On the Optimality of Shifted Laplacian in a Class of Polynomial Preconditioners for the Helmholtz Equation

  • Siegfried CoolsEmail author
  • Wim Vanroose
Part of the Geosystems Mathematics book series (GSMA)


This paper introduces and explores a class of polynomial preconditioners for the Helmholtz equation, denoted as expansion preconditioners EX(m), that form a direct generalization to the classical complex shifted Laplace (CSL) preconditioner. The construction of the EX(m) preconditioner is based on a truncated Taylor series expansion of the original Helmholtz operator inverse. The expansion preconditioner is shown to significantly improve Krylov solver convergence rates for growing values of the number of series terms m. However, the addition of multiple terms in the expansion also increases the computational cost of applying the preconditioner. A thorough cost-benefit analysis of the addition of extra terms in the EX(m) preconditioner proves that the CSL or EX(1) preconditioner is the most efficient member of the expansion preconditioner class for general practical and solver problem settings. Additionally, possible extensions to the expansion preconditioner class that further increase preconditioner efficiency are suggested, and numerical experiments in 1D and 2D are presented to validate the theoretical results.


Helmholtz equation Krylov subspace methods Multigrid methods Preconditioning Shifted Laplacian 



This research in this chapter was funded by the Research Council of the University of Antwerp under BOF grant number 41-FA070300-FFB5342. The scientific responsibility rests with its authors. The authors would like to cordially thank Dr. Bram Reps for fruitful discussions on the subject.


