G. Fleischer; R. Gorenflo; B. Hofmann : On the Autoconvolution Equation and Total Variation Constraints

Author(s) :

G. Fleischer; R. Gorenflo; B. Hofmann

Title :

On the Autoconvolution Equation and Total Variation Constraints

Electronic source :

[gzipped dvi-file] 38 kB
[gzipped ps-file] 90 kB

Preprint series

Technische Universität Chemnitz-Zwickau, Fakultät für Mathematik (Germany). Preprint 97-9, 1997

Mathematics Subject Classification :

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

45G10 [ Nonsingular nonlinear integral equations ]

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

Abstract :

This paper is concerned with the numerical analysis of the autoconvolution equation $x*x=y$ restricted to the interval [0,1]. We present a discrete constrained least squares approach and prove its convergence in $L^p(0,1),1<p<\infinite$ , where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaption to the Sobolev space $H^1(0,1)$ and some remarks on monotone functions are added. The paper is completed by a numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y.

Keywords :

autoconvolution, ill-posed problem, discretization, constrained least squares approach, bounded total variation

Language :

english

Publication time :

3/1997