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Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods

Journal of Scientific Computing volume 39, pages 340–370 (2009)Cite this article

Abstract

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.

References

  1. Bastian, P., Hackbusch, W., Wittum, G.: Additive and multiplicative multigrid: a comparison. Computing 60, 345–364 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  2. Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructuring. Math. Comput. 47, 103–134 (1986)

    MATH  Article  MathSciNet  Google Scholar 

  3. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977)

    MATH  Article  MathSciNet  Google Scholar 

  4. Carpentieri, B., Giraud, L., Gratton, S.: Additive and multiplicative two-level spectral preconditioning for general linear systems. SIAM J. Sci. Comput. 29, 1593–1612 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  5. Dryja, M.: An additive Schwarz algorithm for two- and three-dimensional finite element elliptic problems. In: Domain Decomposition Methods, pp. 168–172. SIAM, Philadelphia (1989)

    Google Scholar 

  6. Dryja, M., Widlund, O.B.: Towards a unified theory of domain decomposition algorithms for elliptic problems. In: Third International Symposium on DDM for PDEs, pp. 3–21. SIAM, Philadelphia (1990)

    Google Scholar 

  7. Dryja, M., Widlund, O.B.: Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Commun. Pure Appl. Math. 48, 121–155 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  8. Eiermann, M., Ernst, O.G., Schneider, O.: Analysis of acceleration strategies for restarted minimal residual methods. J. Comput. Appl. Math. 123, 261–292 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  9. Erlangga, Y.A., Nabben, R.: Multilevel projection-based nested Krylov iterations for boundary value problems. SIAM J. Sci. Comput. 30, 1572–1595 (2008)

    Article  MathSciNet  Google Scholar 

  10. Faber, V., Manteuffel, T.: Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal. 21, 352–362 (1984)

    MATH  Article  MathSciNet  Google Scholar 

  11. Frank, J., Vuik, C.: On the construction of deflation-based preconditioners. SIAM J. Sci. Comput. 23, 442–462 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  12. De Gersem, H., Hameyer, K.: A deflated iterative solver for magnetostatic finite element models with large differences in permeability. Eur. Phys. J. Appl. Phys. 13, 45–49 (2000)

    Article  Google Scholar 

  13. Giraud, L., Ruiz, D., Touhami, A.: A comparative study of iterative solvers exploiting spectral information for SPD systems. SIAM J. Sci. Comput. 27, 1760–1786 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  14. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  15. Graham, I.G., Scheichl, R.: Robust domain decomposition algorithms for multiscale PDEs. Numer. Methods Partial Differ. Equ. 23, 859–878 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  16. Hackbusch, W.: Multigrid Methods and Applications. Springer, Berlin (1985)

    Google Scholar 

  17. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)

    MATH  MathSciNet  Google Scholar 

  18. Kaasschieter, E.F.: Preconditioned conjugate gradients for solving singular systems. J. Comput. Appl. Math. 24, 265–275 (1988)

    MATH  Article  MathSciNet  Google Scholar 

  19. Kolotilina, L.Y.: Twofold deflation preconditioning of linear algebraic systems, I: theory. J. Math. Sci. 89, 1652–1689 (1998)

    Article  MathSciNet  Google Scholar 

  20. MacLachlan, S.P., Tang, J.M., Vuik, C.: Fast and robust solvers for pressure correction in bubbly flow problems. J. Comput. Phys. 227, 9742–9761 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  21. Mandel, J.: Balancing domain decomposition. Commun. Appl. Numer. Methods 9, 233–241 (1993)

    MATH  Article  MathSciNet  Google Scholar 

  22. Mandel, J.: Hybrid domain decomposition with unstructured subdomains. In: Proceedings of the Sixth International Symposium on DDM, Como, Italy, 1992. Contemp. Math., 157, pp. 103–112. AMS, Providence (1994)

    Google Scholar 

  23. Mandel, J., Brezina, M.: Balancing domain decomposition for problems with large jumps in coefficients. Math. Comput. 216, 1387–1401 (1996)

    MathSciNet  Google Scholar 

  24. Mansfield, L.: On the conjugate gradient solution of the Schur complement system obtained from domain decomposition. SIAM J. Numer. Anal. 27, 1612–1620 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  25. Mansfield, L.: Damped Jacobi preconditioning and coarse grid deflation for conjugate gradient iteration on parallel computers. SIAM J. Sci. Stat. Comput. 12, 1314–1323 (1991)

    MATH  Article  MathSciNet  Google Scholar 

  26. Meijerink, J.A., Van der Vorst, H.A.: An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comput. 31, 148–162 (1977)

    MATH  Article  Google Scholar 

  27. Morgan, R.B.: A restarted GMRES method augmented with eigenvectors. SIAM J. Matrix Anal. Appl. 16, 1154–1171 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  28. Nabben, R., Vuik, C.: A comparison of deflation and coarse grid correction applied to porous media flow. SIAM J. Numer. Anal. 42, 1631–1647 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  29. Nabben, R., Vuik, C.: A comparison of deflation and the balancing preconditioner. SIAM J. Sci. Comput. 27, 1742–1759 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  30. Nabben, R., Vuik, C.: A comparison of abstract versions of deflation, balancing and additive coarse grid correction preconditioners. Numer. Linear Algebra Appl. 15, 355–372 (2008)

