Skip to main content

Inverse problems and their regularization

  • Chapter
  • First Online:
Computational Mathematics Driven by Industrial Problems

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1739))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Backus and F. Gilbert, Numerical applications of a formalism for geophysical inverse problems, Geophys. J. R. Astron. Soc., 13(1967) 247–276

    Article  Google Scholar 

  2. A.B. Bakushinskii, The problem of the convergence of the iteratively regularized Gauß-Newton method, Comput. Maths. Math. Phys., 32(1992) 1353–1359

    MathSciNet  Google Scholar 

  3. H. Banks and K. Kunisch, Parameter Estimation Techniques for Distributed Systems, Birkhäuser, Boston, 1989

    Book  MATH  Google Scholar 

  4. J. Beck, B. Blackwell, C.S. Clair, Inverse Heat Conduction: Ill-Posed Problems; Wiley, New York 1985

    MATH  Google Scholar 

  5. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging; Inst. of Physics Publ., Bristol 1998

    Book  MATH  Google Scholar 

  6. B. Blaschke, A. Neubauer, and O. Scherzer, On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numer. Anal., 17(1997) 421–436

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Binder, M. Hanke, and O. Scherzer, On the Landweber iteration for nonlinear ill-posed problems, J. Inverse Ill-Posed Probl., 4(1996) 381–389

    Article  MathSciNet  MATH  Google Scholar 

  8. 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, Appl. Anal., 55(1994) 215–234

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Brakhage, On ill-posed problems and the method of conjugate gradients, in [24] 165–175

    Google Scholar 

  10. M. Burger, Iterative regularization of a parameter identification problem occurring in polymer crystallization, submitted

    Google Scholar 

  11. M. Burger, V. Capasso, H.W. Engl, Inverse problems related to crystallization of polymers, Inverse Problems, 15 (1999) 155–173

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Burger, H.W. Engl, Training neural networks with noisy data as an ill-posed problem, to appear in Advances in Comp. Mathematics

    Google Scholar 

  13. V. Capasso, Mathematical Models for Polymer Crystallization, This volume

    Google Scholar 

  14. G. Chavent, K. Kunisch, On weakly nonlinear inverse problems, SIAM J. Appl. Math., 56(1996) 542–572

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Colton, R. Ewing, and W. Rundell, Inverse Problems in Partial Differential Equations, SIAM, Philadelphia, 1990

    MATH  Google Scholar 

  16. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Berlin, 1992

    Book  MATH  Google Scholar 

  17. D. Colton, A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12(1996) 383–393.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Colton, M. Piana, R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13(1997) 1477–1493

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Deuflhard and G. Heindl, Affine invariant convergence theorems for Newton's method and extensions to related methods, SIAM J. Numer. Anal., 16(1979) 1–10

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Druckenthaner, H. Zeisel, A. Schiefermüller, G. Kolb, A. Ferstl, H. Engl, A. Schatz, Online simulation of the blast furnace, Advanced Steel, (1997–98) 58–61

    Google Scholar 

  21. H.W. Engl, Regularization methods for the stable solution of inverse problems, Surveys on Mathematics for Industry, 3(1993) 71–143

    MathSciNet  MATH  Google Scholar 

  22. H. W. Engl and H. Gfrerer, A posteriori parameter choice for general regularization methods for solving linear ill-posed problems, Appl. Numer. Math. 4(1988) 395–417

    Article  MathSciNet  MATH  Google Scholar 

  23. H. W. Engl and W. Grever, Using the L-curve for determining optimal regularization parameters, Numer. Math., 69(1994) 25–31

    Article  MathSciNet  MATH  Google Scholar 

  24. H. W. Engl and C. W. Groetsch, Inverse and Ill-Posed Problems, Academic Press, Orlando, 1987

    MATH  Google Scholar 

  25. H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems 5(1989) 523–540

    Article  MathSciNet  MATH  Google Scholar 

  26. H.W. Engl, T. Langthaler, Control of the solidification front by secondary cooling in continuous casting of steel, in: H.W. Engl, H. Wacker and W. Zulehner, eds. Case Studies in Industrial Mathematics Teubner, Stuttgart, 1988, 51–77

