B. Hofmann; G. Fleischer : Stability Rates for Linear Ill-Posed Problems with Convolution and Multiplication Operators

Author(s) :

B. Hofmann; G. Fleischer

Title :

Stability Rates for Linear Ill-Posed Problems with Convolution and Multiplication Operators

Electronic source :

[gzipped dvi-file] 34 kB
[gzipped ps-file] 73 kB

Preprint series

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 98-7, 1998

Mathematics Subject Classification :

65J20 [ Improperly posed problems (numerical methods in abstract spaces) ]

47B38 [ Operators on function spaces ]

65R30 [ Improperly posed problems (integral equations, numerical methods) ]

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 [15] the ill-posedness of type I if A is not compact, but we have R(A) 6= 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 noncompact 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 in [10] 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 the multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for the 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 noncompact linear operators in Hilbert spaces, discrete least-squares method, stability rates, singular values, convolution and multiplication operators, Galerkin matrices, condition numbers, increasing rearrangements

Language :

english

Publication time :

5/1998

Notes :

partial support through DFG under Ho 1454/3-2