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
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References
Aki, K., Richards, P.G., 1980, Quantitative seismology: Theory and methods, W.H. Freeman, New York
Baumeister, J., 1987, Stable solutions of inverse problems, Vieweg, Braunschweig
Bjork, A., 1990, Least squares methods, In Ciarlet, P.G., and Lions, J.L., eds, Handbook of Numerical Analysis, North-Holland, Amsterdam
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 85–97
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 67–74
Chavent, G., 1991, New size ×curvature conditions for strict quasi-convexity of sets, SIAM J. Contr. Optim. 29(6), 1348–1372
Chavent, G., 1991, Quasi-convex sets and size ×curvature condition, application to nonlinear inversion, J. Appl. Math. Optim. 24(1), 129–169
Chavent, G., Kunisch, K., 1993, Regularization in state space, M2AN 27, 535–564
Chavent, G., Kunisch, K., 1996, On weakly nonlinear inverse problems, SIAM J. Appl. Math. 56(2), 542–572
Chavent, G., Kunisch, K., 1998, State space regularization: Geometric theory, Appl. Math. Opt. 37, 243–267
Engl, H.W., Kunisch, K., Neubauer, A., 1989, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Probl. 5, 523–540
Engl, H.W., Hanke, M., Neubauer, A., 1996, Regularization of inverse problems, Kluwer, Dordrecht, p 321, (Mathematics and its applications, 375) ISBN 0-7923-4157-0
Groetsch, C.W., 1984, The theory of Tykhonov regularization for Fredholm equations of the first kind, Research Notes in Mathematics 105, Pitman, Boston
Isakov, V., 1998, Inverse problems for partial differential equations, Springer, Berlin, p 284 (Applied mathematical sciences, 127) ISBN 0-387-98256-6
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), 1040–1066
Lavaud, B., Kabir, N., Chavent, G., 1999, Pushing AVO inversion beyond linearized approximation, J. Seismic Explor. 8, 279–302
Levenberg, K., 1944, A method for the solution of certain nonlinear problems in least squares, Appl. Math. 11, 164–168
Lines, L.R., Treitel, S., Tutorial: A review of least-squares inversion and its application to geophysical problems, Geophys. Prospect. 39, 159–181
Lions, J.L., 1969, Quelques Méthodes de Résolution des Problèmes aux limites Non Linéaires, Dunod, Paris
Louis, A.K., 1989, Inverse und Schlecht Gestellte Probleme, Teubner, Stuttgart
Marquardt, D.W., 1963, An algorithm for least squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11, 431–441
Morozov, V.A., 1984, Methods for solving incorrectly posed problems, Springer, New York
Neubauer, A., 1989, Tikhonov regularization for nonlinear ill-posed problems: Optimal convergence rate and finite dimensional approximation, Inverse Probl. 5, 541–558
Richter, G.R., 1981, An inverse problem for the steady state diffusion equation, SIAM J. Math. 4, 210–221
Tikhonov, A.N., Arsenin, V., 1977, Solutions of ill-posed problems, Wiley, New York
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Chavent, G. (2009). Nonlinear Inverse Problems: Examples and Difficulties. In: Nonlinear Least Squares for Inverse Problems. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2785-6_1
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