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

  • Bernd Hofmann

    Technische Universität Chemnitz, Germany
  • G. Fleischer

    Technische Universität Chemnitz, Germany


In this paper we deal with the ’strength’ of ill-posedness for ill-posed linear operator equations Ax=yAx = y in Hilbert spaces, where we distinguish according to M. Z. .Nashed the ill-posedness of type I if AA is not compact, but we have R(A)R(A)ˉR(A) \neq \bar{R(A)} for the range R(A)R(A) of AA, and the ill-posedness of type II for compact operators AA. From our considerations it seems to follow that-the problems with non-compact operators AA are not ingeneral ’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 AA as discussed by B. Hofmann and U. Tautenhahn are derived from the decay rate of the non-increasing sequence of singular values of AA. Since singular values do not exist for non-compact operators AA, 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 thenon-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.

Cite this article

Bernd Hofmann, G. Fleischer, Stability Rates for Linear Ill-Posed Problems with Compact and Non-Compact Operators. Z. Anal. Anwend. 18 (1999), no. 2, pp. 267–286

DOI 10.4171/ZAA/881