Sobolev error estimates and a priori parameter selection for semi-discrete Tikhonov regularization

Abstract

The goal of this paper is to investigate the Tikhonov–Phillips method for semi-discrete linear ill-posed problems in order to determine tight error bounds and to obtain a good parameter choice. We consider the equation Aƒ = g, where the operator A is known, noisy discrete data with can be observed and a solution ƒ* is sought. Assuming that ƒ* is an element of a Sobolev space, we use the well-known theory of optimal recovery in Hilbert spaces for the reconstruction process. We then provide L2-error estimates in terms of the data density and derive an a priori selection for the regularization parameter, which guarantees an optimal compromise between approximation and stability. Finally, we illustrate the parameter selection with a simple example.