Про методи регуляризації некоректних задач загальнофізичного профілю: Fìz.-mat. model. ìnf. tehnol. 2021, 33:148-155

• Date: 2021
• Main Author: Starkov, Vyacheslav
• Format: Article
• Language: Ukrainian
• Published: Інститут прикладних проблем механіки і математики ім. Я. С. Підстригача НАН України 2021
• Subjects: некоректна задача; алгоритм; оператор; квантова оптика; фотон
• Online Access: https://www.fmmit.lviv.ua/index.php/fmmit/article/view/219
• Journal Title: Physico-mathematical modeling and informational technologies

On the example of a specific physical problem of noise reduction associated with losses, dark counts, and background radiation, a summary of methods for regularizing ill-posed problems is given in the statistics of photocounts of quantum light. The mathematical formulation of the problem is presented by an operator equation of the first kind. The operator is generated by a matrix with countable elements. In the sense of Hadamard, the problem of reconstructing the number of photons of quantum light is due to the compactness of the operator of the mathematical model. A rigorous definition of a regularizing operator (regularizer) is given. The problem of stable approximation to the exact solution of the operator equation with inaccurately given initial data can be overcome by one of the most well-known regularization methods, the theoretical foundations of which were laid in the works of A.N. Tikhonov. The selection of an important class of regularizing algorithms is based on the construction of a parametric family of functions that are Borel measurable on the semiaxis and satisfy some additional conditions. The set of regularizers in this family includes most of the known regularization methods. The main ones are given in the work. References Sperling, J., Phillips, D. S., Bulmer, J. F. F., Thekkadath, G. S. (2020). Detector-agnostic phase-space distributions. Phys. Rev. Lett. Hloušek, J., Dudka, M., Straka, I., and Ježek, M. (2019). Accurate Detection of Arbitrary Photon Statistics. Phys. Rev. Lett., 123. DOI https://doi.org/10.1103/physrevlett.123.153604 Kolmogorov, A. N., Fomin, S. V. (1972). Elementyi teorii funktsiy i funktsionalnogo analiza. M.: Nauka. (in Russian). Tihonov, A. N., Arsenin, V. Ya. (1966). Metodyi resheniya nekorrektnyih zadach. M.: Nauka. (in Russian). Vaynikko, G. M., Veretennikov, A. Yu. (1986). Iteratsionnyie protseduryi v nekorrektnyih zadachah. – M.: Nauka. (in Russian).