Wave discrete inverse problems: From boundary control to Gelfand–Levitan–Krein equations | Sciact - Система учета научной деятельности ИМ СО РАН

Wave discrete inverse problems: From boundary control to Gelfand–Levitan–Krein equations Научная публикация

Журнал

Journal of Inverse and Ill-Posed Problems
ISSN: 0928-0219 , E-ISSN: 1569-3945

Вых. Данные

Год: 2026, Том: 34, Номер: 5, Страницы: 929-950 Страниц : 22 DOI: 10.1515/jiip-2026-0092

Ключевые слова

Coefficient inverse problem; boundary control; Gelfand–Levitan equation; Krein equation; discrete inverse problem

Авторы

Kabanikhin Sergey ¹ , Mikhaylov Aleksander ² , Shishlenin Maxim ³

Организации

1	Institute of Computational Mathematics and Mathematical Geophysics , Novosibirsk , Russia
2	St. Petersburg Department of Steklov Mathematical Institute ; and ITMO University; and St. Petersburg State University , St. Petersburg , Russia
3	Sobolev Institute of Mathematics , Novosibirsk , Russia

Информация о финансировании (2)

1	Министерство науки и высшего образования РФ	FWNF-2026-0029
2	Институт вычислительной математики и математической геофизики СО РАН	FWNM-2025-0001

This paper develops a unified discrete framework for dynamic, spectral, and scattering inverse problems associated with the one-dimensional wave equation with a potential on a semi-axis. Using a second-order explicit finite-difference scheme, we reduce the continuous initial-boundary value problem to a discrete dynamical system governed by a tridiagonal Jacobi matrix. We establish explicit Fourier-type integral transforms connecting the dynamic response operator, the Weyl function, and the spectral measure. Our main theoretical result is that, in the discrete one-dimensional setting, the Boundary control method and the Krein integral equation approach are strictly equivalent: their corresponding systems of linear algebraic equations coincide. We construct a discrete analogue of the Gelfand–Levitan equation and prove necessary and sufficient conditions for the existence and uniqueness of the solution to the discrete coefficient inverse problem, namely the unique solvability of an auxiliary linear system for each depth level. Additionally, we analyze the relationship between discrete scattering data (Jost solutions and norming constants) and spectral measures for Jacobi operators. The proposed framework provides a rigorous foundation for layer-stripping numerical inversion algorithms applicable to acoustic flaw detection, seismic tomography, and diagnostic wave imaging.

Kabanikhin S. , Mikhaylov A. , Shishlenin M.
Wave discrete inverse problems: From boundary control to Gelfand–Levitan–Krein equations
Journal of Inverse and Ill-Posed Problems. 2026. V.34. N5. P.929-950. DOI: 10.1515/jiip-2026-0092 WOS Scopus OpenAlex

Поступила в редакцию:	28 июл. 2026 г.
Принята к публикации:	1 сент. 2026 г.
Опубликована в печати:	2 окт. 2026 г.
Опубликована online:	2 окт. 2026 г.

≡ Web of science:	WOS:001893799000009
≡ Scopus:	2-s2.0-105052330486
≡ OpenAlex:	W7215083856