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Continuation problems: Theory, numerics, neural networks and applications Full article

Journal Journal of Inverse and Ill-Posed Problems
ISSN: 0928-0219 , E-ISSN: 1569-3945
Output data Year: 2026, Volume: 34, Number: 3, Pages: 455–479 Pages count : 23 DOI: 10.1515/jiip-2024-0089
Tags Continuation problem; inverse and ill-posed problem; reconstructing a function; regularization; neural networks
Authors Kabanikhin Sergey 1 , Shishlenin Maxim 2 , Bakanov Galitdin 3 , Liu Shuang 4 , Yuan Lele 5 , Zhang Ye 6
Affiliations
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

Funding (2)

1 Sobolev Institute of Mathematics FWNF-2024-0001
2 Institute of Computational Mathematics and Mathematical Geophysics FWNM-2025-0001

Abstract: 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.
Cite: 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
Dates:
Submitted: Dec 14, 2024
Accepted: Mar 16, 2026
Published print: May 29, 2026
Published online: May 29, 2026
Identifiers:
≡ Web of science: WOS:001777736200001
≡ OpenAlex: W7162768938
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