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Regularization Methods for Nonlinear Ill-Posed Problems with Applications to Phase Reconstruction

  • Barbara Blaschke-Kaltenbacher
  • Heinz W. Engl

Abstract

This conference has shown again that many inverse problems arise in medical imaging and nondestructive testing. Mathematically, such problems involve e.g. parameter identification from boundary measurements (see e.g. [36]), inverse scattering (see e.g. [12], [13], [14]), or phase reconstruction (see e.g. [40]). The mathematical formulation of inverse problems usually gives rise to ill-posed problems in Hadamard’s sense, where especially the lack of stability causes numerical difficulties. We take this as a motivation to survey some results about convergence and convergence rates for regularization methods for nonlinear illposed problems, which have to be used to deal with the instability issue. Again motivated by some talks at this conference, we illustrate our results by some phase reconstruction problems.

Keywords

Inverse Problem Convergence Rate Regularization Method Tikhonov Regularization Boundary Measurement 
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.

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References

  1. 1.
    R. Acar, C.R. Vogel, Analysis of total variation penalty methods. Inverse Problems 10 (1994), 1217–1229.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    A.B. Bakushinskii, The problem of the convergence of the iteratively regularized Gauss-Newton method, Comput.Math.Math.Phys. 32 (1992), 1353–1359.MathSciNetGoogle Scholar
  3. 3.
    A.B. Bakushinskii, Iterative methods for solving non-linear operator equations in the absence of regularity. A new approach., Russian Acad.Sci.Dokl.Math. 47 (1993), 451–454.MathSciNetGoogle Scholar
  4. 4.
    A.B. Bakushinskii and A.V. Goncharskii, Iterative Methods for the Solution of Incorrect Problems, Nauka, Moscow, 1989. In Russian.Google Scholar
  5. 5.
    A. Binder, M. Hanke, O. ScherzerOn the Landweber iteration for nonlinear ill-posed problems, to appear in J. Inverse and Ill-Posed Problems.Google Scholar
  6. 6.
    A. Binder, H.W. Engl, C.W. Groetsch, A. Neubauer, and O. Scherzer, Weakly closed nonlinear operators and parameter identification in parabolic equations by Tikhonov regularization, Applicable Analysis 55 (1994), 215–234.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    B. Blaschke, Some Newton type methods for the solution of nonlinear ill-posed problems, PhD thesis, Johannes Kepler Universität Linz, 1996.Google Scholar
  8. 8.
    B. Blaschke, H.W. Engl, W. Grever, M. Klibanov, An application of Tikhonov regularization to phase retrieval, to appear in Nonlinear World.Google Scholar
  9. 9.
    B. Blaschke, A. Neubauer, and O. Scherzer, On convergence rates for the iteratively regularized Gauß-Newton method, to appear in IMA J. Numer. Anal.Google Scholar
  10. 10.
    H. Brakhage, On ill-posed problems and the method of conjugate gradients, in C.W. GROETSCH (eds.), Inverse and Ill-Posed Problems, Academic Press, Orlando, 1987 [19], 165–175.Google Scholar
  11. 11.
    G. Chavent, K. Kunisch, On weakly nonlinear ill-posed problems, to appear in SIAM J. Appl. Math. (1996).Google Scholar
  12. 12.
    K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, New York, 1989.MATHCrossRefGoogle Scholar
  13. 13.
    D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.MATHGoogle Scholar
  14. 14.
    D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Berlin, 1992.MATHGoogle Scholar
  15. 15.
    D.C. Dobson, Phase reconstruction via nonlinear least-squares, Inverse Problems 8 (1992), 541–557.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    B. Eicke, Iteration methods for convexly constrained ill-posed problems in Hilbert space, Numer. Funct. Anal. Optim. 13 (1992), 413–429.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    B. Eicke, A.K. Louis, R. Plato, The instability of some gradient methods for ill-posed problems, Numer. Math. 58 (1990), 129–134.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    H.W. Engl, H. Gfrerer, A posteriori parameter choice for general regularization methods for solving linear ill-posed problems, Appl. Numer. Math. 4 (1988), 395–417.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    H.W. Engl, C.W. Groetsch (eds.), Inverse and Ill-Posed Problems, Academic Press, Orlando, 1987.MATHGoogle Scholar
