Analysis of a Regularization Algorithmfor a Linear Operator Equation Containinga Discontinuous Component of the Solution
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alysis of a Regularization Algorithm for a Linear Operator Equation Containing a Discontinuous Component of the Solution V. V. Vasin1,2,∗ and V. V. Belyaev1,2,∗∗ Received April 18, 2019; revised July 8, 2019; accepted July 15, 2019
Abstract—We study a linear operator equation that does not satisfy the Hadamard wellposedness conditions. It is assumed that the solution of the equation has different smoothness properties in different regions of its domain. More exactly, the solution is representable as the sum of a smooth and discontinuous components. The Tikhonov regularization method is applied for the construction of a stable approximate solution. In this method, the stabilizer is the sum of the Lebesgue norm and the smoothed BV -norm. Each of the functionals in the stabilizer depends only on one component and takes into account its properties. Convergence theorems are proved for the regularized solutions and their discrete approximations. It is shown that discrete regularized solutions can be found with the use of the Newton method and nonlinear analogs of α-processes. Keywords: ill-posed problem, regularization method, discontinuous solution, total variation, discrete approximation.
DOI: 10.1134/S0081543820040197 1. INTRODUCTION Let A be a continuous linear operator acting on a pair of Banach spaces U and F and having a discontinuous inverse A−1 , which implies the ill-posedness in the Hadamard sense of the equation Au = f
(1.1)
with an approximately given right-hand side f δ such that f −f δ ≤ δ. We assume that the solution to the problem (1.1), along with a smooth background, contains regions with discontinuities. It is well known that, in the traditional (single-component) approach, the presence of regions with different smoothness properties in a solution makes it difficult to choose a stabilizing functional for which the solution would be recovered with the required accuracy in all regions with the structure (gaps, kinks) of the solution preserved. One of the possible approaches to solving this problem is based on the idea of representing the solution as a sum of two components and choosing a stabilizer also in the form of a sum of functionals, each of which depends only on one component and takes into account the smoothness property of this particular component. 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia 2 Ural Federal University, Yekaterinburg, 620002 Russia e-mail: ∗ [email protected], ∗∗ beliaev [email protected]
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This technique showed good results in applied research in the processing of noisy signals and images by the Tikhonov regularization method when recovering the continuous and discontinuous components of the solution [1, 2]. The theoretical justification for the convergence of such twoand three-component approach is contained in [3, 4], where the norm of the Lipschitz space was used along with the total variation to approximate the discontinuous component. In this case, the Tikhonov functi
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