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• W56571430 abstract "In this paper, we show the restricted influence of non-compact multiplication operators mapping in L2(0, 1) occurring in linear ill-posed operator equations and in the linearization of nonlinear ill-posed operator equations with compact forward operators. We give examples of nonlinear inverse problems in natural science and stochastic finance that can be written as nonlinear operator equations for which the forward operator is a composition of a simple linear integration operator and a nonlinear Nemytskii operator. Hence, the Frechet derivative of such a forward operator is a composition of integration and multiplication operators. It is shown for power type functions and conjectured for a wider class of multiplier (weight) functions with essential zeros that the unbounded inverse of the injective multiplication operator does not influence the (local) degree of ill-posedness of inverse problems under consideration. In a more general Hilbert space setting, we investigate the role of approximate source conditions in the method of Tikhonov regularization for linear and nonlinear ill-posed operator equations. We introduce a distance function measuring the violation of canonical source conditions and derive convergence rates for regularized solutions in the linear case based on that functions. In this context, we formulate cross-connections to convergence rates in Tikhonov regularization with general source conditions as frequently used in the recent literature. By considering the structure of source conditions in Tikhonov regularization it could be expected for the multiplication operators in L2(0, 1) that different decay rates of multiplier functions near a zero, for example the decay as a power or as an exponential function, would lead to completely different ill-posedness situations. Also based on the studies concerning approximate source conditions we indicate that only integrals of multiplier functions and not the specific character of the decay of multiplier functions in a neighborhood of a zero determine the convergence behavior of regularized solutions. MSC2000 subject classification: 47A52, 47J06, 47B06, 65J20, 47H30, 47B33, 65R30, 45C05" @default.
• W56571430 created "2016-06-24" @default.
• W56571430 creator A5059831151 @default.
• W56571430 date "2005-01-01" @default.
• W56571430 modified "2023-09-23" @default.
• W56571430 title "The potential for ill-posedness of multiplication operators occurring in inverse problems" @default.
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• W56571430 hasPublicationYear "2005" @default.
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