C. W. Groetsch

Inverse problems in the mathematical sciences

Contents: Introduction - Inverse problems modeled by integral equations of the first kind: Causation - Parameter estimation in differential equations: Model identification - Mathematical background for inverse problems - Some methodology for inverse problems - An annotated bibliography on inverse problems.

Nonstationary iterated Tikhonov regularization

A convergence rate is established for nonstationary iterated Tikhonov regularization, applied to ill-posed problems involving closed, densely defined linear operators, under general conditions on the iteration parameters. It is also shown that an order-optimal accuracy is attained when a certain a posteriori stopping rule is used to determine the iteration number.

A note on segmenting Mann iterates

W. R. Mann [5] introduced the following general iterative procedure: Suppose A = (a& is an infinite, lower triangular, regular row-stochastic matrix. If E is a closed convex subset of a Banach space and T is a continuous mapping of E into itself and xi E E, then M(x, , A, T) is the process defined by 71

Differentiation of approximately specified functions

Experienced doodlers know that the graph of a function can wiggle tightly around, while not departing far from, a given smooth curve. And anyone who suffers from motion sickness is painfully aware of the violent accelerations and decelerations a callous cabby can accomplish at low speeds in crowded city traffic. These examples, one geometrical and one physical, are expressions of the mathematical fact that uniformly close functions need not have uniformly close derivatives. An example from elementary calculus of extreme behavior of the derivative is provided by the function

Inverse Scale Space Theory for Inverse Problems

In this paper we derive scale space methods for inverse problems which satisfy the fundamental axioms of fidelity and causality and we provide numerical illustrations of the use of such methods in deblurring. These scale space methods are asymptotic formulations of the Tikhonov-Morozov regularization method. The analysis and illustrations relate diffusion filtering methods in image processing to Tikhonov regularization methods in inverse theory.

Weakly closed nonlinear operators and parameter identification in parabolic equations by tikhonov regularization

This paper is devoted to studying convergence rates for the tikhonov regularization of nonlinear ill–posed problems from a geometrical point of view. Also the non–attainable case is considered. In our theory, the weak closedness of the operator defining the equation plays a central role. We prove the weak closedness of this operator for two parameter estimation problems in parabolic equations.

Spectral methods for linear inverse problems with unbounded operators

Abstract Spectral theory for bounded linear operators is used to develop a general class of approximation methods for the Moore-Penrose generalized inverse of a closed, densely defined linear operator. Issues of convergence and stability are addressed and the methods are modified to provide a stable class of methods for evaluation of unbounded linear operators.

Comments on Morozov’s Discrepancy Principle

The choice of regularization parameter by Morozov’s principle is characterized in a new way and is related to another parameter choice strategy. An asymptotic order of accuracy is derived which is essentially best possible and a discrepancy principle is developed in a finite element context.

Non-stationary iterated Tikhonov-Morozov method and third-order differential equations for the evaluation of unbounded operators

In this paper we analyse the non-stationary iterative Tikhonov-Morozov method analytically and numerically for the stable evaluation of differential operators and for denoizing images. A relationship between non-stationary iterative Tikhonov-Morozov regularization and a filtering technique based on a differential equation of third order is established and both methods are shown to be effective for denoizing images and for the stable evaluation of differential operators. The theoretical results are verified numerically on model problems in ultrasound imaging and numerical differentiation.

Convergence analysis of a regularized degenerate kernel method for Fredholm integral equations of the first kind

We present a simple asymptotic convergence analysis of a degenerate kernel method for Fredholm integral equations of the first kind which is based on a quadrature of the variational equation for the Tikhonov functional. Convergence theorems are proved in both the mean-square and uniform norms and the Tikhonov-regularity of the method for the case of inexact data is established in both norms