Convergence Rates Results for Iterative Methods for Solving Nonlinear Ill-Posed Problems

  • H. W. Engl
  • O. Scherzer


The growth of the area of inverse problems within applied mathematics in recent years has been driven both by the needs of applications and by advances in a rigorous convergence theory of regularization methods for the solution of nonlinear ill-posed problems. There are at least two widely used approaches for solving inverse problems in a stable way: Tikhonov regularization and iterative regularization techniques. In this paper we give an overview over the latter. Moreover, we put the analysis of iterative methods for the solution of ill-posed problems into perspective with the analysis of iterative methods for the solution of well-posed problems.


Inverse Problem Nonexpansive Mapping Conjugate Gradient Method Tikhonov Regularization Inverse Scattering Problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.L. Ageev, T.V. Bolotova, and V.V. Vasin. Solution to the inverse gravity problem for two interfaces in a medium. Izvestiya, Physics of the Solid Earth. 34:225–227, 1998.Google Scholar
  2. 2.
    P. Anselone. Collectively Compact Operatar Approximation Theory. Prentice-Hall, Englewood Cliffs, New Jersey, 1971.MATHGoogle Scholar
  3. 3.
    A.B. Bakushinskii. The problem of the convergence of the iteratively regularized Gauß-Newton method. Comput. Maths. Math. Phys., 32:1353–1359, 1992.MathSciNetGoogle Scholar
  4. 4.
    A.B. Bakushinskii. Universal linear approximations of solutions to nonlinear operator equations and their application. J. Inv. Ill-Posed Problems, 5:501–521, 1997.CrossRefMathSciNetGoogle Scholar
  5. 5.
    A.B. Bakushinskii and A.V. Goncharskii. Iterative Methods far the Solution of Incorrect Problems. Nauka, Moscow, 1989. in Russian.Google Scholar
  6. 6.
    A.B. Bakushinskii and A.V. Goncharskii. Ill-Posed Problems: Theory and Applications. Kluwer Academic Publishers, Dordrecht, Boston, London, 1994.CrossRefGoogle Scholar
  7. 7.
    R.E. Bank and D.J. Rose. Analysis of a multilevel iterative method for nonlinear finite element equations. Math. Comput., 39:453–465, 1982.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Bertero and P. Boccacci. Introductian to Inverse Problems in Imaging. IOP Publishing, London, 1998.MATHCrossRefGoogle Scholar
  9. 9.
    M. Bertero, T.A. Poggio, and V. Torre. Ill-posed problems in early vision. Proc. IEEE, 76:869–889, 1988.CrossRefGoogle Scholar
  10. 10.
    A. Binder, M. Hanke, and O. Scherzer. On the Landweber iteration for nonlinear ill-posed problems. J. Inverse Ill-Posed Probl., 4:381–389, 1996.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    B. Blaschke. Some Newton Type Methods for the Regularization of Nonlinear Ill Posed Problems. Universitätsverlag Rudolf Trauner, Linz, Austria, 1996. PhD-Thesis, Schriften der Johannes-Kepler-Universität.Google Scholar
  12. 12.
    B. Blaschke, A. Neubauer, and O. Scherzer. On convergence rates for the iteratively regularized Gauss-Newton method. IMA J. Numer. Anal., 17:421–436, 1997.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    H. Brakhage. On ill-posed problems and the method of conjugate gradients. In [36], pages 177–185, 1987.Google Scholar
  14. 14.
    F.E. Browder, editor. Nonlinear Functional Analysis, volume 18. Amer. Math. Soc., Providence, 1970. Proc. Symposia in Pure Math.Google Scholar
  15. 15.
