Ill-Posed Inverse Problems and Their Optimal Regularization

The regularization of ill-posed systems of equations is carried out by corrections of the data or the operator. It is shown how the efficiency of regularizations can be calculated by statistical decision principles. The efficiency of nonlinear regularizations depends on the distribution of the admitted disturbances of the data. For the class of linear regularizations optimal corrections are given. Keywords: Ill-posed problems, regularizations, efficiency, smoothing. The research for this paper was carried out within Sonderforschungsbereich 373 at the University of Potsdam and Humboldt-University Berlin and was printed using funds available by Deutsche Forschungsgemeinschaft 1 Introduction If it is not possible to measure certain properties of an object directly and one has to draw conclusions about these properties from indirect observations instead, then this is called an inverse problem or problem of identification. Denoting the measurable observation by g and the parameters des..