M. Thamban Nair

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.

Lavrentiev Regularization for Linear Ill-Posed Problems under General Source Conditions

In this paper we study the problem of identifying the solution x† of linear ill-posed problems Ax = y with non-negative and self-adjoint operators A on a Hilbert space X where instead of exact data y noisy data y ∈ X are given satisfying ‖y − y‖ ≤ δ with known noise level δ. Regularized approximations xα are obtained by the method of Lavrentiev regularization, that is, xα is the solution of the singularly perturbed operator equation Ax + αx = y, and the regularization parameter α is chosen either a priori or a posteriori by the rule of Raus. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximations provide order optimal error bounds on the set M . Our results cover the special case of finitely smoothing operators A and extend recent results for infinitely smoothing operators. In addition, we generalize our results to the method of iterated Lavrentiev regularization of order m and discuss a special ill-posed problem arising in inverse heat conduction.

A modified Newton--Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations

Recently, a new iterative method, called Newton-Lavrentiev regularization (NLR) method, was considered by George (2006) for regularizing a nonlinear ill-posed Hammerstein-type operator equation in Hilbert spaces. In this paper we introduce a modified form of the NLR method and derive order optimal error bounds by choosing the regularization parameter according to the adaptive scheme considered by Pereverzev and Schock (2005).

A simplified generalized Gauss-Newton method for nonlinear ill-posed problems

Iterative regularization methods for nonlinear ill-posed equations of the form F (x) = y, where F : D(F ) ⊂ X → Y is an operator between Hilbert spaces X and Y , usually involve calculation of the Frechet derivatives of F at each iterate and at the unknown solution x†. In this paper, we suggest a modified form of the generalized Gauss-Newton method which requires the Frechet derivative of F only at an initial approximation x0 of the solution x†. The error analysis for this method is done under a general source condition which also involves the Frechet derivative only at x0. The conditions under which the results of this paper hold are weaker than those considered by Kaltenbacher (1998) for an analogous situation for a special case of the source condition.

The trade-off between regularity and stability in Tikhonov regularization

When deriving rates of convergence for the approximations generated by the application of Tikhonov regularization to ill-posed operator equations, assumptions must be made about the nature of the stabilization (i.e., the choice of the seminorm in the Tikhonov regularization) and the regularity of the least squares solutions which one looks for. In fact, it is clear from works of Hegland, Engl and Neubauer and Natterer that, in terms of the rate of convergence, there is a trade-off between stabilization and regularity. It is this matter which is examined in this paper by means of the best-possible worst-error estimates. The results of this paper provide better estimates than those of Engl and Neubauer, and also include and extend the best possible rate derived by Natterer. The paper concludes with an application of these results to first-kind integral equations with smooth kernels.

On a generalized arcangeli's method for tikhonov regularization with inexact data

A class of parameter choice strategies which include an approximated version of the Arcangelis method is considered for ill-posed operator equations when the data are not known exactly. A particular case of the method is a generalization and modification of the Martis method, in which case the results include, and in certain cases improve the conclusions of Engl and Neubauer (1985) under weaker conditions.

An a posteriori parameter choice for simplified regularization of ill-posed problems

An a posteriori parameter choice strategy is proposed for the simplified regularization of ill-posed problems where no information about the smoothness of the unknown solution is required. If the smoothness of the solution is known then, as a particular case, the optimal rate is achieved. Our result also includes a recent result of Guacanme (1990).

A generalization of Arcangeli's method for ill-posed problems leading to optimal rates

Schock (1984) considered a general a posteriori parameter choice strategy for the regularization of ill-posed problems which provide nearly the optimal rate of convergence. We improve the result of Schock and give a class of parameter choice strategies leading to optimal rates As a particular case we prove that the Arcangelis method do give optimal rate of convergence.

On Morozov’s Method for Tikhonov Regularization as an Optimal Order Yielding Algorithm

It is shown that Tikhonov regularization for an ill-posed operator equation Kx = y using a possibly unbounded regularizing operator L yields an order-optimal algorithm with respect to certain stability set when the regularization parameter is chosen according to Morozovs discrepancy principle. A more realistic error estimate is derived when the operators K and L are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also estimates available under the Hilbert scales approach.

Error bounds and parameter choice strategies for simplified regularization in Hilbert scales

Simplified regularization in the setting of Hilbert scales has been considered for obtaining stable approximate solutions for ill-posed operator equations. The derived error estimates using an a posteriori as well as an a priori parameter choice strategy are shown to be of optimal order with respect to certain natural assumptions on the ill-posedness of the equation.