On multidimensional ill-posed problems with discontinuous solutions
Copyright policy )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.
Inverse problems for PDEs, Numerical solutions of ill-posed problems in abstract spaces; regularization, Algorithms for approximation of functions, Nonlinear differential equations in abstract spaces, pseudosolution, regularization method
Inverse problems for PDEs, Numerical solutions of ill-posed problems in abstract spaces; regularization, Algorithms for approximation of functions, Nonlinear differential equations in abstract spaces, pseudosolution, regularization method
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).3 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!