Thamban M Nair

Regularized versions of continuous analogues of Newton's method and modified Newton's method for obtaining approximate solutions to a nonlinear ill-posed operator equation of the form F(u)=f, where F is a monotone operator defined from a Hilbert space H into itself, have been studied in the literature. For such methods, error estimates are available only under Holder-type source conditions on the solution. In this paper, presenting the background materials systematically, we derive error estimates under a general source condition. For the special case of the regularized modified Newton's method under a Holder-type source condition, we also carry out error analysis by replacing the monotonicity of F by a weaker assumption. This analysis facilitates inclusion of certain examples of parameter identification problems, which was not possible otherwise. Moreover, an a priori stopping rule is considered when we have a noisy data f instead of f. This rule yields not only convergence of the regularized approximations to the exact solution as the noise level tends to zero but also provides convergence rates that are optimal under the source conditions considered.

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An "unregularized" use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness. © 2006 Elsevier Inc. All rights reserved.

For solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the operator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions. © 2008, Institute of Mathematics, NAS of Belarus. All rights reserved.

In this paper we study linear ill-posed problems Ax = y in a Hilbert space setting where instead of exact data y noisy data yδ are given satisfying

Let X1 and X2 be complex Banach spaces, and let A1 ∈ BL(X1), A2 ∈ BL(X2), A3 ∈ BL(X1, X2) and A4 ∈ BL(X2, X1). We propose an iterative procedure which is a modified form of Newton's iterations for obtaining approximations for the solution R ∈ BL(X1, X2) of the Riccati equation A2R - RA1 = A3 + RA4R, and show that the convergence of the method is quadratic. The advantage of the present procedure is that the conditions imposed on the operators A1, A2, A3, A4 are weaker than the corresponding conditions for Newton's iterations, considered earlier by Demmel (1987), Nair (1989) and Nair (1990) in the context of obtaining error bounds for approximate spectral elements. Also, we discuss an application of the procedure to spectral approximation under perturbations of the operator.

An iterated version of the Lavrentiev's method, in the setting of a Banach space, is suggested for obtaining stable approximate solutions for the ill-posed operator equation Au = v, when the data A and v are known only approximately. In the setting of a Hilbert space with appropriate a priori parameter choice, the suggested procedure yields order optimal error estimates. An iterated version of Tikhonov regularization yielding order optimal error estimate is a special case of the procedure. The assumption on the approximating operators show that the finite dimensional system arising out of it would be of smaller size for larger iterates. This aspect is compared with an assumption of [3] for a degenerate kernel method for integral equations of the first kind. © 2000, by Walter de Gruyter GmbH & Co. All rights reserved.

We present a new procedure in the finite dimensional setting for the recently developed method of mollifiers for obtaining stable approximate solutions for ill-posed equations. A particular case of the derived error estimate improves upon similar results obtained recently by Rieder and Schuster (2000). © VSP 2004.

Pereverzev (1995) considered Tikhonov regularization combined with a modified form of a projection method for obtaining stable approximate solutions for ill-posed operator equations. He showed, under a certain a priori choice of the regularization parameter and a specific smoothness assumption on the solution, that the method yields the optimal order with less computational information, in the sense of complexity, than the projection method considered by Plato and Vainikko (1990). In this paper we apply a modified form of the Arcangeli's discrepancy principle for choosing the regularization parameter, and show that the conclusions of Pereverzev still hold. In fact, we do the analysis using a modified form of the generalized Arcangeli's method suggested by Schock (1984) under more flexible smoothness assumption on the solution, as has been done by George and Nair (1998), and derive the optimal result as a special case. Moreover, we compare the computational complexity of the present method with two tradit ional projection methods in the case of a priori parameter choice, and also discuss the computational complexity required to implement the suggested discrepancy principle.

We present a. new procedure in the finite dimensional setting for the recently developed method of mollifiers for obtaining stable approximate solutions for ill-posed equations. A particular case of the derived error estimate improves upon similar results obtained recently by Rieder and Schuster (2000).

Tikhonov regularization is one of the widely used procedures for the regularization of nonlinear as well as linear ill-posed problems. The error analysis carried out in most of the works that appeared in last few years on Tikhonov regularization of nonlinear ill-posed problems are under Hölder type source conditions on the unknown solution which is known to be applicable only for mildly ill-posed problems. In this paper we consider Tikhonov regularization of nonlinear ill-posed problems and derive order optimal error estimate under a general source condition together with an a posteriori parameter rule proposed by Scherzer et al., which is applicable for severely ill-posed problems as well. © de Gruyter 2007.