Advertisement

Nonlinear Inverse Problems: Examples and Difficulties

  • Guy ChaventEmail author
Chapter
First Online:
  • 1.9k Downloads
Part of the Scientific Computation book series (SCIENTCOMP)

Abstract

We present in this chapter the nonlinear least-squares (NLS) approach to parameter estimation and inverse problems, and analyze the difficulties associated with their theoretical and numerical resolution.

We begin in Sect. 1.1 with a simple finite dimensional example of nonlinear parameter estimation problem: the estimation of four parameters in the Knott–Zoeppritz equations. This example will reveal the structure of inverse problem, and will be used to set up the terminology.

Then we define in Sect. 1.2 an abstract framework for NLS problems, which contains the structure underlying the example of Sect. 1.1. Next we review in Sect. 1.3 the difficulties associated with the resolution of NLS problems, and hint at possible remedies and their location in the book.

Finally, Sects. 1.4–1.6 describe infinite dimensional parameter estimation problems of increasing difficulty, where the unknown is the source or diffusion coefficient function of an elliptic equation, to which the analysis developed in  Chaps. 2– 5 will be applied.

Examples of time marching problems are given in Sects. 2.8 and 2.9 of  Chap. 2

Keywords

Inverse Problem Reflection Coefficient Elliptic Equation Observation Operator Parameter Estimation Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Aki, K., Richards, P.G., 1980, Quantitative seismology: Theory and methods, W.H. Freeman, New YorkGoogle Scholar
  2. 8.
    Baumeister, J., 1987, Stable solutions of inverse problems, Vieweg, BraunschweigGoogle Scholar
  3. 12.
    Bjork, A., 1990, Least squares methods, In Ciarlet, P.G., and Lions, J.L., eds, Handbook of Numerical Analysis, North-Holland, AmsterdamGoogle Scholar
  4. 16.
    Chavent, G., 1979, Identification of distributed parameter systems: About the output least squares method, its implementation and identifiability, In Proceedings of the IFAC Symposium on Identification, Pergamon, pp 85–97Google Scholar
  5. 17.
    Chavent, G., 1986, Identifiability of parameters in the output least square formulation, In Walter, E., ed, Structural Identifiability of Parametrics Model, chapter 6, Pergamon Press, pp 67–74Google Scholar
  6. 19.
    Chavent, G., 1991, New size ×curvature conditions for strict quasi-convexity of sets, SIAM J. Contr. Optim. 29(6), 1348–1372zbMATHCrossRefMathSciNetGoogle Scholar
  7. 20.
    Chavent, G., 1991, Quasi-convex sets and size ×curvature condition, application to nonlinear inversion, J. Appl. Math. Optim. 24(1), 129–169zbMATHCrossRefMathSciNetGoogle Scholar
  8. 26.
    Chavent, G., Kunisch, K., 1993, Regularization in state space, M2AN 27, 535–564Google Scholar
  9. 28.
    Chavent, G., Kunisch, K., 1996, On weakly nonlinear inverse problems, SIAM J. Appl. Math. 56(2), 542–572zbMATHCrossRefMathSciNetGoogle Scholar
  10. 29.
    Chavent, G., Kunisch, K., 1998, State space regularization: Geometric theory, Appl. Math. Opt. 37, 243–267zbMATHCrossRefMathSciNetGoogle Scholar
  11. 37.
    Engl, H.W., Kunisch, K., Neubauer, A., 1989, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Probl. 5, 523–540zbMATHCrossRefMathSciNetGoogle Scholar
  12. 38.
    Engl, H.W., Hanke, M., Neubauer, A., 1996, Regularization of inverse problems, Kluwer, Dordrecht, p 321, (Mathematics and its applications, 375) ISBN 0-7923-4157-0Google Scholar
  13. 43.
    Groetsch, C.W., 1984, The theory of Tykhonov regularization for Fredholm equations of the first kind, Research Notes in Mathematics 105, Pitman, BostonGoogle Scholar
  14. 46.
    Isakov, V., 1998, Inverse problems for partial differential equations, Springer, Berlin, p 284 (Applied mathematical sciences, 127) ISBN 0-387-98256-6Google Scholar
  15. 47.
    Ito, K., Kunisch, K., 1994, On the injectivity and linearization of the coefficient to solution mapping for elliptic boundary value problems, J. Math. Anal. Appl. 188(3), 1040–1066zbMATHCrossRefMathSciNetGoogle Scholar
  16. 50.
    Lavaud, B., Kabir, N., Chavent, G., 1999, Pushing AVO inversion beyond linearized approximation, J. Seismic Explor. 8, 279–302Google Scholar
  17. 54.
    Levenberg, K., 1944, A method for the solution of certain nonlinear problems in least squares, Appl. Math. 11, 164–168MathSciNetGoogle Scholar
  18. 55.
    Lines, L.R., Treitel, S., Tutorial: A review of least-squares inversion and its application to geophysical problems, Geophys. Prospect. 39, 159–181Google Scholar
  19. 56.
    Lions, J.L., 1969, Quelques Méthodes de Résolution des Problèmes aux limites Non Linéaires, Dunod, PariszbMATHGoogle Scholar
  20. 59.
    Louis, A.K., 1989, Inverse und Schlecht Gestellte Probleme, Teubner, StuttgartzbMATHGoogle Scholar
  21. 62.
    Marquardt, D.W., 1963, An algorithm for least squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11, 431–441zbMATHCrossRefMathSciNetGoogle Scholar
  22. 63.
    Morozov, V.A., 1984, Methods for solving incorrectly posed problems, Springer, New YorkGoogle Scholar
  23. 67.
    Neubauer, A., 1989, Tikhonov regularization for nonlinear ill-posed problems: Optimal convergence rate and finite dimensional approximation, Inverse Probl. 5, 541–558zbMATHCrossRefMathSciNetGoogle Scholar
  24. 70.
    Richter, G.R., 1981, An inverse problem for the steady state diffusion equation, SIAM J. Math. 4, 210–221Google Scholar
  25. 75.
    Tikhonov, A.N., Arsenin, V., 1977, Solutions of ill-posed problems, Wiley, New YorkzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Ceremade, Université Paris-DauphineParis Cedex 16France
  2. 2.Inria-RocquencourtLe Chesnay CedexFrance

Personalised recommendations

Cite chapter

Buy options