Numerische Mathematik

, Volume 60, Issue 1, pp 341–373 | Cite as

Accelerated Landweber iterations for the solution of ill-posed equations

  • Martin Hanke
Article

Summary

In this paper, the potentials of so-calledlinear semiiterative methods are considered for the approximate solution of linear ill-posed problems and ill conditioned matrix equations. Several efficient two-step methods are presented, most of which have been introduced earlier in the literature. Stipulating certain conditions concerning the smoothness of the solution, a notion of optimal speed of convergence may be formulated. Various direct and converse results are derived to illustrate the properties of this concept.

If the problem's right hand side data are contaminated by noise, semiiterative methods may be used asregularization methods. Assuming optimal rate of convergence of the iteration for the unperturbed problem, the regularized approximations will be of order optimal accuracy.

To derive these results, specific properties of polynomials are used in connection with the basic theory of solving ill-posed problems. Rather recent results onfast decreasing polynomials are applied to answer an open question of Brakhage.

Numerical examples are given including a comparison to the method of conjugate gradients.

Mathematics Subject Classification (1991)

65J10 65F10 

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References

  1. 1.
    Achieser, N.I. (1967): Vorlesungen über Approximationstheorie. Akademie-Verlag, BerlinGoogle Scholar
  2. 2.
    Bakushinskii, A.B. (1967): A general method of constructing regularizing algorithms for a linear ill-posed equation in Hilbert space. USSR Comput. Math. Phys.7, 3, 279–287Google Scholar
  3. 3.
    Björck, Å., Eldén, L. (1979): Methods in numerical algebra for ill-posed problems. Tech. Report LiTH-MAT-R-33-1979, Linköping University, LinköpingGoogle Scholar
  4. 4.
    Brakhage, H. (1987) On ill-posed problems and the method of conjugate gradients. In: H.W. Engl, C. W. Groetsch, eds., Inverse and Ill-Posed Problems. Academic Press, Boston, New York, London, pp. 165–175Google Scholar
  5. 5.
    Ditzian, Z., Totik, V. (1987): Moduli of Smoothness. Springer, New York Berlin Heidelberg.Google Scholar
  6. 6.
    Eicke, B., Louis, A.K., Plato, R. (1990): The instability of some gradient methods for ill-posed problems, Numer. Math.58, 129–134Google Scholar
  7. 7.
    Eiermann, M., Varga, R.S., Niethammer, W. (1987): Iterationsverfahren für nichtsymmetrische Gleichungssysteme und Approximationsmethoden im Komplexen. Jahresber. Deutsch. Math.-Verein.89, 1–32Google Scholar
  8. 8.
    Faddeev, D.K., Faddeeva, V.N. (1963): Computational Methods of Linear Algebra. Freeman, San Francisco LondonGoogle Scholar
  9. 9.
    Fridman, V.M. (1956): Method of successive approximations for a Fredholm integral equation of the 1st kind. Uspekhi Mat. Nauk11, 1, 233–234 [in Russian]Google Scholar
  10. 10.
    Gavurin, M.K., Ryabov, V.M. (1973): Application of Chebyshev polynomials to the regularization of ill-posed and ill-conditioned equations in Hilbert space. USSR Comput. Math. Math. Phys.13, 6, 283–287Google Scholar
  11. 11.
    Gradshteyn, I.S., Ryzhik, I.M. (1965): Table of Integrals, Series, and Products. Academic Press, New York San Francisco LondonGoogle Scholar
  12. 12.
    Groetsch, C.W. (1975): On existence criteria and approximation procedures for integral equations of the first kind. Math. Comput.29, 1105–1108Google Scholar
  13. 13.
    Groetsch, C.W. (1977): Generalized Inverses of Linear Operators. Dekker, New York BaselGoogle Scholar
  14. 14.
    Groetsch, C.W. (1984): The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston London MelbourneGoogle Scholar
  15. 15.
    Hanke, M., Niethammer, W. (1990): On the acceleration of Kaczmarz's method for inconsistent linear systems. Linear Algebra Appl.130, 83–98Google Scholar
  16. 16.
    Hansen, P.C. (1990): The discrete Picard condition for discrete ill-posed problems. BIT30, 658–672Google Scholar
  17. 17.
    Ivanov, K.G., Totik, V. (1990): Fast decreasing polynomials, Constr. Approx.6, 1–20Google Scholar
  18. 18.
    Kress, R. (1989): Linear Integral Equations. Springer, Berlin Heidelberg New YorkGoogle Scholar
  19. 19.
    Landweber, L. (1951): An iteration, formula for Fredholm integral equations of the first kind. Amer. J. Math.73, 615–624Google Scholar
  20. 20.
    Lorentz, G.G. (1966): Approximation of Functions. Holt, Rinehart and Winston, New York Chicago San FranciscoGoogle Scholar
  21. 21.
