Regular Algorithms Based on a Separation of Perturbed Function Values for the Localization of Discontinuity Lines - Ageev, A. L.; Antonova, T. V. (Numerical analysis and applications, 2025)
We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables, provided that outside the discontinuity lines the function satisfies a Lipschitz condition, and at each point on the lines there is a discontinuity of the first kind. For a uniform grid with step , it is assumed that at each node the mean values of a perturbed function on a square with side are known, and the perturbed function approximates the exact function in The level of perturbation is assumed to be known. We propose a new approach based on a separation of the original noisy data to constructing regularizing algorithms for localizing the discontinuity lines. New algorithms are constructed for a class of functions with piecewise linear discontinuity lines and a convergence theorem with estimates of approximation accuracy is proved.