Nonlinear Least Squares for Inverse Problems pp 209-270 | Cite as
Regularization of Nonlinear Least Squares Problems
Chapter
First Online:
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).
$$\hat{x}\quad \mbox{ minimizes }\quad J(x) = \frac{1} {2}\|\varphi (x) - {z\|}_{F}^{2}\quad \mbox{ over }\quad C. $$
(5.1)
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.
Keywords
Regularization Parameter Regularity Condition Tangent Cone Observation Operator Deflection Condition
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.
References
- 1.Aki, K., Richards, P.G., 1980, Quantitative seismology: Theory and methods, W.H. Freeman, New YorkGoogle Scholar
- 8.Baumeister, J., 1987, Stable solutions of inverse problems, Vieweg, BraunschweigGoogle Scholar
- 18.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 452–463Google Scholar
- 26.Chavent, G., Kunisch, K., 1993, Regularization in state space, M2AN 27, 535–564Google Scholar
- 27.Chavent, G., Kunisch, K., 1994, Convergence of Tikhonov regularization for constrained ill-posed inverse problems, Inverse Probl. 10, 63–76zbMATHCrossRefMathSciNetGoogle Scholar
- 28.Chavent, G., Kunisch, K., 1996, On weakly nonlinear inverse problems, SIAM J. Appl. Math. 56(2), 542–572zbMATHCrossRefMathSciNetGoogle Scholar
- 30.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, 423zbMATHCrossRefMathSciNetGoogle Scholar
- 31.Chavent, G., Lemonnier, P., 1974, Identification de la non linéarité d’une équation parabolique quasilinéaire, J. Appl. Math. Optim. 1(2), 121–162zbMATHCrossRefMathSciNetGoogle Scholar
- 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
- 39.Girault, V., Raviart, P.A., 1979, Finite element methods for Navier-Stokes equations, Springer, BerlinCrossRefGoogle Scholar
- 54.Levenberg, K., 1944, A method for the solution of certain nonlinear problems in least squares, Appl. Math. 11, 164–168MathSciNetGoogle Scholar
- 56.Lions, J.L., 1969, Quelques Méthodes de Résolution des Problèmes aux limites Non Linéaires, Dunod, PariszbMATHGoogle Scholar
- 62.Marquardt, D.W., 1963, An algorithm for least squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11, 431–441zbMATHCrossRefMathSciNetGoogle Scholar
- 65.Neubauer, A., 1987, Finite dimensional approximation of constrained Tikhonov-regularized solutions of ill-posed linear operator equations, Math. Comput. 48, 565–583zbMATHMathSciNetGoogle Scholar
- 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