基于光滑全变分的修正tikhonov正则化方法求解线性不适定问题

Pub Date : 2021-06-01 DOI:10.32523/2306-6172-2021-9-2-88-100

V. Vasin, V. Belyaev

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引用次数: 0

摘要

研究了一类在Hadamard意义上不适定的线性算子方程。假设其解可表示为光滑和不连续分量的和。为了构造一个稳定的近似解，我们使用改进的Tikhonov方法，将稳定泛函作为光滑分量的Lebesgue范数和不连续分量的光滑bv范数的和。证明了正则解及其有限维近似的存在性、唯一性和收敛性定理。并给出了数值实验结果。

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THE MODIFIED TIKHONOV REGULARIZATION METHOD WITH THE SMOOTHED TOTAL VARIATION FOR SOLVING THE LINEAR ILL-POSED PROBLEMS

We investigate a linear operator equation of the first kind that is ill-posed in the Hadamard sence. It is assumed that its solution is representable as a sum of smooth and discontinuous components. To construct a stable approximate solutions, we use the modified Tikhonov method with the stabilizing functional as a sum of the Lebesgue norm for the smooth component and a smoothed BV-norm for the discontinuous component. Theorems of exis- tence, uniqueness, and convergence both the regularized solutions and its finite-dimentional approximations are proved. Also, results of numerical experiments are presented.

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