    Google Scholar 

  27. H. W. Engl, A. K. Louis and W. Rundell (eds.), Inverse Problems in Geophysics, SIAM, Philadelphia, 1996

    Google Scholar 

  28. H. W. Engl, A. K. Louis and W. Rundell (eds.), Inverse Problems in Medical Imaging and Nondestructive Testing, Springer, Vienna, New York, 1996

    Google Scholar 

  29. H. W. Engl and W. Rundell (eds.), Inverse Problems in Diffusion Processes, SIAM, Philadelphia, 1995

    Google Scholar 

  30. H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996

    Book  MATH  Google Scholar 

  31. H.W. Engl and O. Scherzer, Convergence rates results for iterative methods for solving nonlinear ill-posed problems, to appear in: D. Colton, H.W. Engl, A.K. Louis, J. McLaughlin, W.F. Rundell (eds.), Solution Methods for Inverse Problems, Springer, Vienna/New York, 2000

    Google Scholar 

  32. 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(1998) 1081–1106

    Article  MathSciNet  MATH  Google Scholar 

  33. W. Grever, A. Binder, H.W. Engl, K. Mörwald, Optimal cooling strategies in continuous casting of steel with variable casting speed, Inverse Problems in Engineering, 2 (1996) 289–300

    Article  Google Scholar 

  34. W. Grever A nonlinear parabolic initial boundary value problem modelling the continuous casting of steel with variable casting speed, ZAMM, 78 (1998) 109–119

    Article  MathSciNet  MATH  Google Scholar 

  35. C. W. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg, Braunschweig, 1993

    Book  MATH  Google Scholar 

  36. C. W. Groetsch, Generalized Inverses of Linear Operators: Representation and Approximation, Dekker, New York, 1977

    MATH  Google Scholar 

  37. M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems, Longman Scientific & Technical, Harlow, Essex, 1995

    MATH  Google Scholar 

  38. M. Hanke, A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Probl., 13(1997) 79–95

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Hanke, A. Neubauer, and O. Scherzer. A convergence analysis of Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72(1995) 21–37

    Article  MathSciNet  MATH  Google Scholar 

  40. P. C. Hansen and D. P. O'Leary, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993) 1487–1503

    Article  MathSciNet  MATH  Google Scholar 

  41. F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Probl., 14(1998) 67–82

    Article  MathSciNet  MATH  Google Scholar 

  42. B. Hofmann, Mathematik inverser Problem, Teubner, Stuttgart 1999

    Google Scholar 

  43. T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauß-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems, 13(1997) 1279–1299

    Article  MathSciNet  MATH  Google Scholar 

  44. T. Hohage, Convergence rates of a regularized Newton method in soundhard inverse scattering, SIAM J. Numer. Anal., 36(1998) 125–142

    Article  MathSciNet  MATH  Google Scholar 

  45. D. Isaacson and J.C. Newell, Electrical Impedance Tomography, SIAM Review, 41(1999) 85–101

    Article  MathSciNet  MATH  Google Scholar 

  46. V. Isakov, Inverse Problems in Partial Differential Equations, Springer, Berlin, New York, 1998

    Book  MATH  Google Scholar 

  47. B. Kaltenbacher, Some Newton-type methods for the regularization of nonlinear ill-posed problems, Inverse Probl., 13 (1997) 729–753

    Article  MathSciNet  MATH  Google Scholar 

  48. B. Kaltenbacher, A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems, Numer. Math., 79(1998) 501–528

    Article  MathSciNet  MATH  Google Scholar 

  49. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York 1996

    Book  MATH  Google Scholar 

  50. A. Kirsch, B. Schomburg and G. Berendt, The Backus-Gilbert method, Inverse Problems, 4(1988) 771–783

    Article  MathSciNet  MATH  Google Scholar 

  51. G. Landl, T. Langthaler, H. W. Engl and H. F. Kauffmann, Distribution of event times in time-resolved fluorescence: the exponential series approach — algorithm, regularization, analysis, J. Comput. Phys., 95(1991) 1–28

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Leonov and A. Yagola, Special regularizing methods for ill-posed problems with sourcewise represented solutions, Inverse Problems, 14(1998) 1539–1550