    Article  MathSciNet  Google Scholar 

  31. Nicolaides, R.A.: Deflation of Conjugate Gradients with applications to boundary value problems. SIAM J. Matrix Anal. Appl. 24, 355–365 (1987)

    MATH  MathSciNet  Google Scholar 

  32. Padiy, A., Axelsson, O., Polman, B.: Generalized augmented matrix preconditioning approach and its application to iterative solution of ill-conditioned algebraic systems. SIAM J. Matrix Anal. Appl. 22, 793–818 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  33. Pavarino, L.F., Widlund, O.B.: Balancing Neumann-Neumann methods for incompressible Stokes equations. Commun. Pure Appl. Math. 55, 302–335 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  34. Saad, Y., Yeung, M., Erhel, J., Guyomarc’h, F.: A deflated version of the conjugate gradient algorithm. SIAM J. Sci. Comput. 21, 1909–1926 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  35. Simoncini, V., Szyld, D.B.: Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl. 14, 1–59 (2007)

    Article  MathSciNet  Google Scholar 

  36. Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  37. Tang, J.M., Nabben, R., Vuik, C., Erlangga, Y.A.: Theoretical and numerical comparison of various projection methods derived from deflation, domain decomposition and multigrid methods. Delft University of Technology, DIAM Report 07-04, ISSN 1389-6520 (2007)

  38. Tang, J.M., MacLachlan, S.P., Nabben, R., Vuik, C.: A comparison of two-level preconditioners based on multigrid and deflation. Submitted

  39. Tang, J.M., Vuik, C.: On deflation and singular symmetric positive semi-definite matrices. J. Comput. Appl. Math. 206, 603–614 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  40. Tang, J.M., Vuik, C.: Efficient deflation methods applied to 3-D bubbly flow problems. Electron. Trans. Numer. Anal. 26, 330–349 (2007)

    MathSciNet  Google Scholar 

  41. Tang, J.M., Vuik, C.: New variants of deflation techniques for bubbly flow problems. J. Numer. Anal. Ind. Appl. Math. 2, 227–249 (2007)

    MATH  MathSciNet  Google Scholar 

  42. Tang, J.M., Vuik, C.: Fast deflation methods with applications to two-phase flows. Int. J. Mult. Comput. Eng. 6, 13–24 (2008)

    Article  Google Scholar 

  43. Toselli, A., Widlund, O.B.: Domain Decomposition: Algorithms and Theory. Comput. Math., vol. 34. Springer, Berlin (2005)

    MATH  Google Scholar 

  44. Trottenberg, U., Oosterlee, C.W., Schüller, A.: Multigrid. Academic Press, London (2001)

    MATH  Google Scholar 

  45. Vuik, C., Segal, A., 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. Phys. 152, 385–403 (1999)

    MATH  Article  Google Scholar 

  46. Vuik, C., Nabben, R., Tang, J.M.: Deflation acceleration for domain decomposition preconditioners. In: Proceedings of the 8th European Multigrid Conference on Multigrid, Multilevel and Multiscale Methods, The Hague, The Netherlands, September 27–30, 2005

  47. Wesseling, P.: An Introduction to Multigrid Methods. Wiley, New York (1992). Corrected reprint: Edwards, Philadelphia (2004)

    MATH  Google Scholar 

  48. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, 581–613 (1992)

    MATH  Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics, Delft University of Technology, J.M. Burgerscentrum, Mekelweg 4, 2628 CD, Delft, The Netherlands

    J. M. Tang & C. Vuik

  2. Institut für Mathematik, Technische Universität Berlin, MA 3-3, Straße des 17. Juni 136, 10623, Berlin, Germany

    R. Nabben

  3. Department of Earth and Ocean Sciences, The University of British Columbia, 6339 Stores Road, Vancouver, British Columbia, V6T 1Z4, Canada

    Y. A. Erlangga

Authors
  1. J. M. Tang
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  2. R. Nabben
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  3. C. Vuik
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  4. Y. A. Erlangga
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Correspondence to C. Vuik.

Additional information

Part of this work has been done during the visit of the first, third and fourth author at Technische Universität Berlin. The research is partially funded by the Dutch BSIK/BRICKS project and the Deutsche Forschungsgemeinschaft (DFG), Project NA248/2-2.

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Tang, J.M., Nabben, R., Vuik, C. et al. Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods. J Sci Comput 39, 340–370 (2009). https://doi.org/10.1007/s10915-009-9272-6

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  • DOI: https://doi.org/10.1007/s10915-009-9272-6

Keywords

  • Deflation
  • Domain decomposition
  • Multigrid
  • Conjugate gradients
  • Two-grid schemes
  • Two-level preconditioning
  • SPD matrices
  • Two-level PCG methods