Inverse problems on low-dimensional manifolds

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.

arXiv comment70 pages, 5 figures. This revised version contains the application of the abstract results to the Calderón problem with a triangular inclusion and to the Gel'fand-Calderón problem with piecewise constant potentials on non-intersecting balls

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• Lipschitz stability estimate for the simultaneous recovery of two coefficients in the anisotropic Schrödinger type equation via local Cauchy data
• Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces
• The learned range test method for the inverse inclusion problem
• Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks
• Sampling theorems for inverse problems on Riemannian manifolds