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Continuation problems: Theory, numerics, neural networks and applications Научная публикация

Журнал

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

Вых. Данные

Год: 2026, Том: 34, Номер: 3, Страницы: 455–479 Страниц : 23 DOI: 10.1515/jiip-2024-0089

Ключевые слова

Continuation problem; inverse and ill-posed problem; reconstructing a function; regularization; neural networks

Авторы

Kabanikhin Sergey ¹ , Shishlenin Maxim ² , Bakanov Galitdin ³ , Liu Shuang ⁴ , Yuan Lele ⁵ , Zhang Ye ⁶

Организации

1	Institute of Computational Mathematics and Mathematical Geophysics , Novosibirsk , Russia ; and MSU-BIT-SMBU Joint Research Center of Applied Mathematics, Shenzhen MSU-BIT University, Shenzhen, P. R. China
2	Sobolev Institute of Mathematics , Novosibirsk , Russia
3	Khoja Akhmet Yassawi International Kazakh-Turkish University , Turkestan , Kazakhstan
4	School of Mathematics and Statistics , Beijing Institute of Technology , Beijing , P. R. China
5	School of Mathematics and Systems Science , Liaocheng University , Liaocheng , P. R. China
6	MSU-BIT-SMBU Joint Research Center of Applied Mathematics , Shenzhen MSU-BIT University , Shenzhen ; and School of Mathematics and Statistics, Beijing Institute of Technology, Beijing , P. R. China

Информация о финансировании (2)

1	Институт математики им. С.Л. Соболева СО РАН	FWNF-2024-0001
2	Институт вычислительной математики и математической геофизики СО РАН	FWNM-2025-0001

We study ill-posed continuation problems for partial differential equations, with an emphasis on the mechanisms of ill-posedness and their mitigation via conditional stability and regularization. Three canonical examples of elliptic, parabolic, and hyperbolic type are used to illustrate the underlying ill-posedness. For a second-order elliptic continuation problem, we summarize well-posedness results for the associated direct and adjoint problems, establish conditional stability estimates, and develop an adjoint-based iterative reconstruction method with convergence-rate guarantees. For a parabolic continuation problem, we present corresponding well-posedness results and an adjoint-based iterative scheme. For a hyperbolic continuation problem, we derive a conditional stability result. We further analyze the singular numbers of the continuation operator for a complex-valued Helmholtz equation, thereby characterizing the frequency dependence of the ill-posedness. Finally, we compare Tikhonov regularization with linear neural networks for ill-posed Helmholtz inverse problems, highlighting their complementary strengths.

Kabanikhin S. , Shishlenin M. , Bakanov G. , Liu S. , Yuan L. , Zhang Y.
Continuation problems: Theory, numerics, neural networks and applications
Journal of Inverse and Ill-Posed Problems. 2026. V.34. N3. P.455–479. DOI: 10.1515/jiip-2024-0089 WOS OpenAlex

Поступила в редакцию:	14 дек. 2024 г.
Принята к публикации:	16 мар. 2026 г.
Опубликована в печати:	29 мая 2026 г.
Опубликована online:	29 мая 2026 г.

≡ Web of science:	WOS:001777736200001
≡ OpenAlex:	W7162768938