Regular Algorithms Based on a Separation of Perturbed Function Values for the Localization of Discontinuity Lines

• Published in: Numerical analysis and applications Vol. 18; no. 3; pp. 203 - 215
• Main Authors: Ageev, A. L., Antonova, T. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.09.2025; Springer Nature B.V
• Subjects: Algorithms; Approximation; Discontinuity; Estimates; Ill posed problems; Lipschitz condition; Localization; Mathematics; Mathematics and Statistics; Numerical Analysis; Separation; ill-posed problems; regularization method; global localization; image separation; discontinuity lines; discretization; separability threshold
• ISSN: 1995-4239; 1995-4247
• DOI: 10.1134/S1995423925030012

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.