Abstract:
The ill-posed problem of determining the position (localization) of discontinuity lines of a function of two variables from noisy data is considered. The function is assumed to be smooth outside the discontinuity lines and to have a discontinuity of the first kind on these lines. At each node of a uniform grid with step $\tau$, the mean values of the perturbed function on a square with side $\tau$ are known. The perturbed function approximates the exact function in space $L_2(\mathbb{R}^2)$. It is assumed that the level of perturbation $\delta$ is known. The discontinuity lines are localized using methods based on the separation of the original noisy data that has been constructed and investigated. This paper shows that, compared to the averaging methods in the authors' previous works, separation methods provide a guaranteed estimate of the localization accuracy over a wider class of correctness, i.e., under substantially weaker conditions on the discontinuity line. For the sake of simplicity, it is assumed that the discontinuity lines are polygonal. The requirements for grid step size and angles have been relaxed. Estimates of the approximation accuracy and other important characteristics of the constructed method are obtained. As demonstrated by the provided examples, in the case of nonsmooth boundaries, separation-based methods guarantee better approximation accuracy than averaging methods. The algorithm in this paper can be used in the construction of separation-based methods for approximating fractal discontinuity lines.
Keywords:ill-posed problems, regularization method, discontinuity line, data separation, global localization, Lipschitz condition.
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