Stability Rates for Linear Ill-Posed Problems with Compact and Non-Compact Operators

B. Hofmann and G. Fleischer

Abstract: In this paper we deal with the `strength' of ill-posedness for ill-posed linear operator equations $Ax = y$ in Hilbert spaces, where we distinguish according to M. Z. Nashed the ill-posedness of type I if $A$ is not compact, but we have $R(A) \ne \ol{R(A)}$ for the range $R(A)$ of $A$, and the ill-posedness of type II for compact operators $A$. From our considerations it seems to follow that the problems with non-compact operators $A$ are not in general `less' ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators $A$ as discussed by B. Hofmann and U. Tautenhahn are derived from the decay rate of the non-increasing sequence of singular values of $A$. Since singular values do not exist for non-compact operators $A$, we introduce stability rates in order to have a common measure for the compact and non-compact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in the non-compact case. Moreover, using increasing rearrangements of multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for multiplication operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared for compact operators and multiplication operators.

Keywords: linear ill-posed problems, compact and non-compact linear operators in Hilbert spaces, discrete least-squares method, stability rates, singular values, convolution and multiplication operators, Galerkin matrices, condition numbers, increasing rearrangements

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