Inverse problems on low-dimensional manifolds
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Título: | Inverse problems on low-dimensional manifolds |
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Autor/es: | Alberti, Giovanni S. | Arroyo, Ángel | Santacesaria, Matteo |
Centro, Departamento o Servicio: | Universidad de Alicante. Departamento de Matemáticas |
Palabras clave: | Inverse problems | Calderón problem | Gel’fand-Calderón problem | Machine learning | Manifolds | Lipschitz stability | Reconstruction algorithm |
Fecha de publicación: | 15-dic-2022 |
Editor: | IOP Publishing |
Cita bibliográfica: | Nonlinearity. 2023, 36: 734-808. https://doi.org/10.1088/1361-6544/aca73d |
Resumen: | We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess. These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calderón's inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel'fand-Calderón's problem for the Schrödinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls. |
Patrocinador/es: | This material is based upon work supported by the Air Force Office of Scientific Research under Award Number FA8655-20-1-7027. Á A is partially supported by the Grants MTM2017-85666-P and 2017 SGR 395. |
URI: | http://hdl.handle.net/10045/140158 |
ISSN: | 0951-7715 (Print) | 1361-6544 (Online) |
DOI: | 10.1088/1361-6544/aca73d |
Idioma: | eng |
Tipo: | info:eu-repo/semantics/article |
Derechos: | © 2022 IOP Publishing Ltd & London Mathematical Society |
Revisión científica: | si |
Versión del editor: | https://doi.org/10.1088/1361-6544/aca73d |
Aparece en las colecciones: | INV - GAM - Artículos de Revistas Personal Investigador sin Adscripción a Grupo |
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Alberti_etal_2023_Nonlinearity_final.pdf | Versión final (acceso restringido) | 771,38 kB | Adobe PDF | Abrir Solicitar una copia |
Alberti_etal_2023_Nonlinearity_preprint.pdf | Preprint (acceso abierto) | 665,68 kB | Adobe PDF | Abrir Vista previa |
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