Ill-posed Problems In Hilbert Spaces Research Articles

Saturation of Regularization Methods for Linear Ill-Posed Problems in Hilbert Spaces

We prove the saturation of methods for solving linear ill-posed problems in Hilbert spaces for a wide class of regularization methods. It turns out that, under a certain convexity assumption, saturation must necessarily occur. We provide easy to verify assumptions, which allow us to calculate the rate at which saturation occurs.

On Landweber iteration for nonlinear ill-posed problems in Hilbert scales

In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index \(k\) for exact data and in terms of the noise level \(\delta\) for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented confirming the theoretical ones.

On the discretization and regularization of ill-posed problems with noncompact operators

In this paper, discretization and regularization schemes for linear ill-posed problems in Hilbert spaces with noncompact operators and noisy data are studied. Convergence results are obtained where the limiting processes in discretization and regularization parameters may be performed independently. As an application of an independent interest, the inverse problem for ground water filtration is considered. Actually, this problem gave us an impulse for the general study of this paper.

Iteration methods for convexly constrained ill-posed problems in hilbert space

Minimization problems in Hilbert space with quadratic objective function and closed convex constraint set C are considered. In case the minimum is not unique we are looking for the solution of minimal norm. If a problem is ill-posed, i.e. if the solution does not depend continuously on the data, and if the data are subject to errors then it has to be solved by means of regularization methods. The regularizing properties of some gradient projection methods—i.e. convergence for exact data, order of convergence under additional assumptions on the solution and stability for perturbed data—are the main issues of this paper.

The necessary and sufficient conditions for the approximations of linear ill-posed problems in a Hilbert space to converge

The weakest operational topology is derived, guaranteeing the convergence of permutations of solutions obtained by Tikhonov's regularizations method and the discrepancy method of linearized ill-posed problems in Hilbert spaces. Conditions are also derived for the systems of subspaces and operators which are necessary and sufficient for the approximations of the perturbations considered to converge.

On the quasi-optimality principle for ill-posed problems in Hilbert space

The properties of a quasi-optimal choice of the regularization parameter for the extremum of the smoothing Tikhonov functional for a linear equation of the first kind in an infinitely dimensional Hilbert space are studied; this is done within the limits of absolutely exact calculations. The concept of an automatically regularizing parameter is introduced, and a certain subset in the set of all kinds of initial data of the problem is separated in such a way that an algorithm based on the quasi-optimality principle has automatically regularizing properties on this subset.

The critical level of discrepancy in regularization methods

A class of methods of solving linear ill-posed problems in Hilbert space is studied. The regularization parameter is chosen from the condition for the norm of the discrepancy to be equal to the level of accuracy in specifying the right-hand side of the equation. This choice of regularization parameter leads in certain cases to a convergent order-wise optimal process, and in others to a divergent process. The case of an approximately specified operator is also considered.

Necessary and sufficient conditions for convergence of regularization methods for solving linear operator equations of the first kind

In this paper, we give necessary and sufficient conditions for weak and strong convergence of general regularization methods for the solution of linear ill-posed problems in Hilbert space. As special cases we obtain convergence criteria for Tychonoff regularization, for a regularization method based on Showalter's integral formula, and for regularization by truncated iteration