  20. 20.
    H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.MATHCrossRefGoogle Scholar
  21. 21.
    H.W. Engl, K. Kunisch, A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems 5 (1989), 523–540.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    H.W. Engl, G. Landl, Convergence rates for maximum entropy regularization, SIAM J. Numer. Anal. 30 (1993), 1509–1536.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    H.W. Engl, A. Neubauer, Convergence rates for Tikhonov regularization of implicitly defined nonlinear inverse problems with an application to inverse scattering, in: S. Kubo, ed., Inverse Problems, Techn. Publ. Atlanta, 1993, 90–98.Google Scholar
  24. 24.
    H.W. Engl, W. Rundell, O. Scherzer, A regularization scheme for an inverse problem in age-structured populations, J. Math. Anal. Appl. 182 (1994), 658–679.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    H.W. Engl, O. Scherzer, M. Yamamoto, Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data, Inverse Problems 10 (1994), 1253–1276.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    H. Gfrerer, An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates, Math. of Comp. 49 (1987) 507–522 and S5-S12.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    C.W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.MATHGoogle Scholar
  28. 28.
    C.W. Groetsch, Inverse Problems in Mathematical Sciences, Vieweg, Braunschweig, 1993.MATHGoogle Scholar
  29. 29.
    M. Hanke, Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math. 60 (1991), 341–373.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    M. Hanke, H.W. Engl, An optimal stopping rule for the v-methods for solving ill-posed problems using Christoffel functions, J. Approx. Theory 79 (1994), 89–108.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    M. Hanke, F. Hettlich, O. Scherzer, The Landweber iteration for an inverse scattering problem, in: in K.-W. Wang et al., eds., Proceedings of the 1995 Design Engineering Technical Conferences, Vol. 3, Part C, Vibration Control, Analysis, and Identification, The Americal Society of Mechanical Engineers, New York, 1995, 909–915.Google Scholar
  32. 32.
    M. Hanke, A. Neubauer, and O. Scherzer, A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math. 72 (1995), 21–37.MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    F. Hettlich, J. Morgan, O. Scherzer, On the estimation of interfaces from boundary measurements, to appear in: Proceedings of the SIAM conference “Symposion on Inverse Problems: Geophysical Applications”, Yosemite, USA 1995.Google Scholar
  34. 34.
    B. Hofmann and O. Scherzer, Influence factors of ill-posedness for nonlinear problems, Inverse Problems 10 (1994), 1277–1297.MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    N.E. Hurt, Phase Retrieval and Zero Crossings, Kluwer, Dordrecht, 1989.MATHGoogle Scholar
  36. 36.
    V. Isakov, Inverse Source Problems, Math. Surveys and Monographs 34, Amer. Math. Soc., Providence, 1990.Google Scholar
  37. 37.
    M.V. Klibanov, J. Malinsky, Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data, Inverse Problems 7 (1991), 577–596.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    M.V. Klibanov and P.E. Sacks, Phaseless inverse scattering and the phase problem in optics, J. Math. Phys. 33 (1992), 3913–3821.MathSciNetCrossRefGoogle Scholar
  39. 39.
    M.V. Klibanov and P.E. Sacks, Use of partial knowledge of the potential in the phase problem of inverse scattering, J. Comput. Phys. 112 (1994), 273–281.MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    M.V. Klibanov, P.E. Sacks and A.V. Tikhonravov, The phase retrieval problem, Inverse Problems 11 (1995), 1–28.MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    R. Kohn, A. McKenney, Numerical implementation of a variational method for electrical impedance tomography, Inverse Problems 6 (1990), 389–414MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    C. Kravaris and J.H. Seinfeld, Identification of parameters in distributed parameter systems by regularization, SIAM J. Contr. and Optimiz. 23 (1985), 217–241.