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_7: 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.
@cite_41: 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.
@cite_3: We examine preconditioning techniques for exponentially ill-posed problems which significantly reduce the total complexity of regularized Newton methods. The construction of the preconditioners exploits the close connection of the CG method, which is applied to solve the linear systems, and Lanczos' method, which is used to determine spectral data to construct and update spectral preconditioners. In examples from acoustic and electromagnetic scattering problems we show the superiority of our preconditioned Newton methods when compared with other kinds of Newton methods.
@cite_34: 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.
@cite_25: In this paper we discuss the construction, analysis and implementation of iterative schemes for the solution of inverse problems based on total variation regularization. Via different approximations of the nonlinearity we derive three different schemes resembling three well-known methods for nonlinear inverse problems in Hilbert spaces, namely iterated Tikhonov, Levenberg–Marquardt and Landweber. These methods can be set up such that all arising subproblems are convex optimization problems, analogous to those appearing in image denoising or deblurring. We provide a detailed convergence analysis and appropriate stopping rules in the presence of data noise. Moreover, we discuss the implementation of the schemes and the application to distributed parameter estimation in elliptic partial differential equations.
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 .