Regularization methods in inverse problems

Abstract

Finding solutions to inverse problems by using regularized inverses is a topic of growing interest over the decades. The regularization theory was developed in 1950's- 1960's by Soviet mathematicians A.N.Tikhonov, V.K.Ivanov and M.M.Lavrentiev. This thesis can be treated as an attempt to bring the topic forward in a simple way with a statistical approach.

In our approach, we use the mean integrated squared error to check the quality of the estimator obtained from different Regularization methods. By using that, we look at restrictions on regularization parameters that preserve the quality of regularized estimators.

Our primary interest is in iterated Tikhonov type estimator. But for the sake of completeness, we discuss the spectral cut-off type and the Tikhonov type. The Landweber iteration type is not treated in this thesis.