Optimized wavelet preconditioning

  • Angela KunothEmail author
Conference paper


The numerical solution of linear stationary variational problems involving elliptic partial differential operators usually requires iterative solvers on account of their problem size. Our guiding principle is to devise theoretically and practically efficient iterative solution schemes which are optimal in the number of arithmetic operations, i.e., of linear complexity in the total number of unknowns. For these algorithms, asymptotically optimal preconditioners are indispensable. This article collects the main ingredients for multilevel preconditioners based on wavelets for certain systems of elliptic PDEs with smooth solutions. Specifically, we consider problems from optimal control with distributed or Dirichlet boundary control constrained by elliptic PDEs. Moreover, the wavelet characterization of function space norms will also be used in modelling the control functional, thereby extending the range of applicability over conventional methods. The wavelet preconditioners are optimized for these PDE systems to exhibit small absolute condition numbers and consequently entail absolute low iteration numbers, as numerical experiments show.


Conjugate Gradient Saddle Point Problem Biorthogonal Wavelet Essential Boundary Condition Cancellation Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ba]
    I. Babuška, The finite element method with Lagrange multipliers, Numer. Math. 20 (1973), 179–192. zbMATHCrossRefGoogle Scholar
  2. [B]
    D. Braess, Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 2nd ed., Cambridge University Press, Cambridge, 2001. zbMATHGoogle Scholar
  3. [BH]
    D. Braess, W. Hackbusch, A new convergence proof for the multigrid method including the V-cycle, SIAM J. Numer. Anal. 20 (1983), 967–975. zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BPX]
    J.H. Bramble, J.E. Pasciak, J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), 1–22. zbMATHCrossRefMathSciNetGoogle Scholar
  5. [BF]
    F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, 1991. Google Scholar
  6. [Bu1]
    C. Burstedde, Fast Optimised Wavelet Methods for Control Problems Constrained by Elliptic PDEs, PhD Dissertation, Mathematisch-Naturwissenschaftliche Fakultät, Universität Bonn, Germany, 2005. Google Scholar
  7. [Bu2]
    C. Burstedde, On the numerical evaluation of fractional Sobolev norms, Comm. Pure Appl. Anal. 6(3) (2007), 587–605. zbMATHCrossRefMathSciNetGoogle Scholar
  8. [BK]
    C. Burstedde, A. Kunoth, Fast iterative solution of elliptic control problems in wavelet discretizations, J. Comp. Appl. Math. 196(1) (2006), 299–319. zbMATHCrossRefMathSciNetGoogle Scholar
  9. [CTU]
    C. Canuto, A. Tabacco, K. Urban, The wavelet element method, part I: Construction and analysis, Appl. Comput. Harm. Anal. 6 (1999), 1–52. zbMATHCrossRefMathSciNetGoogle Scholar
  10. [CDP]
    J.M. Carnicer, W. Dahmen, J.M. Peña, Local decomposition of refinable spaces, Appl. Comp. Harm. Anal. 3 (1996), 127–153. zbMATHCrossRefGoogle Scholar
  11. [CF]
    Z. Ciesielski, T. Figiel, Spline bases in classical function spaces on compact C manifolds: Part I and II, Studia Mathematica (1983), 1–58 and 95–136. Google Scholar
  12. [Co]
    A. Cohen, Numerical Analysis of Wavelet Methods, Studies in Mathematics and its Applications 32, Elsevier, 2003. Google Scholar
  13. [CDF]
    A. Cohen, I. Daubechies, J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485–560. zbMATHCrossRefMathSciNetGoogle Scholar
  14. [DDU]
    S. Dahlke, W. Dahmen, K. Urban, Adaptive wavelet methods for saddle point problems — Optimal convergence rates, SIAM J. Numer. Anal. 40 (2002), 1230–1262. zbMATHCrossRefMathSciNetGoogle Scholar
  15. [D1]
    W. Dahmen, Stability of multiscale transformations, J. Four. Anal. Appl. 2 (1996), 341–361. zbMATHMathSciNetGoogle Scholar
  16. [D2]
    W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numerica (1997), 55–228. Google Scholar
  17. [D3]
    W. Dahmen, Wavelet methods for PDEs – Some recent developments, J. Comput. Appl. Math. 128 (2001), 133–185. zbMATHCrossRefMathSciNetGoogle Scholar
  18. [D4]
    W. Dahmen, Multiscale and wavelet methods for operator equations, in: Multiscale Problems and Methods in Numerical Simulation, C. Canuto (ed.), C.I.M.E. Lecture Notes in Mathematics 1825, Springer Heidelberg (2003), 31–96. Google Scholar
  19. [DK1]
    W. Dahmen, A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992), 315–344. zbMATHCrossRefMathSciNetGoogle Scholar
  20. [DK2]
    W. Dahmen, A. Kunoth, Appending boundary conditions by Lagrange multipliers: Analysis of the LBB condition, Numer. Math. 88 (2001), 9–42. zbMATHCrossRefMathSciNetGoogle Scholar
  21. [DK3]
