Studies on two specific inverse problems from imaging and finance

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices