yvan Notay (a1)


About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.


Corresponding author

*Email addresses: (Yvan Notay) Yvan Notay is Research Director of the Fonds de la Recherche Scientifique - FNRS


Hide All
[1]Axelsson, O., Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.
[2]Baker, A.H., Falgout, R.D., Kolev, T.V., and Yang, U.M., Multigrid smoothers for ultraparallel computing, SIAM J. Sci. Comput., 33 (2011), pp. 28642887.
[3]Bank, R. and Douglas, C., Sharp estimates for multigrid rates of convergence with general smoothing and acceleration, SIAM J. Numer. Anal., 22 (1985), pp. 617633.
[4]Bank, R.E., Dupont, T.F., and Yserentant, H., The hierarchical basis multigrid method, Numer. Math., 52 (1988), pp. 427458.
[5]Bramble, J.H., Pasciak, J.E., Wang, J., and Xu, J., Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp., 57 (1991), pp. 121.
[6]Brandt, A., Algebraic multigrid theory: The symmetric case, Appl. Math. Comput., 19 (1986), pp. 2356.
[7]Brandt, A., General highly accurate algebraic coarsening, Electron. Trans. Numer. Anal., 10 (2000), pp. 120.
[8]Brandt, A., McCormick, S.F., and Ruge, J.W., Algebraic multigrid (AMG) for sparse matrix equations, in Sparsity and its Application, Evans, D.J., ed., Cambridge University Press, Cambridge, 1984, pp. 257284.
[9]Brannick, J.J. and Falgout, R.D., Compatible relaxation and coarsening in algebraic multigrid, SIAM J. Sci. Comput., 32 (2010), pp. 13931416.
[10]Brezina, M., Cleary, A.J., Falgout, R.D., Henson, V.E., Jones, J.E., Manteuffel, T.A., McCormick, S.F., and Ruge, J.W., Algebraic multigrid based on element interpolation (AMGe), SIAM J. Sci. Comput., 22 (2000), pp. 15701592.
[11]Brezina, M., Falgout, R., Maclachlan, S., Manteuffel, T., McCormick, S., and Ruge, J., Adaptive smoothed aggregation (αSA), SIAM Review, 47 (2005), pp. 317346.
[12]Brezina, M., Ketelsen, C., Manteuffel, T., McCormick, S., Park, M., and Ruge, J., Relaxation-corrected bootstrap algebraic multigrid (rBAMG), Numer. Linear Algebra Appl., 19 (2012), pp. 178193.
[13]Brezina, M., Manteuffel, T., McCormick, S., Ruge, J., and Sanders, G., Towards adaptive smoothed aggregation (αSA) for nonsymmetric problems, SIAM J. Sci. Comput., 32 (2010), pp. 1439.
[14]Chartier, T., Falgout, R.D., Henson, V.E., Jones, J., Manteuffel, T., McCormick, S., Ruge, J., and Vassilevski, P.S., Spectral AMGe (ρAMGe), SIAM J. Sci. Comput., 25 (2004), pp. 126.
[15]Dendy, J.E., Black box multigrid for nonsymmetric problems, Appl. Math. Comput., 13 (1983), pp. 261283.
[16]Erlangga, Y.A. and Nabben, R. , Multilevel projection-based nested Krylov iteration for boundary value problems, SIAM J. Sci. Comput., 30 (2008), pp. 15721595.
[17]Falgout, R.D. and Vassilevski, P.S., On generalizing the algebraic multigrid framework, SIAM J. Numer. Anal., 42 (2004), pp. 16691693.
[18]Falgout, R.D., Vassilevski, P.S., and Zikatanov, L.T., On two-grid convergence estimates, Numer. Linear Algebra Appl., 12 (2005), pp. 471494.
[19]Frank, J. and Vuik, C., On the construction of deflation-based preconditioners, SIAM J. Sci. Comput., 23 (2001), pp. 442462.
[20]Hackbusch, W., Convergence of multi-grid iterations applied to difference equations, Math. Comp., 34 (1980), pp. 425440.
[21]Hackbusch, W., Multi-grid Methods and Applications, Springer, Berlin, 1985.
[22]Hogben, L., ed., Handbook of Linear Algebra, CRC Press, Boca Raton, 2007.
[23]Horn, R. and Johnson, C., Matrix Analysis, 2nd Ed., Cambridge University Press, New York, 2013.
[24]Livne, O.E., Coarsening by compatible relaxation, Numer. Linear Algebra Appl., 11 (2004), pp. 205227.
[25]Maclachlan, S.P. and Olson, L.N., Theoretical bounds for algebraic multigrid performance: review and analysis, Numer. Linear Algebra Appl., 21 (2014), pp. 194220.
[26]Maitre, J. and Musy, F., Algebraic formalization of the multigrid method in the symmetric and positive definite case – a convergence estimation for the V-cycle, in Multigrid Methods for Integral and Differential Equations (Bristol, 1983), Paddon, D. and Holsein, H., eds., vol. 3 of Institute of Mathematics and Its Applications Conference Series, Oxford, 1985, Clarendon Press, p. 213223.
[27]Mandel, J., Algebraic study of multigrid methods for symmetric, definite problems, Appl. Math. Comput., 25 (1988), pp. 3956.
[28]Mandel, J., Balancing domain decomposition, Comm. Numer. Methods Engrg., 9 (1993), pp. 233241.
[29]Mandel, J. and Brezina, M., Balancing domain decomposition for problems with large jumps in coefficients, Math. Comp., 65 (1996), pp. 13871401.
[30]Mandel, J., McCormick, S.F., and Ruge, J.W., An algebraic theory for multigrid methods for variational problems, SIAM J. Numer. Anal., 25 (1988), pp. 91110.
