Skip to main content
SpringerLink
Log in

Regularization of Inverse Problems

  • Reference work entry
  • First Online:
Encyclopedia of Applied and Computational Mathematics

Introduction

Inverse problems aim at the determination of a cause x from observations y. Let the mathematical model F : X → Y describe the connection between the cause x and the observation y. The computation of y ∈ Y from x ∈ X forms the direct problem. Often the operator F is not directly accessible but given, e.g., via the solution of a differential equation. The inverse problem is to find a solution of the equation

$$\displaystyle{ F(x) = y. }$$
(1)

from given observations. As the data is usually measured, only noisy data with

$$\displaystyle{ \|y - y^{\delta }\| \leq \delta }$$
(2)

are available.

Such problems appear naturally in all kinds of applications.

Usually, inverse problems do not fulfill Hadamard’s definition of well-posedness:

Definition 1

The problem (1) is well posed, if

  1. 1.

    For all admissible data y exists an x with (1),

  2. 2.

    the solution x is uniquely determined by the data,

  3. 3.

    the solution depends continuously on the data.

If one of the above conditions is violated,...

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 999.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 999.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 999.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Acar, R., Vogel, C.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Probl. 10, 1217–1229 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anzengruber, S., Ramlau, R.: Morozovs discrepancy principle for Tikhonov-type functionals with nonlinear operator. Inverse Probl. 26(2), 1–17 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bakushinskii, A.W.: The problem of the convergence of the iteratively regularized Gauss–Newton method. Comput. Math. Math. Phys. 32, 1353–1359 (1992)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonesky, T.: Morozov’s discrepancy principle and Tikhonov-type functionals. Inverse Probl. 25, 015,015 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bredies, K., Lorenz, D., Maass, P.: A generalized conditional gradient method and its connection to an iterative shrinkage method. Comput. Optim. Appl. 42, 173–193 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Probl. 20(5), 1411–1421 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chavent, G., Kunisch, K.: Regularization of linear least squares problems by total bounded variation. ESAIM, Control Optim. Calc. Var. 2, 359–376 (1997)

    Google Scholar 

  10. Daubechies, I., Defriese, M., DeMol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 51, 1413–1541 (2004)

    Article  MathSciNet  Google Scholar 

  11. Deuflhard, P., Engl, H., Scherzer, O.: A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Probl. 14, 1081–1106 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dobson, D., Scherzer, O.: Analysis of regularized total variation penalty methods for denoising. Inverse Probl. 12, 601–617 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eggermont, P.: Maximum entropy regularization for Fredholm integral equations of the first kind. SIAM J. Math. Anal. 24, 1557–1576 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Engl, H., Landl, G.: Convergence rates for maximum entropy regularization. SIAM J. Numer. Anal. 30, 1509–1536 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Engl, H., Kunisch, K., Neubauer, A.: Convergence rates for Tikhonov regularization of nonlinear ill-posed problems. Inverse Probl. 5, 523–540 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  17. Engl, H., Flamm, C., Kügler, P., Lu, J., Müller, S., Schuster, P.: Inverse problems in systems biology. Inverse Probl. 25, 123,014 (2009)

    Article  MATH  Google Scholar 

  18. Grasmair, M., Haltmeier, M., Scherzer, O.: Sparse regularization with \(\ell^{q}\) penalty term. Inverse Probl. 24(5), 1–13 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Griesse, R., Lorenz, D.: A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inverse Probl. 24(3), 035,007 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems. Longman Scientific & Technical, Harlow (1995)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Hanke, M.: Regularizing properties of a truncated Newton–cg algorithm for nonlinear ill-posed problems. Numer. Funct. Anal. Optim. 18, 971–993 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Hegland, M.: Variable Hilbert scales and their interpolation inequalities with applications to Tikhonov regularization. Appl. Anal. 59(1–4), 207–223 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hofmann, B., Kaltenbacher, B., Pöschl, C., Scherzer, O.: A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Probl. 23, 987–1010 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hohage, T.: Logarithmic convergence rates of the iteratively regularized Gauss–Newton method for an inverse potential and an inverse scattering problem. Inverse Probl. 13, 1279–1299 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ito, K., Kunisch, K.: Lagrange multiplier approach to variational problems and applications. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  29. Kaltenbacher, B.: On Broyden’s method for ill-posed problems. Num. Funct. Anal. Optim. 19, 807–833 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kaltenbacher, B., Hofmann, B.: Convergence rates for the iteratively regularized Gauss-Newton method in Banach spaces. Inverse Probl. 26, 035,007 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. de Gruyter, Berlin (2008)

