Heinz W. Engl

Convergence rates for Tikhonov regularisation of non-linear ill-posed problems

The authors consider non-linear ill-posed problems in a Hilbert space setting, they show that Tikhonov regularisation is a stable method for solving non-linear ill-posed problems and give conditions that guarantee the convergence rate O( square root delta ) for the regularised solutions, where delta is a norm bound for the noise in the data. They illustrate these conditions for several examples including parameter estimation problems. In an appendix, they study the connection between the ill-posedness of a non-linear problem and its linearisation and show that this connection is rather weak. A sufficient condition for ill-posedness is given in the case that the non-linear operator is compact.

Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates

This paper investigates the stable identification of local volatility surfaces σ(S, t) in the Black–Scholes/Dupire equation from market prices of European Vanilla options. Based on the properties of the parameter-to-solution mapping, which assigns option prices to given volatilities, we show stability and convergence of approximations gained by Tikhonov regularization. In the case of a known term-structure of the volatility surface, in particular, if the volatility is assumed to be constant in time, we prove convergence rates under simple smoothness and decay conditions on the true volatility. The convergence rate analysis sheds light onto the importance of an appropriate a priori guess for the unknown volatility and the nature of the ill-posedness of the inverse problem, caused by smoothing properties and the nonlinearity of the direct problem. Finally, the theoretical results are illustrated by numerical experiments.

Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems

The authors propose an a-posteriors strategy for choosing the regularization parameter in Tikhonov regularization for solving nonlinear ill-posed problems and show that under certain conditions, the convergence rate obtained with this strategy is optimal. As a by-product, a new stability estimate for the regularized solutions is given which applies to a class of parameter identification problems. The authors compare the parameter choice strategy with Morozov’s Discrepancy Principle. Finally, numerical results are presented.

A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions

For iterative methods for well-posed problems, invariance properties have been used to provide a unified framework for convergence analysis. We carry over this approach to iterative methods for nonlinear ill-posed problems and prove convergence with rates for the Landweber and the iteratively regularized Gauss-Newton methods. The conditions needed are weaker as far as the nonlinearity is concerned than those needed in earlier papers and apply also to severely ill-posed problems. With no additional effort, we can also treat multilevel versions of our methods.

Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates

We propose a class ofa posteriori parameter choice strategies for Tikhonov regularization (including variants of Morozovs and Arcangelis methods) that lead to optimal convergence rates toward the minimal-norm, least-squares solution of an ill-posed linear operator equation in the presence of noisy data.

A posteriori parameter choice for general regularization methods for solving linear ill-posed problems

Abstract For continuous and iterative regularization methods for solving linear ill-posed problems, we propose an a posteriori parameter choice strategy and stopping rule, respectively, that always leads to optimal convergence rates and does not require the knowledge of the smoothness of the exact solution.

A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction

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 theory for nonlinear inverse problems is difficult to apply for the convergence rate analysis here. By introducing some new techniques, we are able to relax these regularity requirements and derive a much simpler and easily interpretable source condition but still achieve the same convergence rates as the standard Tikhonov regularization theory does.

Surveys on solution methods for inverse problems

Convergence Rates Results for Iterative Methods for Solving Nonlinear III-Posed Problems.- Iterative Regularization Techniques in Image Reconstruction.- A Survey of Regularization Methods for First-Kind Volterra Equations.- Layer Stripping.- The Linear Sampling Method in Inverse Scattering Theory.- Carleman Estimates and Inverse Problems in the Last Two Decades.- Local Tomographic Methods in Sonar.- Efficient Methods in Hyperthermia Treatment Planning.- Solving Inverse Problems with Spectral Data.- Low Frequency Electromagnetic Fields in High Contrast Media.- Inverse Scattering in Anisotropic Media.- Inverse Problems as Statistics.

RANDOM FIXED POINT THEOREMS

Publisher Summary This chapter presents random fixed point theorems. The study of random operator equations was initiated by the Prague school of probabilists around Spacek and Hans in the 1950s. As it seems to be a current trend to use stochastic models rather than deterministic ones, it is not surprising that the interest in random operator equations has been revived in the last years. The basic questions asked about random operator equations contain all problems that are interesting for deterministic operator equations, such as existence, uniqueness, stability, and approximation of solutions. However, the randomization leads to several new questions, such as the measurability of solutions and their statistical properties. The chapter presents the question of single- and multivalued random operators on randomly varying domains of definition.

Convergence rates for maximum entropy regularization

The authors prove results about stability, convergence, and convergence rates for the maximum entropy method by interpreting it as a nonlinear regularization method. The results are applied to a widely studied problem from physical chemistry, namely, the reconstruction of fluorescence lifetime distributions from the distribution of the delays between absorption and emission of photons.