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.
Here the problem is to solve the nonlinear equation eq:eq F(x)=y in the ill-posed case. Typically a solution is found by introducing a regularization term weighted with a positive parameter. There are two possible approaches. The first one employs iterative methods which deal directly with problem , see @cite_25 . In this case an iterative process is set up by minimizing at each step a simplified regularized problem (generally linear) having the structure of @math --- with a weight for @math varying at each iteration. Within this class of methods, one popular choice is the iteratively regularized Gauss-Newton method , see @cite_7 @cite_34 @cite_41 @cite_3 . Anyway, we remark that, despite the name, that algorithm is different from any kind of Gauss-Newton optimization method above, since it is not designed to optimize'' any objective functional @math , but, in fact, it directly looks for an exact solution of the equation .
@cite_7 @cite_41 @cite_3 @cite_34 @cite_25
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In this paper we prove that the iteratively regularized Gauss-Newton method is a locally convergent method for solving nonlinear ill-posed problems, provided the nonlinear operator satisfies a certain smoothness condition. For perturbed data we propose a priori and a posteriori stopping rules that guarantee convergence of the iterates, if the noise level goes to zero. Under appropriate closeness and smoothness conditions on the exact solution we obtain the same convergence rates as for linear ill-posed problems. In this paper we consider the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed inverse problems. Under merely the Lipschitz condition, we prove that this method together with an a posteriori stopping rule defines an order optimal regularization method if the solution is regular in some suitable sense. Iterative algorithms of Gauss–Newton type for the solution of nonlinear least squares problems are considered. They separate the variables into two sets in such a way that in each iteration, optimization with respect to the first set is performed first, and corrections to those of the second after that. The linear-nonlinear case, where the first set consists of variables that occur linearly, is given special attention, and a new algorithm is derived which is simpler to apply than the variable projection algorithm as described by Golub and Pereyra, and can be performed with no more arithmetical operations than the unseparated Gauss–Newton algorithm. A detailed analysis of the asymptotical convergence properties of both separated and unseparated algorithms is performed. It is found that they have comparable rates of convergence, and all converge almost quadratically for almost compatible problems. Simpler separation schemes, on the other hand, converge only linearly. An efficient and simple computer impleme... In this paper we consider the iteratively regularized Gauss–Newton method (IRGNM) in a Banach space setting and prove optimal convergence rates under approximate source conditions. These are related to the classical concept of source conditions that is available only in Hilbert space. We provide results in the framework of general index functions, which include, e.g. Holder and logarithmic rates. Concerning the regularization parameters in each Newton step as well as the stopping index, we provide both a priori and a posteriori strategies, the latter being based on the discrepancy principle.