Calderón's inverse problem with a finite number of measurements II: independent data

Alberti, Giovanni S.; Santacesaria, Matteo

doi:10.1080/00036811.2020.1745192

Cited by 12 publications

(8 citation statements)

References 34 publications

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“…All these results require infinitely many measurements, even though the number of degrees of freedom to recover are finite. Similar results have been obtained with a finite number of measurements, mostly regarding inverse boundary value problems [27,3,2,29,42,4,30,5,38] and scattering problems in the case when the unknown has a periodic, polygonal or polyhedral structure [25,10,14,31,21,22,37].…”

Section: Introductionsupporting

confidence: 77%

Inverse problems on low-dimensional manifolds

Alberti¹,

Arroyo²,

Santacesaria³

2020

Preprint

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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 finitedimensional 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 and derive a globally-convergent reconstruction algorithm from a finite number of measurements. Several examples are discussed.

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Section: Introductionsupporting

confidence: 77%

Inverse problems on low-dimensional manifolds

Alberti¹,

Arroyo²,

Santacesaria³

2020

Preprint

Self Cite

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“…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.…”

Section: Requires the Associated Neumann Boundary Values To Be In Lsupporting

confidence: 61%

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

Garde¹,

Hyvönen²

2022

Preprint

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We investigate a linearised Calderón problem in a two-dimensional bounded simply connected C 1,α domain Ω. After extending the linearised problem for L 2 (Ω) perturbations, we orthogonally decompose L 2 (Ω) = ⊕ ∞ k=0 H k and prove Lipschitz stability on each of the infinite-dimensional H k subspaces. In particular, H 0 is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearised) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmictype in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general L 2 (Ω) perturbation onto the H k spaces. If the perturbation is in the subspace ⊕ K k=0 H k , then the reconstruction method only requires data corresponding to 2K + 2 specific Neumann boundary values.

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“…Results on uniqueness and Lipschitz stability for finitely many unknowns from infinitely or finitely many measurements have been obtained in, e.g., [2,3,11]. Let us stress that, for most problems, it is still an open question, how many (and which) measurements are required to uniquely determine an unknown PDE coefficient with a given resolution, how to explicitly quantify the error amplification, and how to obtain globally convergent reconstruction algorithms.…”

Section: Further Readingmentioning

confidence: 99%

An Introduction to Finite Element Methods for Inverse Coefficient Problems in Elliptic PDEs

Harrach

2021

Jahresber. Dtsch. Math. Ver.

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Several novel imaging and non-destructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, the unknown coefficient is often assumed to be piecewise constant on a given pixel partition (corresponding to the desired resolution), and only finitely many measurement can be made. This leads to the problem of inverting a finite-dimensional non-linear forward operator $\mathcal{F}:\ \mathcal{D}(\mathcal{F})\subseteq \mathbb{R}^{n}\to \mathbb{R}^{m}$ F : D ( F ) ⊆ R n → R m , where evaluating ℱ requires one or several PDE solutions.Numerical inversion methods require the implementation of this forward operator and its Jacobian. We show how to efficiently implement both using a standard FEM package and prove convergence of the FEM approximations against their true-solution counterparts. We present simple example codes for Comsol with the Matlab Livelink package, and numerically demonstrate the challenges that arise from non-uniqueness, non-linearity and instability issues. We also discuss monotonicity and convexity properties of the forward operator that arise for symmetric measurement settings.This text assumes the reader to have a basic knowledge on Finite Element Methods, including the variational formulation of elliptic PDEs, the Lax-Milgram-theorem, and the Céa-Lemma. Section 3 also assumes that the reader is familiar with the concept of Fréchet differentiability.

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Calderón's inverse problem with a finite number of measurements II: independent data

Cited by 12 publications

References 34 publications

Inverse problems on low-dimensional manifolds

Inverse problems on low-dimensional manifolds

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

An Introduction to Finite Element Methods for Inverse Coefficient Problems in Elliptic PDEs

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