Abstract
We consider in this chapter various approaches for the regularization of the general NLS problem (1.10), recalled here for convenience:
and we suppose throughout the chapter that it satisfies the minimum set of hypothesis (1.12) or (4.2).
We develop three of the five approaches described in Sect. 1.3.4 of the introduction: Levenberg–Marquardt–Tychonov (LMT), state-space, and adapted regularization. The two remaining approaches, regularization by parameterization and regularization by size reduction of the admissible parameter set, have been already addressed in Chaps. 3 and 4, respectively.
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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
Chavent, G., 1990, A new sufficient condition for the wellposedness of nonlinear least-squares problems arising in identification and control, In Bensoussan, A., and Lions, J.L., eds, Lecture Notes in Control and Information Sciences 144, Springer, Berlin, pp 452–463
Chavent, G., Kunisch, K., 1993, Regularization in state space, M2AN 27, 535–564
Chavent, G., Kunisch, K., 1994, Convergence of Tikhonov regularization for constrained ill-posed inverse problems, Inverse Probl. 10, 63–76
Chavent, G., Kunisch, K., 1996, On weakly nonlinear inverse problems, SIAM J. Appl. Math. 56(2), 542–572
Chavent, G., Kunisch, K., 2002, The output least square identifiability of the diffusion coefficient from an H 1 observation in a 2-D elliptic equation, ESAIM: Contr. Optim. Calculus Variations 8, 423
Chavent, G., Lemonnier, P., 1974, Identification de la non linéarité d’une équation parabolique quasilinéaire, J. Appl. Math. Optim. 1(2), 121–162
Engl, H.W., Kunisch, K., Neubauer, A., 1989, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Probl. 5, 523–540
Girault, V., Raviart, P.A., 1979, Finite element methods for Navier-Stokes equations, Springer, Berlin
Levenberg, K., 1944, A method for the solution of certain nonlinear problems in least squares, Appl. Math. 11, 164–168
Lions, J.L., 1969, Quelques Méthodes de Résolution des Problèmes aux limites Non Linéaires, Dunod, Paris
Marquardt, D.W., 1963, An algorithm for least squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11, 431–441
Neubauer, A., 1987, Finite dimensional approximation of constrained Tikhonov-regularized solutions of ill-posed linear operator equations, Math. Comput. 48, 565–583
Tikhonov, A.N., Arsenin, V., 1977, Solutions of ill-posed problems, Wiley, New York
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Chavent, G. (2009). Regularization of Nonlinear Least Squares Problems. In: Nonlinear Least Squares for Inverse Problems. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2785-6_5
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DOI: https://doi.org/10.1007/978-90-481-2785-6_5
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