    F.E. Browder and W.V. Petryshyn. Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl., 20:197–228, 1967.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Burger, V. Capasso, and H.W. Engl. Inverse problems related to crystallization of polymers. Inverse Probl., 15:155–173, 1999.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    G. Chavent. New size x curvature conditions for strict quasiconvexity of sets. SIAM J. Cantrol and Optimiz., 29:1348–1372, 1991.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    G. Chavent. Quasi-convex sets and size x curvature condition, applications to nonlincar invcrsion. Appl. Math. Optimiz., 24:129–169, 1991.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    G. Chavent and K. Kunisch. A geometric theory for l 1-stabilization of the inverse problem in a one-dimensional elliptic equation from an h 2-observation. Appl. Math. Optimiz., 27:231–260, 1993.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    G. Chavent and K. Kunisch. Regularization in state space. J. Numer. Anal., 27:535–564, 1993.MATHMathSciNetGoogle Scholar
  21. 2l.
    G. Chavent and K. Kunisch. State space regularization: geometric theory. Appl. Math. Optimiz., 37:243–267, 1998.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    G. Chavent and P. C. Sabatier, editors. Inverse Problems of Wave Propagation and Diffraction. Springer, Berlin. 1997.MATHGoogle Scholar
  23. 23.
    X. Chen, M.Z. Nashed, and L. Qi. Convergence of Newton’s method for singular smooth and nonsmooth equations using adaptive outer inverses. SIAM J. Optim., 7:445–462, 1997.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    X. Chen and L. Qi. A parameterized Newton method and a quasi-Newton method for nonsmooth equations. Comput. Optim. Appl., 3:157–179, 1994.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    X. Chen and T. Yamamoto. On the convergence of some quasi-Newton methods for nonlinear equations with nondifferentiable operators. Computing, 49:87–94, 1992.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    D. Colton, R. Ewing, and W. Rundell, editors. Inverse Problems in Partial Differential Equations. SIAM, Philadelphia, 1990.MATHGoogle Scholar
  27. 27.
    R.S. Dembo, S.C. Eisenstat, and T. Steihaug. Inexact Newton methods. SIAM J. Numer. Anal., 19:400–408, 1982.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    P. Deuflhard, H.W. Engl, and O. Scherzer. A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Probl., 14:1081–1106, 1998.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    P. Deuflhard and G. Heindl. Affine invariant convergence theorems for Newton’s method and extensions to related methods. SIAM J. Numer. Anal., 16:1–10, 1979.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    P. Deuflhard and A. Hohmann. Numerical Analysis. A First Course in Scientific Computation. De Gruyter, Berlin, 1991. Transl. from the German by F.A. Potra and F. Schulz.Google Scholar
  31. 31.
    P. Deuflhard and A. Hohmann. Numerische Mathematik I. Eine algorithmisch orientierte Einführung (Numerical Mathematics I. An algorithmically oriented Introduction). De Gruyter, Berlin, 1993. 2., überarb. Aufl. (German).MATHGoogle Scholar
  32. 32.
    P. Deuflhard and M. Weiser. Local inexact Newton multilevel FEM for nonlinear elliptic problems. In M-O. Bristeau, G. Etgen, W. Fitzgibbon, J.-L. Lions, J. Periaux, and M. Wheeler, editors, Computational Science for the 21st Century, Tours, France, pages 129–138. Wiley-Interscience-Europe, 1997.Google Scholar
  33. 33.
    W.G. Dotson Jr. On the Mann iterative process. Trans. Amer. Math. Soc., 149:65–73, 1970.MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    H.W. Engl. Weak convergence of asymptotically regular sequences for nonexpansive mappings and connectors with certain Chebyshef-centers. Nonlinear Analysis, Theory & Applications, 1:495–501, 1977.MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    H.W. Engl. Weak convergence of Mann iteration for nonexpansive mappings without convexity assumptions. Bollettino U.M.I, 14:471–475, 1977.MATHMathSciNetGoogle Scholar
  36. 36.
    H.W. Engl and C.W. Groetsch, editors. Inverse and Ill-Posed Problems. Academic Press, Boston, 1987.MATHGoogle Scholar
  37. 37.