    Book  MATH  Google Scholar 

  32. Kravaris, C., Seinfeld, J.H.: Identification of parameters in distributed parameter systems by regularization. SIAM J. Control Optim. 23, 217–241 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kress, R.: Linear Integral Equations. Springer, New York (1989)

    Book  MATH  Google Scholar 

  34. Louis, A.: Approximate inverse for linear and some nonlinear problems. Inverse Probl. 12, 175–190 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Louis, A., Maass, P.: A mollifier method for linear operator equations of the first kind. Inverse Probl. 6, 427–440 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. Louis, A.K.: Inverse und Schlecht Gestellte Probleme. Teubner, Stuttgart (1989)

    Book  MATH  Google Scholar 

  37. Luecke, G.R., Hickey, K.R.: Convergence of approximate solutions of an operator equation. Houst. J. Math. 11, 345–353 (1985)

    MathSciNet  MATH  Google Scholar 

  38. Mathe, P., Pereverzev, S.V.: Geometry of linear ill-posed problems in variable Hilbert scales. Inverse Probl. 19(3), 789803 (2003)

    MathSciNet  MATH  Google Scholar 

  39. Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)

    Book  Google Scholar 

  40. Natterer, F.: Regularisierung schlecht gestellter Probleme durch Projektionsverfahren. Numer. Math. 28, 329–341 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  41. Natterer, F.: The Mathematics of Computerized Tomography. Teubner, Stuttgart (1986)

    MATH  Google Scholar 

  42. Nemirovskii, A.S.: The regularizing properties of the adjoint gradient method in ill posed problems. USSR Comput. Math. Math. Phys. 26(2), 7–16 (1986)

    Article  Google Scholar 

  43. Pereverzev, S.V.: Optimization of projection methods for solving ill-posed problems. Computing 55, 113–124 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  44. Plato, R., Vainikko, G.: On the regularization of projection methods for solving ill-posed problems. Numer. Math. 57, 63–79 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ramlau, R.: A modified Landweber–method for inverse problems. Numer. Funct. Anal. Optim. 20(1&2), 79–98 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ramlau, R.: Morozov’s discrepancy principle for Tikhonov regularization of nonlinear operators. Numer. Funct. Anal. Optim. 23(1&2), 147–172 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ramlau, R.: TIGRA–an iterative algorithm for regularizing nonlinear ill–posed problems. Inverse Probl. 19(2), 433–467 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ramlau, R., Resmerita, E.: Convergence rates for regularization with sparsity constraints. Electron. Trans. Numer. Anal. 37, 87–104 (2010)

    MathSciNet  MATH  Google Scholar 

  49. Ramlau, R., Teschke, G.: A Tikhonov-based projection iteration for non-linear ill-posed problems with sparsity constraints. Numer. Math. 104(2), 177–203 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ramlau, R., Teschke, G.: Sparse recovery in inverse problems. In: Fornasier, M. (ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery, pp. 201–262. De Gruyter, Berlin/New York (2010)

    Google Scholar 

  51. Resmerita, E.: Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Probl. 21, 1303–1314 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Resmerita, E., Scherzer, O.: Error estimates for non-quadratic regularization and the relation to enhancement. Inverse Probl. 22, 801–814 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  53. Rieder, A.: On convergence rates of inexact Newton regularizations. Numer. Math. 88, 347–365 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  54. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  55. Scherzer, O.: The use of Morozov’s discrepancy principle for Tikhonov regularization for solving nonlinear ill–posed problems. Computing 51, 45–60 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  56. Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Springer, Berlin (2008)

    MATH  Google Scholar 

  57. Schock, E.: Approximate solution of ill-posed problems: arbitrary slow convergence vs. superconvergence. In: Hämmerlin, G., Hoffmann, K. (eds.) Constructive Methods for the Practical Treatment of Integral Equation, pp. 234–243. Birkhäuser, Basel/Boston (1985)

    Chapter  Google Scholar 

  58. Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. V.H. Winston & Sons, Washington (1977)

    MATH  Google Scholar 

  59. Vogel, C.R., Oman, M.: Iterative methods for total variation denoising. SIAM J. Sci. Comput. 17, 227–238 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zarzer, C.A.: On Tikhonov regularization with non-convex sparsity constraints. Inverse Probl. 25, 025,006 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria

    Heinz W. Engl

  2. Institute for Industrial Mathematics, Kepler University Linz, Linz, Austria

    Ronny Ramlau

Authors
  1. Heinz W. Engl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ronny Ramlau
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Texas at Austin, Austin, TX, USA

    Björn Engquist

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this entry

Cite this entry

Engl, H.W., Ramlau, R. (2015). Regularization of Inverse Problems. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_52

Download citation

Publish with us

Policies and ethics

Search

Navigation