On the Localization of Fractal Lines of Discontinuity from Noisy Data

Abstract

An ill-posed problem of localization (determining the position) of discontinuity lines of a function of two variables is considered: outside the discontinuity lines, the function is smooth, and, at each point on the line, it has a discontinuity of the first kind. Under the Lipschitz conditions on the discontinuity line, averaging procedures are constructed and global discrete regularizing algorithms of localization are studied. A parametric family of fractal lines is constructed for which all conditions can be checked analytically. A fractal having a large fractal dimension is indicated for which the efficiency of the constructed methods can be guaranteed.