    H.W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht, 1996.MATHCrossRefGoogle Scholar
  38. 38.
    H.W. Engl, K. Kunisch, and A. Neubauer. Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Probl., 5:523–540, 1989.MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    H.W. Engl, A.K. Louis, and W. Rundell, editors. Inverse Problems in Geophysical Applications. SIAM, Philadelphia, 1996.Google Scholar
  40. 40.
    H.W. Engl, A.K. Louis, and W. Rundell, editors. Inverse Problems in Medical Imaging and Nondestructive Testing. Springer, Wien, New York, 1996.Google Scholar
  41. 41.
    H.W. Engl and J. McLaughlin, editors. Inverse Problems and Optimal Design in Industry.. B.G. Teubner, Stuttgart, 1994.Google Scholar
  42. 42.
    H.W. Engl and W. Rundell, editors. Inverse Problems in Diffusion Processes. SIAM, Philadelphia, 1995.Google Scholar
  43. 43.
    S.F. Gilyazov. Iterative solution methods for inconsistent linear equations with non self-adjoint operators. Moscov University Computational Mathematics and Cybernetics, pages 8–13, 1977.Google Scholar
  44. 44.
    S.F. Gilyazov. Regularizing algorithms based on the conjugate-gradient method. U.S.S.R. Comput. Math. Math. Phys., 26:8–13, 1986.MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    G.H. Golub and D.P. O’Leary. Some history of the conjugate radient method and Lanczos algorithms: 1948–1976. SIAM Review, 31:50–102, 1989.MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    C.W. Groetsch. A note on segmenting Mann iterates. J. Math. Anal. Appl., 40:369–372, 1972.MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    C.W. Groetsch. A nonstationary iterative process for nonexpansive mappings. Proc. Am. Math. Soc., 43:155–158, 1974.MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    C.W. Groetsch. The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston, 1984.MATHGoogle Scholar
  49. 49.
    W. Hackbusch. Multi-Grid Methods and Applications. Springer-Verlag, Berlin, Heidelberg, New York, 1985.MATHGoogle Scholar
  50. 50.
    W. Hackbusch. Iterative Solution of Large Sparse Systems of Equations. Springer-Verlag, New York, 1994. Applied Mathematical Sciences 95.Google Scholar
  51. 51.
    W. Hackbusch. Integral Equations. Theory and Numerical Treatment. Birkhauser, Basel, 1995.Google Scholar
  52. 52.
    G. Hammerlein and K.-H. Hoffmann, editors. Constructive Methods for the Practical Treatment of Integral Equations. Birkhauser Verlag, Basel, Boston, Stuttgart, 1985. Proceedings of the Conference at the Mathematisches Forschungsinstitut Oberwolfach, June 24–30, 1984. International Series of Numerical Mathematics, Vol. 73.Google Scholar
  53. 53.
    M. Hanke. Conjugate Gradient Type Methods for Ill-Posed Problems. Longman Scientific & Technical, Harlow, 1995. Pitman Research Notes in Mathematics Series.Google Scholar
  54. 54.
    M. Hanke. A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Probl., 13:79–95, 1997.MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    M. Hanke. Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems. Numer. Fund. Anal. Optimiz., 18:971–993, 1997.MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    M. Hanke, F. Hettlich, and O. Scherzer. The Landweber iteration for an inverse scattering problem. In [135], pages 909–915, 1995.Google Scholar
  57. 57.
    M. Hanke, A. Neubauer, and O. Scherzer. A convergence analysis of Landweber iteration for nonlinear ill-posed problems. Numer. Math., 72:21–37, 1995.MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    M. Hanke and C. Vogel. Two-level preconditioners for regularized inverse prohlems 1: theory. Numer. Math., 1999. to appear.Google Scholar
  59. 59.
    M. Hanke and C. Vogel. Two-level preconditioners for regularized inverse problems II: implementation and numerical results, submitted.Google Scholar
  60. 60.