  1. 1.
    J. Aguilar and J.M. Combes. A class of analytic perturbations for one-body Schrödinger Hamiltonians. Communications in Mathematical Physics, 22(4):269–279, 1971.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G.B. Arfken and H.J. Weber. Mathematical methods for physicists: A comprehensive guide. Academic press, 2011.zbMATHGoogle Scholar
  3. 3.
    A. Bayliss, C.I. Goldstein, and E. Turkel. An iterative method for the Helmholtz equation. Journal of Computational Physics, 49(3):443–457, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A. Bayliss, C.I. Goldstein, and E. Turkel. On accuracy conditions for the numerical computation of waves. Journal of Computational Physics, 59(3):396–404, 1985.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J.P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. Journal of computational physics, 114(2):185–200, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R. Bhatia. Matrix analysis. Springer, 1997.CrossRefzbMATHGoogle Scholar
  7. 7.
    A. Brandt. Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation, 31(138):333–390, 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Brandt and S. Ta’asan. Multigrid method for nearly singular and slightly indefinite problems. Multigrid Methods II, Lecture Notes in Math., 1228:99–121, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    W.L. Briggs, V.E. Henson, and S.F. McCormick. A Multigrid Tutorial. Society for Industrial Mathematics, Philadelphia, 2000.CrossRefzbMATHGoogle Scholar
  10. 10.
    H. Calandra, S. Gratton, R. Lago, X. Pinel, and X. Vasseur. Two-level preconditioned Krylov subspace methods for the solution of three-dimensional heterogeneous Helmholtz problems in seismics. Numerical Analysis and Applications, 5(2):175–181, 2012.CrossRefzbMATHGoogle Scholar
  11. 11.
    W.C. Chew and W.H. Weedon. A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microwave and optical technology letters, 7(13):599–604, 1994.CrossRefGoogle Scholar
  12. 12.
    S. Cools and W. Vanroose. Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems. Numerical Linear Algebra with Applications, 20(4):575–597, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    H.C. Elman, O.G. Ernst, and D.P. O’Leary. A multigrid method enhanced by Krylov subspace iteration for discrete Helmholtz equations. SIAM Journal on scientific computing, 23(4):1291–1315, 2002.CrossRefzbMATHGoogle Scholar
  14. 14.
    Y.A. Erlangga and R. Nabben. On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian. Electronic Transactions on Numerical Analysis, 31(403–424):3, 2008.Google Scholar
  15. 15.
    Y.A. Erlangga, C.W. Oosterlee, and C. Vuik. On a class of preconditioners for solving the Helmholtz equation. Applied Numerical Mathematics, 50(3–4):409–425, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Y.A. Erlangga, C.W. Oosterlee, and C. Vuik. A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM Journal on Scientific Computing, 27(4):1471–1492, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Y.A. Erlangga, C. Vuik, and C.W. Oosterlee. Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Applied Numerical Mathematics, 56(5):648–666, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    O.G. Ernst and M.J. Gander. Why it is difficult to solve Helmholtz problems with classical iterative methods. Lecture Notes in Computational Science and Engineering, 83:325–363, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    R.P. Fedorenko. The speed of convergence of one iterative process. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 4(3):559–564, 1964.Google Scholar
  20. 20.
    M.J. Gander, I.G. Graham, and E.A. Spence. Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed? Numerische Mathematik, pages 1–48, 2015.zbMATHGoogle Scholar
  21. 21.
    C. Greif, T. Rees, and D.B. Szyld. Multi-preconditioned GMRES. Technical report - UBC CS TR-2011-12, 2011.Google Scholar
  22. 22.
    A. Laird and M. Giles. Preconditioned iterative solution of the 2D Helmholtz equation. Technical report, NA-02/12, Comp. Lab. Oxford University UK, 2002.Google Scholar
  23. 23.
    J. Liesen and Z. Strakos. Krylov subspace methods: principles and analysis. Oxford University Press, 2012.CrossRefzbMATHGoogle Scholar
  24. 24.
    Q. Liu, R.B. Morgan, and W. Wilcox. Polynomial preconditioned GMRES and GMRES-DR. SIAM Journal on Scientific Computing, 37(5):S407–S428, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    M.M.M. Made. Incomplete factorization-based preconditionings for solving the Helmholtz equation. Int. J. Numer. Meth. Eng., 50(5):1077–1101, 2001.CrossRefzbMATHGoogle Scholar
  26. 26.
    P.M. Morse and H. Feshbach. Methods of theoretical physics, International series in pure and applied physics. New York: McGraw-Hill, 1(1953):29, 1953.Google Scholar
  27. 27.
    D. Osei-Kuffuor and Y. Saad. Preconditioning Helmholtz linear systems. Applied Numerical Mathematics, 60(4):420–431, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    B. Reps and W. Vanroose. Analyzing the wave number dependency of the convergence rate of a multigrid preconditioned Krylov method for the Helmholtz equation with an absorbing layer. Numerical Linear Algebra with Applications, 19(2):232–252, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    B. Reps, W. Vanroose, and H. Zubair. On the indefinite Helmholtz equation: Complex stretched absorbing boundary layers, iterative analysis, and preconditioning. Journal of Computational Physics, 229(22):8384–8405, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Y. Saad. FGMRES: A flexible inner-outer preconditioned GMRES algorithm. SIAM Journal on Scientific Computing, 14(3):461–469, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Y. Saad and M.H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7:856–869, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    A.H. Sheikh, D. Lahaye, and C. Vuik. On the convergence of shifted Laplace preconditioner combined with multilevel deflation. Numerical Linear Algebra with Applications, 20(4):645–662, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    G.H. Shortley and R. Weller. The numerical solution of Laplace’s equation. Journal of Applied Physics, 9(5):334–348, 1938.CrossRefzbMATHGoogle Scholar
  34. 34.
    B. Simon. The definition of molecular resonance curves by the method of Exterior Complex Scaling. Physics Letters A, 71(2):211–214, 1979.CrossRefGoogle Scholar
  35. 35.
    V. Simoncini and D.B. Szyld. Flexible inner-outer Krylov subspace methods. SIAM Journal on Numerical Analysis, pages 2219–2239, 2003.Google Scholar
  36. 36.
    V. Simoncini and D.B. Szyld. Recent computational developments in Krylov subspace methods for linear systems. Numerical Linear Algebra with Applications, 14(1):1–59, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    K. Stüben and U. Trottenberg. Multigrid methods: Fundamental algorithms, model problem analysis and applications. Multigrid methods, Lecture Notes in Math., 960:1–176, 1982.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    L.N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, 2005.zbMATHGoogle Scholar
  39. 39.
    U. Trottenberg, C.W. Oosterlee, and A. Schüller. Multigrid. Academic Press, New York, 2001.zbMATHGoogle Scholar
  40. 40.
    H.A. Van der Vorst. BiCGSTAB: A fast and smoothly converging variant of BiCG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific Computing, 13(2):631–644, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    H.A. van Gijzen, Y. Erlangga, and C. Vuik. Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM Journal on Scientific Computing, 29(5):1942–1985, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    D. Werner. Funktionalanalysis. Springer, 2006.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of AntwerpAntwerpBelgium

Personalised recommendations