In the recent past the authors, with collaborators, have published convergence rate results for regularized solutions of linear ill-posed operator equations by avoiding the usual assumption that the solutions satisfy prescribed source conditions. Instead the degree of violation of such source conditions is expressed by distance functions d(R) depending on a radius R 0 which is an upper bound of the norm of source elements under consideration. If d(R) tends to zero as R → ∞ an appropriate balancing of occurring regularization error terms yields convergence rates results. This approach was called the method of approximate source conditions, originally developed in a Hilbert space setting. The goal of this paper is to formulate chances and limitations of an application of this method to nonlinear ill-posed problems in reflexive Banach spaces and to complement the field of low order convergence rates results in nonlinear regularization theory. In particular, we are going to establish convergence rates for a variant of Tikhonov regularization. To keep structural nonlinearity conditions simple, we update the concept of degree of nonlinearity in Hilbert spaces to a Bregman distance setting in Banach spaces.

Inverse problems in option pricing are frequently regarded as simple and resolved if a formula of Black-Scholes type defines the forward operator. However, precisely because the structure of such problems is straightforward, they may serve as benchmark problems for studying the nature of ill-posedness and the impact of data smoothness and no arbitrage on solution properties. In this paper, we analyse the inverse problem (IP) of calibrating a purely timedependent volatility function from a term-structure of option prices by solving an ill-posed nonlinear operator equation in spaces of continuous and powerintegrable functions over a finite interval. The forward operator of the IP under consideration is decomposed into an inner linear convolution operator and an outer nonlinear Nemytskii operator given by a Black-Scholes function. The inversion of the outer operator leads to an ill-posedness effect localized at small times, whereas the inner differentiation problem is ill posed in a global manner. Several aspects of regularization and their properties are discussed. In particular, a detailed analysis of local ill-posedness and Tikhonov regularization of the complete IP including convergence rates is given in a Hilbert space setting. A brief numerical case study on synthetic data illustrates and completes the paper.

In this paper we deal with convergence rates for regularizing ill-posed problems with operator mapping from a Hilbert space into a Banach space. Since we cannot transfer the well-established convergence rates theory in Hilbert spaces, only few convergence rates results are known in the literature for this situation. Therefore we present an alternative approach for deriving convergence rates. Hereby we deal with so-called distance functions which quantify the violation of a reference source condition. With the aid of these functions we present error bounds and convergence rates for regularized solutions of linear and nonlinear problems when the reference source condition is not satisfied. We show that the approach of applying distance functions carries over the idea of considering generalized source conditions in Hilbert spaces to inverse problems in Banach spaces in a natural way. Introducing this topic for linear ill-posed problems we additionally show that this theory can be easily extended to nonlinear problems.

In this paper, we consider an iterative regularization scheme for linear ill-posed equations in Banach spaces. As opposed to other iterative approaches, we deal with a general penalty functional from Tikhonov regularization and take advantage of the properties of the regularized solutions which where supported by the choice of the specific penalty term. We present convergence and stability results for the presented algorithm. Additionally, we demonstrate how these theoretical results can be applied to L 1-and TV-regularization approaches and close the paper with a short numerical example.

We investigate a method of accelerated Landweber type for the iterative regularization of nonlinear ill-posed operator equations in Banach spaces. Based on an auxiliary algorithm with a simplified choice of the step-size parameter we present a convergence and stability analysis of the algorithm under consideration. We will close our discussion with the presentation of a numerical example.