    W.M. Haussler. A Kantorovich-type convergence analysis for the Gauss-Newton-method. Numer. Math., 48:119–125, 1986.CrossRefMathSciNetGoogle Scholar
  61. 61.
    M. Heinkenschloss, C.T. Kelley, and H.T. Tran. Fast algorithms for nonsmooth compact fixed-point problems. SIAM J. Numer. Anal., 29:1769–1792, 1992.MATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    F. Hettlich. An iterative method for the inverse scattering problem from soundhard obstacles. Z. Angew. Math. Mech., 76:165–168, 1996.MATHGoogle Scholar
  63. 63.
    F. Hettlich, J. Morgan, and O. Scherzer. On the estimation of interfaces from boundary measurements. In [40], pages 163–178, 1996.Google Scholar
  64. 64.
    F. Hettlich and W. Rundell. Iterative methods for the reconstruction of an inverse potential problem. Inverse Probl, 12:251–266, 1996.MATHCrossRefMathSciNetGoogle Scholar
  65. 65.
    F. Hettlich and W. Rundell. Recovery of the support of a source term ill all elliptic differential equation. Inverse Probl, 13:959–976, 1997.MATHMathSciNetGoogle Scholar
  66. 66.
    F. Hettlich and W. Rundell. The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Probl, 14:67–82, 1998.MATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    B. Hofmann and O. Scherzer. Local ill-posedness and source conditions of operator equations in Hilbert spaces. Inverse Probl, 14:1189–1206, 1998.MATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    T. Holiage. Logarithmic convergence rates of the iteratively regularized Gaufj-Newton method for an inverse potential and an inverse scattering problem. Inverse Probl, 13:1279–1300, 1997.CrossRefGoogle Scholar
  69. 69.
    T. Hohage. Convergence rates of a regularized Newton method in sound-hard inverse scattering. SIAM Numer. Anal., 36:125–142, 1999.CrossRefMathSciNetGoogle Scholar
  70. 70.
    T. Hohage and C. Schormann. A Newton-type method for a transmission problem in inverse scattering. Inverse Probl, 14:1207–1228, 1998.MATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    A. Hohmann. Inexact Gauss Newton methods for parameter dependent nonlinear problems. Shaker, Aachen, 1994. Berichte aus der Mathematik.Google Scholar
  72. 72.
    S. Kabanikhin, R. Kowar, and O. Scherzer. On the Landweber iteration for the solution of a parameter identification problem in a hyperbolic partial differential equation of second order. J. Inv. Ill-Posed Problems, 6:403–430, 1998.MATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    B. Kaltenbacher. Some Newton-type methods for the regularization of nonlinear ill-posed problems. Inverse Probl, 13:729–753, 1997.MATHCrossRefMathSciNetGoogle Scholar
  74. 74.
    B. Kaltenbacher. A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems. Numer. Math., 79:501–528, 1998.MATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    W.J. Kammerer and M.Z. Nashed. Steepest descent for singular linear operators with nonclosed range. Applicable Analysis, 1:143–159, 1971.MATHCrossRefMathSciNetGoogle Scholar
  76. 76.
    W.J. Kammerer and M.Z. Nashed. On the convergence of the conjugate gradient method for singular linear operator equations. SIAM J. Numer. Anal, 9:165–181, 1972.MATHCrossRefMathSciNetGoogle Scholar
  77. 77.
    L.W. Kantorowitsch and G.P. Akilow. Funktionalanalysis in normierten Räumen. Akademie Verlag, Berlin, 1964.MATHGoogle Scholar
  78. 78.
    A. Kaplan and R. Tichatschke. Stable Methods for Ill-Posed Problems. Akademie Verlag, Berlin, 1994.MATHGoogle Scholar
  79. 79.
