New accuracy estimates for methods for localizing discontinuity lines of a noisy function
A.L. Ageev, T.V. Antonova
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
Keywords: ill-posed problems, regularization method, discontinuity lines, global localization, discretization, separability threshold
Abstract
We consider the ill-posed problem of localizing (finding the position) the discontinuity lines of a function of two variables, provided that the function of two variables is smooth outside of the discontinuity lines, and at each point on the line has a discontinuity of the first kind. There is a uniform grid with the step τ . It is assumed that we know the averages on the square τ x τ of the perturbed function at each node of the grid. The perturbed function approximates the exact one the in space L 2( mathbb R 2). The perturbation level δ is known. Earlier, the authors investigated (obtained accuracy estimates) the global discrete regularizing algorithms for approximating a set of discontinuity lines of a noisy function. However, stringent smoothness conditions were superimposed on the discontinuity line. The main result of this paper is the improvement of localizing the accuracy estimation methods, which allows replacing the smoothness requirement with a weaker Lipschitz condition. Also, the conditions of separability are formulated in a more general form, as compared to previous studies. In particular, it is established that the proposed algorithm make it possible to obtain the localization accuracy of the order O( δ ). Also, estimates of other important parameters characterizing the localization algorithm are given.
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