On multidimensional ill-posed problems with discontinuous solutions
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.
scientific article; zbMATH DE number 1225783
scientific article