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 ...

In this paper, we explore a robust method for calibration of the local volatility surface for European options. Assuming the the volatility surface is smooth, we apply a second order Tikhonov regularization to the calibration problem. Additionally we propose a new approach for choosing the Tikhonov regularization parameter. Using the TAPENADE automatic differentiation tool in order to obtain ad...

Model-based elastography is fraught with problems owing to the ill-posed nature of the inverse elasticity problem. To overcome this limitation, we have recently developed a novel inversion scheme that incorporates a priori information concerning the mechanical properties of the underlying tissue structures, and the variance incurred during displacement estimation in the modulus image reconstruc...

In this paper we investigate the stability and convergence rates of the widely used output least-squares method with Tikhonov regularization for the identification of the conductivity distribution in a heat conduction system. Due to the rather restrictive source conditions and regularity assumptions on the nonlinear parameter-to-solution operator concerned, the existing Tikhonov regularization ...

In this paper we investigate stability and convergence rates of the Tikhonov regularization method for the identiication of the diiusion parameter in a multi{dimensional parabolic equation. By choosing a problem{adapted approach, we obtain much better results than by applying the general theory of Tikhonov regularization of nonlinear inverse problems.

Two numerical methods for the restoration of noisy images from the Hubble Space Telescope are presented. The first one stems from a B-spline expansion followed by the Tikhonov regularization method. The second one is based on a coarse smoothing coupled with the Tikhonov regularization. The effectiveness of these methods is illustrated by the restoration of some images already proposed in the li...

In this paper we consider the a posteriori parameter choice strategy proposed by Scherzer et al in 1993 for the Tikhonov regularization of nonlinear ill-posed problems and obtain some results on the convergence and convergence rate of Tikhonov regularized solutions under suitable assumptions. Finally we present some illustrative examples.