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Localization method for lines of discontinuity of approximately defined in the small function of two variables. (Russian, English) Zbl 1299.65021

Sib. Zh. Vychisl. Mat. 15, No. 4, 345-357 (2012); translation in Numer. Analysis Appl. 5, No. 4, 285-296 (2012).
Summary: A function of two variables with a line of discontinuity is considered, which has a discontinuity of the first kind. It is assumed that outside discontinuity lines the function to be measured is smooth and has a bounded partial derivative. Instead of the accurate function its approximation in \(L_2\) and perturbation level are known. The problem in question belongs to the class of nonlinear ill-posed problems, for whose solution it is required to construct regularizing algorithms. We propose a reduced theoretical approach to solving the problem of localizing the discontinuity line of the function that is noisy in the space \(L_2\). This is done in the case when conditions of an exact function are imposed ‘in the small’. Methods of averaging have been constructed, the estimations of localizing the line (‘in the small’) have been obtained.

MSC:

65D15 Algorithms for approximation of functions
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