Abstract:
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 $\tau$, it is assumed that at each node the mean values of the perturbed function on a square with side $\tau$ are known, and the perturbed function approximates the exact function in $L_2(\mathbb{R}^2)$. The level of perturbation $\delta$ is assumed to be known. We propose a new approach to construct regularizing algorithms for localizing the discontinuity lines based on a separation of the original noisy data. 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.
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