Email this article 
A STABILITY ESTIMATE IN THE SOURCE PROBLEM FOR THE RADIATIVE TRANSFER EQUATION
- Authors: Romanov V.G.1
-
Affiliations:
- Sobolev Institute of Mathematics
- Issue: Vol 514, No 1 (2023)
- Pages: 34-38
- Section: MATHEMATICS
- URL: https://journal-vniispk.ru/2686-9543/article/view/247081
- DOI: https://doi.org/10.31857/S2686954323600271
- EDN: https://elibrary.ru/CQRKFI
- ID: 247081
Cite item
Open Access
Access granted
Subscription Access
Abstract
It is given a stability estimate of a solution of a source problem for the stationary radiative transfer equation. It is suppose that the source is an isotropic distribution. Earlier stability estimates for this problem were known in a partial case of the emission tomography problem only, when the scattering operator vanishes, and for the complete transfer equation under additional and difficult in checking conditions for the absorption coefficient and the scattering kernel. In the present work, we suggest a new and enough simple approach for obtaining a stability estimate for the problem under the consideration. The transfer equation is considered in a circle of the two-dimension space. In the forward problem, it is assumed that incoming radiation is absent. In the inverse problem for recovering the unknown source some data for solutions of the forward problem related to outgoing radiation are given. The obtained result can be used for an estimation of the summary density of distributed sources of the radiation.
About the authors
V. G. Romanov
Sobolev Institute of Mathematics
Author for correspondence.
Email: romanov@math.nsc.ru
Russian Federation, Novosibirsk
References
- Наттерер Ф. Математические аспекты компьютерной томографии. М.: Мир. 1990. 279 с.
- Finch D.V. Uniqueness for attenuated X-ray transformin the physical range // Inverse Problems. 1986. V. 2. P. 197–203.
- Мухометов Р.Г. Оценка устойчивости решения одной задачи компьютерной томографии // Вопросы корректности задач анализа. Новосибирск. Изд-во: Институт математики СО АН СССР. 1989. С. 122–124.
- Шарафутдинов В.А. О задаче эмиссионной томографии для неоднородных сред // Доклады РАН. 1992. Т. 326. № 2. С. 446–448.
- Арбузов Е.М., Бухгейм А.Л., Казанцев С.Г. Двумерная проблема томографии и теория А-аналитических функций // Алгебра, геометрия, анализ и математическая физика (ред: Решетняк Ю.Г., Бокуть Л.А., Водопьянов С.К., Тайманов И.А.). Новосибирск. Изд-во: Институт математики СО РАН. 1997. С. 6–20.
- Novikov R.G. An inversion formula for attenuated X-ray transformation // Ark. Mat. 2002. V. 40. P. 145–167.
- Natterer F. Inversion of the attenuated Radon transform // Inverse Problems. 2001. V. 17. P. 113–119.
- Larsen E.W. The inverse source problem in radiative transfer // J. Quant. Spect. Radiat. Transfer. 1975. V. 15. P. 1–5.
- Siewert C.E. An inverse source problem in radiative transfer // J. Quant. Spect. Radiat. Transfer. 1993. V. 50. P. 603–609.
- Шарафутдинов В.А. Обратная задача об определении источника в стационарном уравнении переноса на римановом многообразии // Зап. научн. сем. ПОМИ. 1997. Т. 239. С. 236–242.
- Bal G., Tamasan A. Inverse source problem in transport equations. Siam J. Math. Anal. 2007. V. 39. № 1. P. 57–76.
- Stefanov P., Uhlmann G. An inverse source problem in optical molecular imagin // Analysis and PDE. 2008. V. 1. № 1. P. 115–126.
- Мухометов Р.Г. Задача восстановления двумерной римановой метрики и интегральная геометрия // Доклады АН СССР. 1977. Т. 232. № 1. С. 32–35.
Supplementary files