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    R. Kress and W. Rundell, A quasi-Newton method in inverse obstacle scattering, Inverse Problems 10 (1994), 1145–1158.MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    K. Kunisch, W. Ring, Regularization of nonlinear illposed problems with closed operators, Num. Fune. Anal. Optim. 14 (1993), 389–404.MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    G. Landl, R.S. Anderssen, Non-negative differentially constrained entropy-like regularization, Inverse Problems 12 (1996), 35–53.MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    L. Landweber, An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math. 73 (1951), 615–624.MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    A. K. Louis, Convergence of the conjugate gradient method for compact operators, in C.W. GROETSCH (eds.), Inverse and Ill-Posed Problems, Academic Press, Orlando, 1987 [19], 177–183.Google Scholar
  48. 48.
    A. K. Louis, Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989.MATHCrossRefGoogle Scholar
  49. 49.
    A.K. Louis, Parametric reconstruction in biomagnetic imaging, in: M. Bertero, E.R. Pike, eds., Inverse Problems in Scattering and Imaging, Hilger, Bristol, 1992, 156–163.Google Scholar
  50. 50.
    V.A. Morozov, On the solution of the functional equations by the method of regularization, Sov. Math. Dokl. 7 (1966), 414–417.MATHGoogle Scholar
  51. 51.
    V.A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, New York, 1984.CrossRefGoogle Scholar
  52. 52.
    V.A. Morozov, Regularization Methods for Ill-Posed Problems, CRC Press, Boca Raton, 1993.MATHGoogle Scholar
  53. 53.
    A. Neubauer, Tikhonov regularization for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation, Inverse Problems 5 (1989), 541–557.MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    A. Neubauer, Tikhonov regularization for non-linear ill-posed problems in Hilbert scales, Applicable Analysis 46 (1992), 59–72.MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    P. Sacks, F. Santosa, A simple computational scheme for determing the sound speed of an acoustic medium from surface impulse response, SIAM J. Scient. Stat. Comput. 8 (1987), 501–520MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    O. Scherzer, A convergence analysis of a method of steepest descent and a twostep algorithm for nonlinear ill-posed problems, to appear in Num. Func. Anal. Optim.Google Scholar
  57. 57.
    O. Scherzer, Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. Math. Anal. Appl. 194 (1995), 911–933.MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    O. Scherzer, A modified Landweber iteration for solving parameter estimation problems, to appear in: Appl. Math. Optim.Google Scholar
  59. 59.
    O. Scherzer, The use of Tikhonov regularization in the identification of electrical conductivities from overdetermined boundary data, Results of Math. 22 (1992), 598–618.MathSciNetMATHGoogle Scholar
  60. 60.
    O. Scherzer, H.W. Engl, R.S. Anderssen, Parameter identification from boundary measurements in a parabolic equation arising from geophysics, Nonlinear Anal. 20 (1993), 127–156.MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    O. Scherzer, H.W. Engl, K. Kunisch, Optimal a-posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems, SIAM J. Numer. Anal. 30 (1993), 1796–1838.MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    E. Schock, 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, 1985, 234–243.CrossRefGoogle Scholar
  63. 63.
    T. Seidman, C.R. Vogel, Well posedness and convergence for some regularisation methods for non-linear ill posed problems, Inverse Problems 5 (1989), 227–238.MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    A.N. Tikhonov, V.A. Arsenin, Methods for Solving Ill-Posed Problems, Nauka, Moscow, 1979.Google Scholar
  65. 65.
    G.M. Vainikko, A.Y. Veterennikov, Iteration Procedures in Ill-Posed Problems, Nauka, Moscow, 1986. In Russian.Google Scholar
  66. 66.
    V.V. Vasin, Iterative methods for solving ill-posed problems with a priori information in Hilbert spaces, USSR Comput. Math. Math. Phys. 28 (1988), 6–13.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1997

Authors and Affiliations

  • Barbara Blaschke-Kaltenbacher
    • 1
  • Heinz W. Engl
    • 1
  1. 1.Institut für MathematikJohannes Kepler UniversitätLinzAustria

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