We consider a regularized Levenberg–Marquardt method for solving nonlinear ill-posed inverse problems. We use the discrepancy principle to terminate the iteration. Under certain conditions, we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source-wise representations. Mathematics Subject Classification (2000) 65J15 ·...

Many questions in science and engineering give rise to ill-posed inverse problems whose solution is known to satisfy box constrains, such as nonnegativity. The solution of discretized versions of these problems is highly sensitive to perturbations in the data, discretization errors, and round-off errors introduced during the computations. It is therefore often beneficial to impose known constra...

On the Parametrization of Ill-posed Inverse Problems Arising from Elliptic Partial Differential Equations by Fernando Guevara Vasquez Electric impedance tomography (EIT) consists in finding the conductivity inside a body from electrical measurements taken at its surface. This is a severely ill-posed problem: any numerical inversion scheme requires some form of regularization. We present inversi...

Design of experiments for discrete ill-posed problems is a relatively new area of research. While there has been some limited work concerning the linear case, little has been done to study design criteria and numerical methods for ill-posed nonlinear problems. We present an algorithmic framework for nonlinear experimental design with an efficient numerical implementation. The data are modeled a...

In this paper we consider the regularity of the trust region-cg algorithm, when it is applied to nonlinear ill-posed iverse problems. The trust region algorithm can be viewed as a regularization method, but it diiers from the traditional regularization method, because no penalty term is need. Thus, the determing of the so-called regular-ization parameter in a standard regularization method is a...

We compare the performance of several robust large-scale minimization algorithms for the unconstrained minimization of an ill-posed inverse problem. The parabolized Navier–Stokes equation model was used for adjoint parameter estimation. The methods compared consist of three versions of nonlinear conjugate-gradient (CG) method, quasiNewton Broyden–Fletcher–Goldfarb–Shanno (BFGS), the limited-mem...

Saquib, Suhail S. Ph. D., Purdue University, May 1997. Edge-Preserving Models and Efficient Algorithms for Ill-Posed Inverse Problems in Image Processing. Major Professor: Charles A. Bouman. The goal of this research is to develop detail and edge-preserving image models to characterize natural images. Using these image models, we have developed efficient unsupervised algorithms for solving ill-...

Ill-posed inverse problems are widely en countered in computer vision, examples include shape from shading, surface reconstruction from sparse data and optic flow. Unique sol utions to these problems are conventionally found by minimizing an objective function regu larized by a smoothness constraint. However, objective functions of this form often contain many local minima, making it difficult ...

We introduce a reconstruction framework that can account for shape related a priori information in ill-posed linear inverse problems in imaging. It is a variational scheme that uses a shape functional defined using deformable templates machinery from shape theory. As proof of concept, we apply the proposed shape based reconstruction to 2D tomography with very sparse measurements, and demonstrat...