    J.T. King. A minimal error conjugate gradient method for ill-posed problems. Journal of Optimization Theory and Applications, 60:297–304, 1989.MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    K. Kumsch and G. Geymayer. Convergence rates for regularized nonlinear illposed problem. In [81], pages 81–92, 1991.Google Scholar
  81. 81.
    A. Kurzhanski and I. Lasiecka, editors. Modelling and Inverse Problems of Control for Distributed Parameter Systems. Springer, Berlin, 1991. Lecture Notes in Control and Information Sciences, Vol. 154.Google Scholar
  82. 82.
    A.K. Louis. Convergence of the conjugate gradient method for compact operators. In H.W. Engl and C.W. Groetsch, editors, [36], pages 177–185, 1987.Google Scholar
  83. 83.
    A.K. Louis. Inverse und Schlecht Gestellte Probleme. Teubner, Stuttgart, 1989.MATHGoogle Scholar
  84. 84.
    W.R. Mann. Mean value methods in iteration. Proc. Am. Math. Soc, 4:506–510, 1953.MATHCrossRefGoogle Scholar
  85. 85.
    St. Maruster. Quasi-nonexpansity and two classical methods for solving nonlinear equations. Proc. Amer. Math. Soc, 62:119–123, 1977.MATHMathSciNetGoogle Scholar
  86. 86.
    S.F. McCormick. An iterative procedure for the solution of constrained nonlinear equations with applications to optimization problems. Numer. Math., 23:371–385, 1975.MATHCrossRefMathSciNetGoogle Scholar
  87. 87.
    S.F. McCormick. The methods of Kacmarcz and row orthogonalization for solving linear equations and least squares problems in Hilbert space. Indiana University Mathematics Journal, 26:1137–1150, 1977.MATHCrossRefMathSciNetGoogle Scholar
  88. 88.
    K.H. Meyn. Solution of underdetermined nonlinear equations by stationary iteration methods. Numer. Math., 42:161–172, 1983.MATHCrossRefMathSciNetGoogle Scholar
  89. 89.
    V.A. Morozov. On the solution of functional equations by the method of regularization. Soviet Math. Dokl, 7:414–417, 1966.MATHMathSciNetGoogle Scholar
  90. 90.
    V.A. Morozov. Methods for Solving Incorrectly Posed Problems. Springer Verlag, New York, Berlin, Heidelberg, 1984.CrossRefGoogle Scholar
  91. 91.
    V.A. Morozov. Regularization Methods for Ill-Posed Problems. CRC Press, Boca Raton, 1993.MATHGoogle Scholar
  92. 92.
    M.Z. Nashed, editor. Generalized Inverses and Applications. Academic Press, New York, 1976.MATHGoogle Scholar
  93. 93.
    M.Z. Nashed and O. Scherzer. Least squares and bounded variation regularization with nondifferentiable functional. Num. Fund. Anal, and Optimiz., 19:873–901, 1998.MATHCrossRefMathSciNetGoogle Scholar
  94. 94.
    F. Natterer. Numerical solution of bilinear inverse problems. Universtät Münster, Germany, 1998. preprint.Google Scholar
  95. 95.
    A. Neubauer. Tikhonov regularization for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation. Inverse Probl., 5:541–557, 1989.MATHCrossRefMathSciNetGoogle Scholar
  96. 96.
    A. Neubauer and O. Scherzer. Finite-dimensional approximation of Tikhonov regularized solutions of non-linear ill-posed problems. Numer. Fund. Anal, and Optimiz., 11:85–99, 1990.MATHCrossRefMathSciNetGoogle Scholar
  97. 97.
    A. Neubauer and O. Scherzer. A convergence rate result for a steepest descent method and a minimal error method for the solution of nonlinear ill-posed problems. Z. Anal. Anwend., 14:369–377, 1995.MATHMathSciNetGoogle Scholar
  98. 98.
    Z. Opial. Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Amer. Math. Soc., 67:591–597, 1967.CrossRefMathSciNetGoogle Scholar
  99. 99.
