2022
DOI: 10.48550/arxiv.2204.10164
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Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations

Abstract: 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 (linearis… Show more

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Cited by 2 publications
(4 citation statements)
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“…In our next example, we discuss an inverse problem for which the HSE holds on a certain infinite dimensional spaces. For these infinite dimensional spaces [13], showed that the Lipschitz constants occurring are independent of the discretization and therefore, the limitation mentioned in remark 1 no longer exists for this inverse problem.…”
Section: Remarkmentioning
confidence: 99%
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“…In our next example, we discuss an inverse problem for which the HSE holds on a certain infinite dimensional spaces. For these infinite dimensional spaces [13], showed that the Lipschitz constants occurring are independent of the discretization and therefore, the limitation mentioned in remark 1 no longer exists for this inverse problem.…”
Section: Remarkmentioning
confidence: 99%
“…Under the assumed boundary regularity, the ND map belongs to L HS (L 2 (∂Y)) [12, theorem A.2]. We see that with respect to the complex valued L ∞ (Y) perturbations, the forward map γ → Λ(γ) is Fréchet differentiable at γ (see [13]). Moreover, the Fréchet derivative (1).…”
Section: Example 3 (Linearized Calder óN's Problem [13]mentioning
confidence: 99%
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“…Similar iterative algorithms have been used in scattering theory, see [BCR16,BCLV18]. In the context of the Calderón problem, see [GH22] for a recent reconstruction algorithm for small conductivities in dimension 2 in which convergence and stability are proved. Note that, in order to compute the new iteration γ n+1 in (1.9), we have to approximate the Born approximation of γ n and this requires its DtN map.…”
Section: Introductionmentioning
confidence: 99%