On multidimensional ill-posed problems with discontinuous solutions
type: Item
instance of: scholarly article
title: On multidimensional ill-posed problems with discontinuous solutions
zbMATH Open document ID: 0903.35085
DOI: 10.1007/BF02732361
published in: Siberian Mathematical Journal
publication date: 1998-11-22T00:00:00Z
review text: The problem of finding discontinuous solutions to some multidimensional inverse problems is studied. For example, this situation appears in the theory of image processing for improving the quality of images. The relevant operator equation takes the form \(Az = u,\) and it is supposed that \(A\) is a continuous (in general nonlinear) operator from the space \(\nu_A(B)\subset L_1(B)\) of functions with bounded Arzelà variation into a normed space \(U\) and the equation has pseudosolutions constituting a set \(Z^*\). The following problem of finding normal pseudosolutions to the operator equation is studied: Find functions \(\overline{z}(x)\in Z^*\) such that \[ \| \overline{z}\| = \inf\{\| z\| : z\in Z^*\} \equiv\overline{\Omega}. \] The author proves a stable approximation result for normal pseudosolutions and also develops a numerical algorithm based on Tikhonov's approach in the class of functions of bounded variation. The so obtained approximation solutions converge to an exact solution piecewise uniformly.
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 65D15
zbMATH DE Number: 1225783
zbMATH Keywords: regularization method
zbMATH Keywords: pseudosolution
author: A. S. Leonov
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1225783