    J.M. Ortega and W.C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.MATHGoogle Scholar
  100. 100.
    C. Outlaw and C.W. Groetsch. Averaging iteration in a Banach space. Bull. Am. Math. Soc., 75:430–432, 1969.MATHCrossRefMathSciNetGoogle Scholar
  101. 101.
    W.V. Petryshyn and T.E. Williamson. Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings. J. Math. Anal. Appl., 43:459–497, 1973.MATHCrossRefMathSciNetGoogle Scholar
  102. 102.
    L. Qi and X. Chen. A preconditioning proximal Newton method for nondifferentiable convex optimization. Math. Program., 76B:411–429, 1997.MathSciNetGoogle Scholar
  103. 103.
    J. Qi-Nian. On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems. 1998. Preprint, University of Nanjing, China.Google Scholar
  104. 104.
    R. Ramlau. A modified Landweber-method for inverse problems. Num. Fund. Anal. Opt., 20:79–98, 1999.MATHCrossRefMathSciNetGoogle Scholar
  105. 105.
    A. Rieder. A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization. Numer. Math., 75:501–522, 1997.MATHCrossRefMathSciNetGoogle Scholar
  106. 106.
    A. Rieder. On convergence rates of inexact Newton regularizations. Preprint, Universität Saarbrücken, 1998.Google Scholar
  107. 107.
    A. Rieder. On the regularization of nonlinear ill-posed problems via inexact Newton iterations. Inverse Probl, pages 309–327, 1999.Google Scholar
  108. 108.
    P.C. Sabatier (ed.). Some Topics in Inverse Problems. World Scientific, Singapore, 1988.Google Scholar
  109. 109.
    P.C. Sabatier (ed.). Inverse Methods in Action. Spinger, Berlin, Heidelberg, New York, 1990.MATHGoogle Scholar
  110. 110.
    O. Scherzer. A posteriori error estimates for nonlinear ill-posed problems. submitted.Google Scholar
  111. 111.
    O. Scherzer. Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl., 194:911–933, 1995.MATHCrossRefMathSciNetGoogle Scholar
  112. 112.
    O. Scherzer. A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems. Numer. Fund. Anal. Optimization, 17:197–214, 1996.MATHCrossRefMathSciNetGoogle Scholar
  113. 113.
    O. Scherzer. An iterative multi level algorithm for solving nonlinear ill-posed problems. Numer. Math., 80:579–600, 1998.MATHCrossRefMathSciNetGoogle Scholar
  114. 114.
    O. Scherzer. A modified Landweber iteration for solving parameter estimation problems. Appl. Math. Optimiz.. 38:45–68, 1998.MATHCrossRefMathSciNetGoogle Scholar
  115. 115.
    O. Scherzer. An note on Kacmarz’s method for the solution of ill posed problems. in preparation, 1999.Google Scholar
  116. 116.
    O. Scherzer, H.W. Engl, and K. Kunisch. Optimal a-posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J. Numer. Anal., 30:1796–1838. 1993.MATHCrossRefMathSciNetGoogle Scholar
  117. 117.
    O. Scherzer and M. Gullikson. An adaptive strategy for updating the damping parameters in an iteratively regularized Gauss-Newton method. JOTA, 100, 1999. to appear.Google Scholar
  118. 118.
    E. Schock. Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence. In [52], pages 234–243, 1985.Google Scholar
  119. 119.
    T.I. Seidman and C.R. Vogel. Well posedness and convergence of some regularisation methods for non-linear ill posed problems. Inverse Probl. 5:227–238, 1989.MATHCrossRefMathSciNetGoogle Scholar
  120. 120.
    V.P. Tanana. Methods for Solution of Nonlinear Operator Equations. VSP, Utrecht, 1997.MATHGoogle Scholar
  121. 121.
    U. Tautenhahn. On the asymptotical regularization of nonlinear ill-posed problems. Inverse Probl., 10:1405–1418, 1994.MATHCrossRefMathSciNetGoogle Scholar
  122. 122.
