An extension of the Gauss-Newton algorithm is proposed to find local minimizers of penalized nonlinear least squares problems, under generalized Lipschitz assumptions. Convergence results of local type are obtained, as well as an estimate of the radius of the convergence ball. Some applications for solving constrained nonlinear equations are discussed and the numerical performance of the method is assessed on some significant test problems.
@cite_35: Preface. 1. Introduction: Examples of Inverse Problems. 2. Ill-Posed Linear Operator Equations. 3. Regularization Operators. 4. Continuous Regularization Methods. 5. Tikhonov Regularization. 6. Iterative Regularization Methods. 7. The Conjugate Gradient Method. 8. Regularization with Differential Operators. 9. Numerical Realization. 10. Tikhonov Regularization of Nonlinear Problems. 11. Iterative Methods for Nonlinear Problems. A. Appendix: A.1. Weighted Polynomial Minimization Problems. A.2. Orthogonal Polynomials. A.3. Christoffel Functions. Bibliography. Index.
@cite_27: In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by a one-homogeneous (typically weighted l p ) penalty on the coefficients (or isometrically transformed coefficients) of such expansions. For (p < 2), the regularized solution will have a sparser expansion with respect to the basis or frame under consideration. The computation of the regularized solution amounts in our setting to a Landweber-fixed-point iteration with a projection applied in each fixed-point iteration step. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse single photon emission computerized tomography (SPECT) problem.
@cite_17: We shall be concerned with the construction of Tikhonov-based iteration schemes for solving nonlinear operator equations. In particular, we are interested in algorithms for the computation of a minimizer of the Tikhonov functional. To this end, we introduce a replacement functional, that has much better properties than the classical Tikhonov functional with nonlinear operator. Namely, the replacement functional is globally convex and can effectively be minimized by a fixed point iteration. On the basis of the minimizers of the replacement functional, we introduce an iterative algorithm that converges towards a critical point of the Tikhonov functional, and under additional assumptions for the nonlinear operator F, to a global minimizer. Combining our iterative strategy with an appropriate parameter selection rule, we obtain convergence and convergence rates. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse SPECT (single photon emission computerized tomography) problem.
An alternative approach is the classical Tikhonov method @cite_35 replacing by the minimization of the associated Tikhonov functional. A problem of the same type of @math arises, with @math weighted by a properly chosen parameter. Up to our knowledge, besides our method, the papers by Ramlau and Teschke, e.g. @cite_17 @cite_27 , are the only ones providing algorithms for the minimization of such type of functionals. In addition, the scheme they propose is different from ours and in general it converges only weakly. On the other hand, our algorithm assumes the derivative @math to be injective with closed range --- a hypothesis not suitable for handling the ill-posed case.