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Ill-Posed Problems

  • David Colton
  • Rainer Kress
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 93)

Abstract

As previously mentioned, for problems in mathematical physics Hadamard postulated three requirements: a solution should exist, the solution should be unique, and the solution should depend continuously on the data. The third postulate is motivated by the fact that in all applications the data will be measured quantities. Therefore, one wants to make sure that small errors in the data will cause only small errors in the solution. A problem satisfying all three requirements is called well-posed. Otherwise, it is called ill-posed.

References

  1. 23.
    Bakushinskii, A.B.: The problem of the convergence of the iteratively regularized Gauss–Newton method. Comput. Maths. Maths. Phys. 32, 1353–1359 (1992).Google Scholar
  2. 27.
    Baumeister, J.: Stable Solution of Inverse Problems. Vieweg, Braunschweig 1986.zbMATHGoogle Scholar
  3. 32.
    Blaschke, B., Neubauer, A., and Scherzer, O: On convergence rates for the iteratively regularized Gauss–Newton method. IMA J. Numerical Anal. 17, 421–436 (1997).MathSciNetCrossRefGoogle Scholar
  4. 47.
    Burger, M., Kaltenbacher, B., and Neubauer, A.: Iterative solution methods. In: Handbook of Mathematical Methods in Imaging (Scherzer, ed). Springer, Berlin, 345–384 (2011).Google Scholar
  5. 138.
    Engl, H.W., Hanke, M. and Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publisher, Dordrecht 1996.CrossRefGoogle Scholar
  6. 156.
    Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston 1984.zbMATHGoogle Scholar
  7. 165.
    Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven 1923.zbMATHGoogle Scholar
  8. 179.
    Hanke, M.: A regularization Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems 13, 75–95 (1997).CrossRefGoogle Scholar
  9. 182.
    Hanke, M., Neubauer, A., and Scherzer, O.: A convergence analysis for the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21–37 (1995).MathSciNetCrossRefGoogle Scholar
  10. 205.
    Ivanov, K.V.: Integral equations of the first kind and an approximate solution for the inverse problem of potential. Soviet Math. Doklady 3, 210–212 (1962) (English translation).Google Scholar
  11. 206.
    Ivanov, K.V.: On linear problems which are not well-posed. Soviet Math. Doklady 3, 981–983 (1962) (English translation).Google Scholar
  12. 226.
    Kabanikhin, S.I.: Inverse and Ill-posed Problems: Theory and Applications. de Gruyter, Berlin-Boston 2011Google Scholar
  13. 227.
    Kaltenbacher, B., Neubauer, A., and Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. de Gruyter, Berlin, 2008.CrossRefGoogle Scholar
  14. 238.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. 2nd ed, Springer, Berlin 2011.CrossRefGoogle Scholar
  15. 268.
    Kress, R.: Linear Integral Equations. 3rd ed, Springer, Berlin 2014.CrossRefGoogle Scholar
  16. 308.
    Louis, A.K.: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart 1989.CrossRefGoogle Scholar
  17. 322.
    Morozov, V.A.: On the solution of functional equations by the method of regularization. Soviet Math. Doklady 7, 414–417 (1966) (English translation).Google Scholar
  18. 323.
    Morozov, V.A.: Choice of parameter for the solution of functional equations by the regularization method. Soviet Math. Doklady 8, 1000–1003 (1967) (English translation).Google Scholar
  19. 324.
    Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, Berlin 1984.CrossRefGoogle Scholar
  20. 406.
    Tikhonov, A.N.: On the solution of incorrectly formulated problems and the regularization method. Soviet Math. Doklady 4, 1035–1038 (1963) (English translation).Google Scholar
  21. 407.
    Tikhonov, A.N.: Regularization of incorrectly posed problems. Soviet Math. Doklady 4, 1624–1627 (1963) (English translation).Google Scholar
  22. 408.
    Tikhonov, A.N., and Arsenin, V.Y.: Solutions of Ill-posed Problems. Winston and Sons, Washington 1977.zbMATHGoogle Scholar
  23. 418.
    Wang, Y., Yagola, A.G., and Yang, C.: Optimization and Regularization for Computational Inverse Problems and Applications. Springer, Berlin 2011.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Colton
    • 1
  • Rainer Kress
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Institut für Numerische und Angewandte MathematikGeorg-August-Universität GöttingenGöttingenGermany

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