Alberti, Giovanni S., Arroyo, Ángel, Santacesaria, Matteo
Inverse problems on low-dimensional manifolds
Nonlinearity. 2023, 36: 734-808. https://doi.org/10.1088/1361-6544/aca73d
URI: http://hdl.handle.net/10045/140158
DOI: 10.1088/1361-6544/aca73d
ISSN: 0951-7715 (Print)
Abstract: 
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.
Keywords:Inverse problems, Calderón problem, Gel’fand-Calderón problem, Machine learning, Manifolds, Lipschitz stability, Reconstruction algorithm
IOP Publishing
info:eu-repo/semantics/article