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2019 | OriginalPaper | Chapter

4. Ill-Posed Problems

Authors : David Colton, Rainer Kress

Published in: Inverse Acoustic and Electromagnetic Scattering Theory

Publisher: Springer International Publishing

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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.

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previous chapter Direct Acoustic Obstacle Scattering
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Literature
23.
Bakushinskii, A.B.: The problem of the convergence of the iteratively regularized Gauss–Newton method. Comput. Maths. Maths. Phys. 32, 1353–1359 (1992).
27.
Baumeister, J.: Stable Solution of Inverse Problems. Vieweg, Braunschweig 1986.MATH
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).MathSciNetCrossRef
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).
138.
Engl, H.W., Hanke, M. and Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publisher, Dordrecht 1996.CrossRef
156.
Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston 1984.MATH
165.
Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven 1923.MATH
179.
Hanke, M.: A regularization Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems 13, 75–95 (1997).CrossRef
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).MathSciNetCrossRef
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).
206.
Ivanov, K.V.: On linear problems which are not well-posed. Soviet Math. Doklady 3, 981–983 (1962) (English translation).
226.
Kabanikhin, S.I.: Inverse and Ill-posed Problems: Theory and Applications. de Gruyter, Berlin-Boston 2011
227.
Kaltenbacher, B., Neubauer, A., and Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. de Gruyter, Berlin, 2008.CrossRef
238.
Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. 2nd ed, Springer, Berlin 2011.CrossRef
268.
308.
Louis, A.K.: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart 1989.CrossRef
322.
Morozov, V.A.: On the solution of functional equations by the method of regularization. Soviet Math. Doklady 7, 414–417 (1966) (English translation).
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).
324.
Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, Berlin 1984.CrossRef
406.
Tikhonov, A.N.: On the solution of incorrectly formulated problems and the regularization method. Soviet Math. Doklady 4, 1035–1038 (1963) (English translation).
407.
Tikhonov, A.N.: Regularization of incorrectly posed problems. Soviet Math. Doklady 4, 1624–1627 (1963) (English translation).
408.
Tikhonov, A.N., and Arsenin, V.Y.: Solutions of Ill-posed Problems. Winston and Sons, Washington 1977.MATH
418.
Wang, Y., Yagola, A.G., and Yang, C.: Optimization and Regularization for Computational Inverse Problems and Applications. Springer, Berlin 2011.CrossRef
Metadata
Title
Ill-Posed Problems
Authors
David Colton
Rainer Kress
Copyright Year
2019
Book
Inverse Acoustic and Electromagnetic Scattering Theory

Print ISBN: 978-3-030-30350-1

Electronic ISBN: 978-3-030-30351-8

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-30351-8_4

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