    W. Dahmen, A. Kunoth, Adaptive wavelet methods for linear–quadratic elliptic control problems: Convergence Rates, SIAM J. Contr. Optim. 43(5) (2005), 1640–1675. zbMATHCrossRefMathSciNetGoogle Scholar
  22. [DKS]
    W. Dahmen, A. Kunoth, R. Schneider, Wavelet least squares methods for boundary value problems, SIAM J. Numer. Anal. 39 (2002), 1985–2013. zbMATHCrossRefMathSciNetGoogle Scholar
  23. [DKU]
    W. Dahmen, A. Kunoth, K. Urban, Biorthogonal spline wavelets on the interval – Stability and moment conditions, Appl. Comput. Harm. Anal. 6 (1999), 132–196. zbMATHCrossRefMathSciNetGoogle Scholar
  24. [DS1]
    W. Dahmen, R. Schneider, Wavelets with complementary boundary conditions — Function spaces on the cube, Results in Mathematics 34 (1998), 255–293. zbMATHMathSciNetGoogle Scholar
  25. [DS2]
    W. Dahmen, R. Schneider, Composite wavelet bases for operator equations, Math. Comp. 68 (1999), 1533–1567. zbMATHCrossRefMathSciNetGoogle Scholar
  26. [DS3]
    W. Dahmen, R. Schneider, Wavelets on manifolds I: Construction and domain decomposition, SIAM J. Math. Anal. 31 (1999), 184–230. zbMATHCrossRefMathSciNetGoogle Scholar
  27. [DSt]
    W. Dahmen, R. Stevenson, Element–by–element construction of wavelets satisfying stability and moment conditions, SIAM J. Numer. Anal. 37 (1999), 319–325. zbMATHCrossRefMathSciNetGoogle Scholar
  28. [Dau]
    I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909–996. zbMATHCrossRefMathSciNetGoogle Scholar
  29. [Gr]
    P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, 1985. Google Scholar
  30. [J]
    S. Jaffard, Wavelet methods for fast resolution of elliptic problems, Siam J. Numer. Anal. 29 (1992), 965–986. zbMATHCrossRefMathSciNetGoogle Scholar
  31. [KK]
    P. Kantartzis, A. Kunoth, A wavelet approach for a problem in electrical impedance tomography formulated by means of a domain embedding method, Manuscript, in preparation. Google Scholar
  32. [Kr]
    J. Krumsdorf, Finite Element Wavelets for the Numerical Solution of Elliptic Partial Differential Equations on Polygonal Domains, Diploma Thesis, Universität Bonn, 2004. Google Scholar
  33. [K1]
    A. Kunoth, Wavelet Methods — Elliptic Boundary Value Problems and Control Problems, Advances in Numerical Mathematics, Teubner, 2001. Google Scholar
  34. [K2]
    A. Kunoth, Wavelet techniques for the fictitious domain—Lagrange multiplier approach, Numer. Algor. 27 (2001), 291–316. zbMATHCrossRefMathSciNetGoogle Scholar
  35. [K3]
    A. Kunoth, Fast iterative solution of saddle point problems in optimal control based on wavelets, Comput. Optim. Appl. 22 (2002), 225–259. zbMATHCrossRefMathSciNetGoogle Scholar
  36. [K4]
    A. Kunoth, Adaptive wavelet schemes for an elliptic control problem with Dirichlet boundary control, Numer. Algor. 39(1-3) (2005), 199–220. zbMATHCrossRefMathSciNetGoogle Scholar
  37. [KS]
    A. Kunoth, J. Sahner, Wavelets on manifolds: An optimized construction, Math. Comp. 75 (2006), 1319–1349. zbMATHCrossRefMathSciNetGoogle Scholar
  38. [Li]
    J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. zbMATHGoogle Scholar
  39. [MB]
    J. Maes, A. Bultheel, A hierarchical basis preconditioner for the biharmonic equation on the sphere, IMA J. Numer. Anal. 26(3) (2006), 563–583. zbMATHCrossRefMathSciNetGoogle Scholar
  40. [MKB]
    J. Maes, A. Kunoth, A. Bultheel, BPX-type preconditioners for 2nd and 4th order elliptic problems on the sphere, SIAM J. Numer. Anal. 45(1) (2007), 206–222. zbMATHCrossRefMathSciNetGoogle Scholar
  41. [O]
    P. Oswald, On discrete norm estimates related to multilevel preconditioners in the finite element method, in: Constructive Theory of Functions, K.G. Ivanov, P. Petrushev, B. Sendov, (eds.), Proc. Int. Conf. Varna 1991, Bulg. Acad. Sci., Sofia (1992), 203–214. Google Scholar
  42. [Pa]
    R. Pabel, Wavelet Methods for PDE Constrained Elliptic Control Problems with Dirichlet Boundary Control, Diploma Thesis, Universität Bonn, 2006. doi:  10.2370/236_232
  43. [Stv]
    R. Stevenson, Locally supported, piecewise polynomial biorthogonal wavelets on non-uniform meshes, Constr. Approx. 19 (2003), 477–508. zbMATHCrossRefMathSciNetGoogle Scholar
  44. [U]
    K. Urban, Wavelet Methods for Elliptic Partial Differential Equations, Oxford University Press, 2009. Google Scholar
  45. [X1]
    J. Xu, Theory of multilevel methods, Report AM 48, Department of Mathematics, Pennsylvania State University, 1989. Google Scholar
  46. [X2]
    J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34(4) (1992), 581–613. zbMATHCrossRefMathSciNetGoogle Scholar
  47. [Y]
    H. Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), 379–412. zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany

Personalised recommendations