Solving discrete ill-posed problems via Tikhonov regularization introduces the problem of determining a regularization parameter. There are several methods available for choosing such a parameter, yet, in general, the uniqueness of this choice is an open question. Two empirical methods for determining a regularization parameter (which appear in the biomedical engineering literature) are the com...

We propose and analyze variational source conditions (VSC) for the Tikhonov regularization method with $L^p$-penalties applied to an ill-posed operator equation in a Banach space. Our analysis is b...

In this paper, we discuss a classical ill-posed problem– numerical differentiation by the Tikhonov regularization. Based on the conditional stability estimate for this ill-posed problem, a new simple method for choosing regularization parameters is proposed. We show that it has an almost optimal convergence rate when the exact solution is in H2. The advantages of our method are: 1. We can get a...

and Applied Analysis 3 Definition 2.1. Let F : K → 2Rn be a set-valued mapping. F is said to be i monotone on K if for each pair of points x, y ∈ K and for all x∗ ∈ F x and y∗ ∈ F y , 〈y∗ − x∗, y − x〉 ≥ 0, ii maximal monotone on K if, for any u ∈ K, 〈ξ − x∗, u − x〉 ≥ 0 for all x ∈ K and all x∗ ∈ F x implies ξ ∈ F u , iii quasimonotone on K if for each pair of points x, y ∈ K and for all x∗ ∈ F ...

We develop a general convergence analysis for a class of inexact Newtontype regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis w...

This paper considers an efficient iterative approach to solve separable nonlinear least squares problems that arise in large scale inverse problems. A variable projection GaussNewton method is used to solve the nonlinear least squares problem, and Tikhonov regularization is incorporated using an iterative Lanczos hybrid scheme. Regularization parameters are chosen automatically using a weighted...

The L-curve method was developed for the selection of regularization parameters in the solution of discrete systems obtained from ill-posed problems. An analysis of this method is given for selecting a parameter for Tikhonov regularization. This analysis, which is carried out in a semi-discrete, semi-stochastic setting, shows that the L-curve approach yields regularized solutions which fail to ...

The unique continuation on quadratic curves for harmonic functions is discussed in this paper. By using complex extension method, the conditional stability of along illustrated. numerical algorithm provided based collocation method and Tikhonov regularization. estimates parabolic hyperbolic are demonstrated by examples respectively.

We propose a discretized Tikhonov regularization for a Cauchy problem for an elliptic equation by a reproducing kernel Hilbert space. We prove the convergence of discretized regularized solutions to an exact solution. Our numerical results demonstrate that our method can stably reconstruct solutions to the Cauchy problems even in severe cases of geometric configurations.