    Louis, A. K. (1987): Convergence of the conjugate gradient method for compact operators. In: H. W. Engl, C. W. Groetsch, eds., Inverse and Ill-Posed Problems. Academic Press, Boston New York London, pp. 177–183Google Scholar
  22. 22.
    Louis, A.K. (1989). Inverse und schlecht gestellte Probleme. Teubner, StuttgartGoogle Scholar
  23. 23.
    Lubinsky, D.S., Saff, E.B. (1989): Szegö asymptotics for non-Szegö weights on [−1,1]. In: C.K. Chui, L.L. Schumaker, J.D. Wards, eds., Approximation Theory VI, Vol. 2. Academic Press, Boston New York London, pp. 409–412Google Scholar
  24. 24.
    Morozov, V.A. (1966): On the solution of functional equations by the method of regularization. Soviet Math. Dokl.7, 414–417Google Scholar
  25. 25.
    Natterer, F. (1986): The Mathematics of Computerized Tomography. Wiley, Chichester New York BrisbaneGoogle Scholar
  26. 26.
    Nemirovskii, A.S. (1986): The regularizing properties of the adjoint gradient method in ill-posed problems. USSR Comput. Math. Math. Phys.,26, 2, 7–16Google Scholar
  27. 27.
    Nemirovskii, A.S., Polyak, B.T. (1984): Iterative methods for solving linear ill-posed problems under precise information, I. Engrg. Cybernetics22, 3, 1–11Google Scholar
  28. 28.
    Nemirovskii, A.S., Polyak, B.T. (1984): Iterative methods for solving linear ill-posed problems under precise information. II, Engrg. Cybernetics22, 4, 50–56Google Scholar
  29. 29.
    Nevai, P.G. (1979): Orthogonal Polynomials. Mem. Amer. Math. Soc., Vol. 213. Amer. Math. Soc., Providence, Rhode IslandGoogle Scholar
  30. 30.
    Nevai, P.G., Totik, V. (1986): Weighted polynomial inequalities, Constr. Approx.2, 113–127Google Scholar
  31. 31.
    Plato, R. (1990): Optimal algorithms for linear ill-posed problems yield regularization methods. Numer. Funct. Anal. Optim.11, 111–118Google Scholar
  32. 32.
    Potapov, M.K. (1983): Approximation by algebraic polynomials in an integral metric with Jacobi weight. Moscow Univ. Math. Bull.38, 4, 48–57Google Scholar
  33. 33.
    Richardson, L.F. (1910): The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philos. Trans. Roy. Soc. London Ser. A,210, 307–357Google Scholar
  34. 34.
    Schock, E. (1985): Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence. In: G. Hämmerlin, K.-H. Hoffmann, eds., Constructive Methods for the Practical Treatment of Integral Equations. Birkhäuser, Basel Boston Stuttgart, pp. 234–243Google Scholar
  35. 35.
    Schock, E. (1987): Semi-iterative methods for the approximate solution of ill-posed problems. Numer. Math.50, 263–271Google Scholar
  36. 36.
    Schock, E. (1988): Pointwise rational approximation and iterative methods for ill-posed problems. Numer. Math.54, 91–103Google Scholar
  37. 37.
    Stiefel, E. (1955): Relaxationsmethoden bester Strategie zur Lösung linearer Gleichungssysteme. Comment. Math. Helv.29, 157–179Google Scholar
  38. 38.
    Strand, O.N. (1974): Theory and methods related to the singular-function expansion and Landweber's iteration for integral equations of the first kind. SIAM J. Numer. Anal.11, 798–825Google Scholar
  39. 39.
    Szegö, G. (1967): Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ., Vol. 23. Amer. Math. Soc., Providence, Rhode IslandGoogle Scholar
  40. 40.
    Tikhonov, A.N., Arsenin, V.Y. (1977): Solutions of Ill-Posed Problems. Wiley, New York Toronto LondonGoogle Scholar
  41. 41.
    Vainikko, G.M. (1980): Error estimates of the successive approximation method for ill-posed problems. Automat. Remote Control,40, 356–363Google Scholar
  42. 42.
    Vainikko, G.M. (1982): The discrepancy principle for a class of regularization methods. USSR Comput. Math. Math. Phys.22, 3, 1–19Google Scholar
  43. 43.
    Vainikko, G.M. (1983): The critical level of discrepancy, in regularization methods. USSR Comput. Math. Math. Phys.22, 6, 1–9Google Scholar
  44. 44.
    Vainikko, G.M., Veretennikov, A.Y. (1986): Iteration Procedures in Ill-Posed Problems. Nauka, Moscow [in Russian]Google Scholar
  45. 45.
    Van Der Sluis, A., Van Der Vorst, H.A. (1990): SIRT- and CG-type methods for the iterative solution of sparse linear least squares problems. Linear Algebra Appl.130, 257–303Google Scholar
  46. 46.
    Varah, J.M. (1979): A practical examination of some numerical methods for linear discrete ill-posed problems. SIAM Rev.21, 100–111Google Scholar
  47. 47.
    Varga, R.S. (1962): Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Martin Hanke
    • 1
  1. 1.Institut für Praktische MathematikUniversität KarlsruheKarlsruheFederal Republic of Germany

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