Algorithms for localizing discontinuity lines with a new type of averaging

A. L. Ageev   ; T. V. Antonova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 4, pp. 5-18

Résumé

We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables. It is assumed that the function is smooth outside the discontinuity lines and has a discontinuity of the first kind at each point of these lines. The average values of the perturbed function on a square $\tau\times\tau$ are assumed to be known at each node of a uniform grid with step $\tau$. The perturbed function with a given perturbation level $\delta$ approximates the exact function in the space $L_2(\mathbb{R}^2)$. Global discrete regularizing algorithms are constructed for the localization of the discontinuity lines from noisy data. A new approach to the construction of averaging methods for solving the localization problem is proposed. The use of a new type of averaging allows one to construct regularizing algorithms without using the derivative of the averaging function. A new technique is developed and used for deriving estimates. This technique is applicable to a wide range of new methods with a nonclassical averaging domain. On classes of functions with piecewise linear discontinuity lines, estimates of the localization error and other important characteristics of the regularizing algorithm are obtained. It is shown that the new algorithms in some situations are more economical in terms of the number of operations compared to the methods that were investigated by the authors in previous works.

Détail

Keywords: ill-posed problem, regularization method, discontinuity lines, global localization, discretization, separability threshold.

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