Eberhard Schock

On the Adaptive Selection of the Parameter in Regularization of Ill-Posed Problems

We study the possibility of using the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko [Z. Vychisl. Mat. Mat. Fiz., 4 (1964), pp. 564-571] and to an adaptation scheme proposed in a statistical context by Lepskii [Theory Probab. Appl., 36 (1990), pp. 454-466] is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for nonlinear ill-posed problems, and an inverse problem of the two-dimensional profile reconstruction.

Morozov's Discrepancy Principle under General Source Conditions

In this paper we study linear ill-posed problems Ax = y in a Hilbert space setting where instead of exact data y noisy data y^delta are given satisfying |y - y^delta| <= delta with known noise level delta. Regularized approximations are obtained by a general regularization scheme where the regularization parameter is chosen from Morozovs discrepancy principle. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximation provides order optimal error bounds on the set M. Our results cover the special case of finitely smoothing operators A and extends recent results for infinitely smoothing operators.

Morozov's Discrepancy Principle for Tikhonov Regularization of Severely Ill-Posed Problems in Finite-Dimensional Subspaces

In this paper severely ill-posed problems are studied which are represented in the form of linear operator equations with infinitely smoothing operators but with solutions having only a finite smoothness. It is well known, that the combination of Morozovs discrepancy principle and a finite dimensional version of the ordinary Tikhonov regularization is not always optimal because of its saturation property. Here it is shown, that this combination is always order-optimal in the case of severely ill-posed problems.

PARAMETER CHOICE BY DISCREPANCY PRINCIPLES FOR THE APPROXIMATE SOLUTION OF ILL-POSED PROBLEMS

A general a posteriori strategy for choosing the regularization parameter as a function of the error level is given which provides nearly the optimal rate of convergence.

Ill-posed equations with transformed argument

We discuss the operator transforming the argument of a function in the L2-setting. Here this operator is unbounded and closed. For the approximate solution of ill-posed equations with closed operators, we present a new view on the Tikhonov regularization.

Semi-iterative methods for the approximate solution of Ill-posed problems

SummaryWe study semi-iterative (two step interative) methods of the form

Über die Konvergenzgeschwindigkeit projektiver Verfahren. II

x_{n + 1} = T^* T(\alpha x_n + \gamma x_{n - 1} ) + \beta x_n + (1 - \beta )x_{n - 1} - (\alpha + \gamma )T^* y

COMPARISON PRINCIPLES FOR ITERATIVE METHODS

for the approximate solution of ill-posed or ill-conditioned linear equationsTx=y in (infinite or finite dimensional) Hilbert spaces. We present results on convergence, convergence rates, the influence of perturbed data, and on the comparison of different methods.