[31]McCormick, S., An algebraic interpretation of multigrid methods, SIAM J. Numer. Anal., 19 (1982), pp. 548560.
[32]McCormick, S., Multigrid methods for variational problems: Further results, SIAM J. Numer. Anal., 21 (1984), pp. 255263.
[33]McCormick, S., Multigrid methods for variational problems: general theory for the V-cycle, SIAM J. Numer. Anal., 22 (1985), pp. 634643.
[34]Nabben, R. and Vuik, C., A comparison of deflation and coarse grid correction applied to porous media flow, SIAM J. Numer. Anal., 42 (2004), pp. 16311647.
[35]Nabben, R. and Vuik, C., A comparison of deflation and the balancing preconditioner, SIAM J. Sci. Comput., 27 (2006), pp. 17421759.
[36]Nabben, R. and Vuik, C., A comparison of abstract versions of deflation, balancing and additive coarse grid correction preconditioners, Numer. Linear Algebra Appl., 15 (2008), pp. 355372.
[37]Napov, A. and Notay, Y., Comparison of bounds for V-cycle multigrid, Appl. Numer. Math., 60 (2010), pp. 176192.
[38]Napov, A. and Notay, Y., Algebraic analysis of aggregation-based multigrid, Numer. Linear Algebra Appl., 18 (2011), pp. 539564.
[39]Napov, A. and Notay, Y., An algebraic multigrid method with guaranteed convergence rate, SIAM J. Sci. Comput., 34 (2012), pp. A1079–A1109.
[40]Notay, Y., Algebraic multigrid and algebraic multilevel methods: A theoretical comparison, Numer. Linear Algebra Appl., 12 (2005), pp. 419451.
[41]Notay, Y., Convergence analysis of perturbed two-grid and multigrid methods, SIAM J. Numer. Anal., 45 (2007), pp. 10351044.
[42]Notay, Y., Algebraic analysis of two-grid methods: The nonsymmetric case, Numer. Linear Algebra Appl., 17 (2010), pp. 7396.
[43]Notay, Y., Aggregation-based algebraic multigrid for convection-diffusion equations, SIAM J. Sci. Comput., 34 (2012), pp. A2288–A2316.
[44]Notay, Y. and Napov, A., Further comparison of additive and multiplicative coarse grid correction, Appl. Numer. Math., 65 (2013), pp. 5362.
[45]Notay, Y. and Vassilevski, P.S., Recursive Krylov-based multigrid cycles, Numer. Linear Algebra Appl., 15 (2008), pp. 473487.
[46]Oswald, P., Multilevel Finite Element Approximation: Theory and Applications, Teubner Skripte zur Numerik, Teubner, Stuttgart, 1994.
[47]Quarteroni, A. and Valli, A., Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, Oxford, 1999.
[48]Ruge, J.W. and Stüben, K., Algebraic multigrid (AMG), in Multigrid Methods, McCormick, S.F., ed., vol. 3 of Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, 1987, pp. 73130.
[49]Sala, M. and Tuminaro, R.S., A new Petrov-Galerkin smoothed aggregation preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 31 (2008), pp. 143166.
[50]Smith, B., Bjørstad, P., and Gropp, W., Domain Decomposition, Cambridge University Press, Cambridge, 1996.
[51]Stüben, K., An introduction to algebraic multigrid, in Trottenberg, et al. [55], 2001, pp. 413532. Appendix A.
[52]Tang, J., Maclachlan, S., Nabben, R., and Vuik, C., A comparison of two-level preconditioners based on multigrid and deflation, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 17151739.
[53]Tang, J., Nabben, R., Vuik, C., and Erlangga, Y., Comparison of two-level preconditioners derived fromdeflation, domain decomposition and multigrid methods, J. Sci. Comput., 39 (2009), pp. 340370.
[54]Toselli, A. and Widlund, W., Domain Decomposition, Springer Ser. Comput. Math. 34, Springer-Verlag, Berlin, 2005.
[55]Trottenberg, U., Oosterlee, C.W., and Schüller, A., Multigrid, Academic Press, London, 2001.
[56]Vaněk, P., Brezina, M., and Mandel, J., Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math., 88 (2001), pp. 559579.
[57]Vassilevski, P.S., Multilevel Block Factorization Preconditioners, Springer, New York, 2008.
[58]Vuik, C., Segal, A., El YAAKOUBI, L., and Dufour, E., A comparison of various deflation vectors applied to elliptic problems with discontinuous coefficients, Appl. Numer. Math., 41 (2002), pp. 219233.
[59]Vuik, C., Segal, A., and Meijerink, J.A., An efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients, J. Comput. Physics, 152 (1999), pp. 385403.
[60]Vuik, C., Segal, A., Meijerink, J.A., and Wijma, G.T., The construction of projection vectors for a deflated ICCG method applied to problems with extreme contrasts in the coefficients, J. Comput. Physics, 172 (2001), pp. 426450.
[61]Wienands, R. and Oosterlee, C.W., On three-grid Fourier analysis for multigrid, SIAM J. Sci. Comput., 23 (2001), pp. 651671.
[62]Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Review, 34 (1992) , pp. 581613.
[63]Yserentant, H., Old and new convergence proofs for multigrid methods, Acta Numer., 2 (1993) , pp. 285326.


Algebraic Theory of Two-Grid Methods

  • Yvan Notay (a1)


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