Infinite-Dimensional Inverse Problems with Finite Measurements

Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations

“…To our knowledge, Theorem 1.4 is the first infinite-dimensional Lipschitz stability result in the linearised Calderón problem, which nicely complements the long tradition for Lipschitz stability results in finite-dimensional settings for the nonlinear Calderón problem; see e.g. [22,1,2,3] for some recent general results. This suggests that it may be beneficial to consider stability in terms of ND maps, instead of DN maps, due to the more desirable topological properties of the former.…”

A Bernstein–von-Mises theorem for the Calderón problem with piecewise constant conductivities

“…Statements of this form were proved e.g. in [2,3,15] and in fact hold true for a larger class of inverse problems, essentially as long as the forward map (here γ → Λ γ , mapping between appropriate spaces) and its linearisation are known to be injective. One aim of this article is to bridge the gap between the just mentioned advances in deterministic inverse problems and the statistical literature; this comes with the following message: if there are good reasons to model an inverse problem on a parameter space of fixed finite dimension D, then under natural injectivity assumptions, the analysis of the noisy problem falls into a wellunderstood regime, where excellent statistical guarantees-including Le Cam's version of the Bernstein-von-Mises theorem-become available.…”

“…This is a widely studied variant of the problem (see e.g. [2,3,11,15,17]) that yields a model with parameter space of fixed dimension D. Postponing precise definitions to section 1.2, our setting can be described as follows: we take the viewpoint of electrical impedance tomography (EIT), where one seeks to recover an unknown conductivity γ inside a body Ω ⊂ R d from voltage-to-current measurements Λ γ at the boundary ∂Ω. We make the assumption that γ is piecewise constant with respect to a given partition Ω 1 ∪ • • • ∪ Ω D ⊂ Ω and obeys bounds 0 < γ min γ γ max , such that it can be described by a parameter…”

Continuous Generative Neural Networks

“…Let Ψl and σl be defined as in (7) and (8), respectively. Let Ψ 1 be defined as in (4). If Hypotheses 1, 3, 4 and 5 are satisfied, then the generator G defined in (6) is injective.…”

“…Indeed, another useful property of CGNNs is dimensionality reduction. For ill-posed inverse problems, it is well known that imposing finite-dimensional priors improves the stability and the quality of the reconstruction [6,14,13,5,15], also working with finitely-many measurements [3,31,2,4,1]. In practice, these priors are unknown or cannot be analytically described: yet, they can be approximated by a (trained) CGNN.…”