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

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.

[1]  M. Hanke.  Accelerated Landweber iterations for the solution of ill-posed equations , 1991 .

[2]  A. Bakushinskii.  The problem of the convergence of the iteratively regularized Gauss-Newton method , 1992 .

[3]  H. Gfrerer.  An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates , 1987 .

[4]  George Weiss,et al.  Integral Equation Methods , 1969 .

[5]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[6]  V. A. Morozov,et al.  Methods for Solving Incorrectly Posed Problems , 1984 .

[7]  L. Landweber.  An iteration formula for Fredholm integral equations of the first kind , 1951 .

[8]  P. Sabatier,et al.  Inverse Problems in Quantum Scattering Theory , 1977 .

[9]  Heinz W. Engl,et al.  Inverse and Ill-Posed Problems , 1987 .

[10]  Otmar Scherzer,et al.  A Modified Landweber Iteration for Solving Parameter Estimation Problems , 1998 .

[11]  Bertolt Eicke.  Iteration methods for convexly constrained ill-posed problems in hilbert space , 1992 .

[12]  Scherzer Otmar,et al.  A convergence analysis of a method of steepest descent and a two–step algorothm for nonlinear ill–posed problems , 1996 .

[13]  Otmar Scherzer,et al.  The Use of Tikhonov Regularization in the Identification of Electrical Conductivities from Overdetermined Boundary Data , 1992 .

[14]  Michael V. Klibanov,et al.  Use of partial knowledge of the potential in the phase problem of inverse scattering , 1994 .

[15]  H. Engl,et al.  Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems , 1993 .

[16]  Norman E. Hurt,et al.  Phase Retrieval and Zero Crossings , 1989 .

[17]  William Rundell,et al.  A Regularization Scheme for an Inverse Problem in Age-Structured Populations , 1994 .

[18]  O. SCHERZER,et al.  On the Landweber iteration for nonlinear ill-posed problems , 1996 .

[19]  Michael V. Klibanov,et al.  Phaseless inverse scattering and the phase problem in optics , 1992 .

[20]  Curtis R. Vogel,et al.  Well posedness and convergence of some regularisation methods for non-linear ill posed problems , 1989 .

[21]  H. Engl,et al.  A posteriori parameter choice for general regularization methods for solving linear ill-posed problems , 1988 .

[22]  Fadil Santosa,et al.  A Simple Computational Scheme for Determining the Sound Speed of an Acoustic Medium from Its Surface Impulse Response , 1987 .

[23]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[24]  Andreas Neubauer,et al.  Tikhonov regularisation for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation , 1989 .

[25]  Karl Kunisch,et al.  On Weakly Nonlinear Inverse Problems , 1996, SIAM J. Appl. Math..

[26]  H. Engl,et al.  Convergence rates for Tikhonov regularisation of non-linear ill-posed problems , 1989 .

[27]  Otmar Scherzer,et al.  Factors influencing the ill-posedness of nonlinear problems , 1994 .

[28]  Eberhard Schock,et al.  Approximate Solution of Ill-Posed Equations: Arbitrarily Slow Convergence vs. Superconvergence , 1985 .

[29]  H. Brakhage.  ON ILL-POSED PROBLEMS AND THE METHOD OF CONJUGATE GRADIENTS , 1987 .

[30]  Alfred K. Louis.  CONVERGENCE OF THE CONJUGATE GRADIENT METHOD FOR COMPACT OPERATORS , 1987 .

[31]  William Rundell,et al.  A quasi-Newton method in inverse obstacle scattering , 1994 .

[32]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[33]  Andreas Neubauer,et al.  Tikhonov regularization of nonlinear III-posed problems in hilbert scales , 1992 .

[34]  A. Neubauer,et al.  On convergence rates for the Iteratively regularized Gauss-Newton method , 1997 .

[35]  Karl Kunisch,et al.  Regularization of nonlinear illposed problems with closed operators , 1993 .

[36]  Heinz W. Engl,et al.  Convergence rates for maximum entropy regularization , 1993 .

[37]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[38]  Alexander V. Tikhonravov,et al.  The phase retrieval problem , 1995 .

[39]  C. Kravaris,et al.  Identification of parameters in distributed parameter systems by regularization , 1983, The 22nd IEEE Conference on Decision and Control.

[40]  M. Hanke,et al.  A convergence analysis of the Landweber iteration for nonlinear ill-posed problems , 1995 .

[41]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[42]  Otmar Scherzer,et al.  Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data , 1994 .

[43]  Robert V. Kohn,et al.  Numerical implementation of a variational method for electrical impedance tomography , 1990 .

[44]  D. Dobson.  Phase reconstruction via nonlinear least-squares , 1992 .

[45]  Otmar Scherzer,et al.  Convergence Criteria of Iterative Methods Based on Landweber Iteration for Solving Nonlinear Problems , 1995 .

[46]  Heinz W. Engl,et al.  Weakly closed nonlinear operators and parameter identification in parabolic equations by tikhonov regularization , 1994 .

[47]  Otmar Scherzer,et al.  Parameter identification from boundary measurements in a parabolic equation arising from geophysics , 1993 .

[48]  V. V. Vasin.  Iterative methods for solving ill-posed problems with a prior information in Hilbert spaces , 1990 .

[49]  Alfred K. Louis,et al.  The instability of some gradient methods for ill-posed problems , 1990 .

[50]  Michael V. Klibanov,et al.  Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data , 1991 .

[51]  Robert S. Anderssen,et al.  Non-negative differentially constrained entropy-like regularization , 1996 .

[52]  Victor Isakov,et al.  Inverse Source Problems , 1990 .

[53]  G. Hämmerlin,et al.  Constructive methods for the practical treatment of integral equations : proceedings of the conference held at the Mathematisches Forschungsinstitut, Oberwolfach, June 24-30, 1984 , 1985 .

[54]  Heinz W. Engl,et al.  An optimal stopping rule for the n-method for solving ill-posed problems, using Christoffel functions , 1994 .

[55]  A. Louis.  Inverse und schlecht gestellte Probleme , 1989 .