    Article  MathSciNet  MATH  Google Scholar 

  53. A. K. Louis, Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989

    Book  MATH  Google Scholar 

  54. A. K. Louis, Convergence of the conjugate gradient method for compact operators, in [24]

    Google Scholar 

  55. A. K. Louis and P. Maass, A mollifier method for linear operator equations of the first kind, Inverse Problems, 6(1990) 427–440

    Article  MathSciNet  MATH  Google Scholar 

  56. M. Z. Nashed, Generalized Inverses and Applications, Academic Press, New York, 1976

    MATH  Google Scholar 

  57. F. Natterer, The Mathematics of Computerized Tomography, Teubner, Stuttgart, 1986

    MATH  Google Scholar 

  58. 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(1995) 369–377

    Article  MathSciNet  MATH  Google Scholar 

  59. E. Radmoser, R. Wincor, Determining the inner contour of a furnace from temperature measurements, Industrial Mathematics Institute, Johannes Kepler Universität Linz, Technical Report 12 (1998)

    Google Scholar 

  60. R. Ramlau, A modified Landweber-method for inverse problems Num. Funct. Anal. Opt., 20(1999) 79–98

    Article  MathSciNet  MATH  Google Scholar 

  61. A. Ramm, Scattering by Obstacles, Reidel, Dordrecht, 1986

    Book  MATH  Google Scholar 

  62. E. Ratajski, H. Janeschitz-Kriegl, How to determine high growth speeds in polymer crystallization, Colloid Polym. Sci., 274 (1996) 938–951

    Article  Google Scholar 

  63. T. Raus, Residue principle for ill-posed problems, Acta et Comment. Univers. Tartuensis, 672(1984) 16–26

    MathSciNet  Google Scholar 

  64. A. Rieder, A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization, Numer. Math., 75(1997) 501–522

    Article  MathSciNet  MATH  Google Scholar 

  65. O. Scherzer, A convergence analysis of a method of steepest descent and a two-step algorithm for nonlinear ill-posed problems, Numer. Funct. Anal. Optimization, 17(1996) 197–214

    Article  MathSciNet  MATH  Google Scholar 

  66. O. Scherzer, An iterative multi level algorithm for solving nonlinear ill-posed problems, Numer. Math., 80(1998) 579–600

    Article  MathSciNet  MATH  Google Scholar 

  67. E. Schock, Approximate solution of ill-posed equations: arbitrarily slow convergence vs. superconvergence, in: G. Hämmerlin and K. H. Hoffmann, eds., Constructive Methods for the Practical Treatment of Integral Equations, Birkhäuser, Basel, 1985, 234–243

    Chapter  Google Scholar 

  68. T. I. Seidman and C. R. Vogel, Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems, Inverse Problems, 5(1989) 227–238

    Article  MathSciNet  MATH  Google Scholar 

  69. M. Tanaka, T. Matsumoto, S. Oida, Identification of unknown boundary shape of rotationally symmetric body in steady heat conduction via BEM and filter theories, in: M. Tanaka and G.S. Dulikravich eds., Inverse Problems in Engineering Mechanics, Elsevier Science B.V., Tokyo, 1998, 121–130

    Chapter  Google Scholar 

  70. C. R. Vogel, Non-convergence of the L-curve regularization parameter selection method, Inverse Problems, 12(1996) 535–547

    Article  MathSciNet  MATH  Google Scholar 

  71. G. Wahba, Spline Models for Observational Data, SIAM, Philadelphia, 1990

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Rainer E. Burkard Antony Jameson Gilbert Strang Peter Deuflhard Jacques-Louis Lions Vincenzo Capasso Jacques Periaux Heinz W. Engl

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag

About this chapter

Cite this chapter

Engl, H.W. (2000). Inverse problems and their regularization. In: Burkard, R.E., et al. Computational Mathematics Driven by Industrial Problems. Lecture Notes in Mathematics, vol 1739. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103919

Download citation

  • DOI: https://doi.org/10.1007/BFb0103919

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67782-6

  • Online ISBN: 978-3-540-44976-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics