Approximation of discontinuity lines for a noisy function of two variables with countably many singularities

Abstract

We construct and study the methods of localization (positioning) of the lines in whose neighborhoods a measured function of two variables is smooth, whereas at each point of the lines it has discontinuity of the first kind (i.e., lines of discontinuity). Assume that the function has countably many discontinuity lines: on finitely many of them the function has a “large” jump, whereas the jump values on the other lines satisfy some smallness condition. Given a noise-contaminated function and an error level in L2, it is required to determine the number and localize the position of discontinuity lines from the first set for the exact function. This problem belongs to the class of nonlinear ill-posed problems and, to solve it, we need to construct some regularizing algorithms. Some simplified theoretical approach is proposed in the case when the conditions on the exact function are imposed in a narrow strip intersecting the discontinuity lines.We constructed the averaging methods and obtained accuracy estimates for the localization of discontinuity lines.