B. Hofmann : On Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert Space

Author(s) :

B. Hofmann

Title :

On Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert Space

Electronic source :

[gzipped dvi-file] 25 kB
[gzipped ps-file] 63 kB

Preprint series

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 97-8, 1997

Mathematics Subject Classification :

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

47H15 [ Equations involving nonlinear operators ]

65J10 [ Equations with linear operators (numerical methods) ]

65J15 [ Equations with nonlinear operators (numerical methods) ]

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

45G10 [ Nonsingular nonlinear integral equations ]

Abstract :

In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems in a Hilbert space setting. We define local ill-posedness of a nonlinear operator equation $F(x) = y_0$ in a solution point $x_0$ and the interplay between the nonlinear problem and its linearization using the Frechet derivative $F\acent(x_0)$ . To find an appropriate ill-posedness concept for the linarized equation we define intrinsic ill-posedness for linear operator equations $Ax = y$ and compare this approach with the ill-posedness definitions due to Hadamard and Nashed.

Keywords :

Nonlinear inverse problems, oerator equations, ill-posedness, stability estimates, local ill-posedness, Frechet derivative, linearized problem, compact operators, integral equations, parameter identification

Language :

english

Publication time :

8/1997