We consider linear inverse problems that are formulated in the continuous domain. The object of recovery is a function that is assumed to minimize a convex objective functional. The solutions are constrained by imposing a continuousdomain regularization. We derive the parametric form of the solution (representer theorems) for Tikhonov (quadratic) and generalized total-variation (gTV) regulariza...

Although the residual method, or constrained regularization, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existin...

In a recent paper [24] an algorithm for large-scale Tikhonov regularization in standard form called GKB-FP was proposed and numerically illustrated. In this paper, further insight into the convergence properties of this method is provided and extensions to general-form Tikhonov regularization are introduced. In addition, as alternative to Tikhonov regularization, a preconditioned LSQR method co...

This paper reports studies on the influence of the regularization parameter and the first estimate on the performance of iterative image restoration algorithms. We discuss regularization parameter estimation methods that have been developed for the linear Tikhonov-Miller filter to restore images distorted by additive Gaussian noise. We have performed experiments on synthetic data to show that t...

in this paper the 3d inversion of gravity data using two different regularization methods, namely tikhonov regularization and truncated singular value decomposition (tsvd), is considered. the earth under the survey area is modeled using a large number of rectangular prisms, in which the size of the prisms are kept fixed during the inversion and the values of densities of the prisms are the mode...

We consider the problem of finding regularized solutions to ill-posed Volterra integral equations. The method we consider is a sequential form of Tikhonov regularization that is particularly suited to problems of Volterra type. We prove that when this sequential regularization method is coupled with several standard discretizations of the integral equation (collocation, rectangular and midpoint...

Tikhonov regularization is one of the most popular methods for the solution of linear discrete ill-posed problems. In many applications the desired solution is known to lie in the nonnegative cone. It is then natural to require that the approximate solution determined by Tikhonov regularization also lies in this cone. The present paper describes two iterative methods, that employ modulus-based ...

We shall be concerned with the construction of Tikhonov–based iteration schemes for solving nonlinear operator equations. In particular, we are interested in algorithms for the computation of a minimizer of the Tikhonov functional. To this end, we introduce a replacement functional, that has much better properties than the classical Tikhonov functional with nonlinear operator. Namely, the repla...

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of ...