    A. N. Tikhonov and V. Y. Arsenin. Solutions of Ill-Posed Problems. John Wiley & Sons, Washington, D.C., 1977. Translation editor: Fritz John.Google Scholar
  123. 123.
    A.N. Tikhonov. Regularization of incorrectly posed problems. Soviet Math. Dokl., 4:1624–1627, 1963.MATHGoogle Scholar
  124. 124.
    A.N. Tikhonov. Solution of incorrectly formulated problems and the regularization methods. Soviet Math. Dokl., 4:1035–1038, 1963.Google Scholar
  125. 125.
    A.N. Tikhonov, A. Goncharsky, V. Stepanov, and A. Yagola. Numerical Methods for the Solution of Ill-Posed Problems. Kluwer, Dordrecht, 1995.MATHGoogle Scholar
  126. 126.
    A.N. Tikhonov, A.S. Leonov, A.I. Prilepko, I.A. Vasin, V.A. Vatutin, and A.G. Yagola, editors. Ill-Posed Problems in Natural Sciences. VSP, Utrecht, 1992.Google Scholar
  127. 127.
    G.M. Vainikko. Error estimates of the successive approximation method for ill-posed problems. Automat. Remote Control, 40:356–363, 1980.Google Scholar
  128. 128.
    V.V. Vasin. Iterative methods for the approximate solution of ill posed problems with a priori informations and their applications. In [36], pages 211–229, 1987.Google Scholar
  129. 129.
    V.V. Vasin. Ill-posed problems and iterative approximation of fixed points of pseudo-contractive mappings. In [126], pages 214–223, 1992.Google Scholar
  130. 130.
    V.V. Vasin. Monotone iterative processes for nonlinear operator equations and their applications to volterra equations. J. Inverse Ill-Posed Probl., 4:331–340, 1996.MATHCrossRefMathSciNetGoogle Scholar
  131. 131.
    V.V. Vasin. Monotonic iterative processes for operator equations in semiordered spaces. Dokl. Math. 54, 53:487–489, 1996. Translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 349, No. 1. 7–9.Google Scholar
  132. 132.
    V.V. Vasin. On the convergence of gradient-type methods for nonlinear equations. Doklady Mathematics, 57:173–175, 1998. Translated from Dokladv Akademii Nauk Vol. 359 (1998), pp. 7–9.Google Scholar
  133. 133.
    V.V. Vasin and A.L. Ageev. Ill-Posed Problems with A-Priori Information. VSP, Utrecht, 1995.MATHGoogle Scholar
  134. 134.
    V.V. Vasin, I.L. Prutkin, and L.Yu Timerkhanova. Retrieval of a three-dimensional relief of geological boundary from gravity data. Izvestiya, Physics of the Solid Earth, 32:901–905, 1996.Google Scholar
  135. 135.
    K.-W. Wang, B. Yang, J.Q. Sun, K. Seto, K. Nonami, H.-S. Tzou, S.S. Rao, G.R. Tomlinson, B. Yang, H.T. Banks, G.M.L. Gladwell, M. Link, G. Lallement, T.E. Alberts, C.-A. Tan, and Y.Y. Hung, editors. Proceedings of the 1995 Design Engineering Technical Conferences. The American Society of Mechanical Engineers, New York, 1995. Vibration Control, Analysis, and Identification, Vol. 3, Part C.Google Scholar
  136. 136.
    J. Weidmann. Lineare Operatoren in Hilberträumen. Teubner, Stuttgart, 1976.MATHGoogle Scholar

Copyright information

© Springer-Verlag/Wien 2000

Authors and Affiliations

  • H. W. Engl
    • 1
  • O. Scherzer
    • 1
  1. 1.Industrial Mathematics InstituteJohannes Kepler UniversitätLinzAustria

Personalised recommendations