Zeitschrift für Analysis und ihre Anwendungen Vol. 18, No. 2, pp. 267  286 (1999) 

Stability Rates for Linear IllPosed Problems with Compact and NonCompact OperatorsB. Hofmann and G. FleischerBoth authors: Techn. Univ. of Chemnitz, Fac. Math., D09107 ChemnitzAbstract: In this paper we deal with the `strength' of illposedness for illposed linear operator equations $Ax = y$ in Hilbert spaces, where we distinguish according to M. Z. Nashed the illposedness 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 illposedness of type II for compact operators $A$. From our considerations it seems to follow that the problems with noncompact operators $A$ are not in general `less' illposed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete leastsquares solutions and the growth rate of Galerkin matrices in both cases. Illposedness measures for compact operators $A$ as discussed by B. Hofmann and U. Tautenhahn are derived from the decay rate of the nonincreasing sequence of singular values of $A$. Since singular values do not exist for noncompact operators $A$, we introduce stability rates in order to have a common measure for the compact and noncompact 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 noncompact case. Moreover, using increasing rearrangements of multiplier functions specific measures of illposedness called illposedness 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 illposed problems, compact and noncompact linear operators in Hilbert spaces, discrete leastsquares method, stability rates, singular values, convolution and multiplication operators, Galerkin matrices, condition numbers, increasing rearrangements Full text of the article:
Electronic fulltext finalized on: 31 Jul 2001. This page was last modified: 9 Nov 2001.
© 2001 Heldermann Verlag
