Approximation of discontinuity lines of a noisy function of two variables

Abstract

In the paper, the methods are constructed and investigated of localization (determination of the location) of the line in a neighborhood of which a measured function of two variables is smooth, whereas it has a discontinuity of the first kind on this line. Instead of the exact function, there is available some of its approximations in L2 and the perturbation level. The problem under consideration belongs to the class of nonlinear ill-posed problems, and to solve it some regularization algorithms are needed. A simplified theoretical approach is suggested to the problem of localization of the discontinuity line of a noisy function, when the conditions on the exact function are imposed on an arbitrary small strip intersecting the discontinuity line. Some averaging methods are constructed, and estimates for the precision of the line localization are obtained.