Limiting Carleman weights and anisotropic inverse problems
TLDR
In this article, the authors considered the anisotropic Calderon problem and related inverse problems, and characterized those Riemannian manifolds which admit limiting Carleman weights, and gave a complex geometrical optics construction for a class of such manifolds.Abstract:
In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.read more
Citations
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Uniform stability estimates for the discrete Calderon problems
TL;DR: In this article, the authors focus on the analysis of discrete versions of the Calderon problem in dimension d ≥ 3 and obtain stability estimates for the discrete Calderon problems that hold uniformly with respect to the discretization parameter.
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Calderón problem for connections
TL;DR: In this article, the problem of identifying a connection ∇ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ∇*∇ over conformally transversally anisotropic (CTA) manifolds was considered.
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Reconstructing electromagnetic obstacles by the enclosure method
TL;DR: In this paper, the boundary measurements are encoded in the impedance map that sends the tangential component of the electric field to tangential components of the magnetic field, and the boundary measurement is obtained by probing the medium with complex geometrical optics solutions to the corresponding Maxwell's equations and extend the enclosure method to this case.
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Recovering a quasilinear conductivity from boundary measurements
TL;DR: In this article, the authors consider the inverse problem of recovering an isotropic quasilinear conductivity from the Dirichlet-to-Neumann map when the conductivity depends on the solution and its gradient.
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Reconstructions from boundary measurements on admissible manifolds
TL;DR: In this paper, it was shown that a potential can be reconstructed from the Dirichlet-to-Neumann map for the Schrodinger operator in a fixed admissible 3-dimensional Riemannian manifold.
References
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The Analysis of Linear Partial Differential Operators I
TL;DR: In this article, the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certain products using instruction manuals, which are clearlybuilt to give step-by-step information about how you ought to go ahead in operating certain equipments.
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TL;DR: A very readable introduction to Riemannian geometry and geometric analysis can be found in this paper, where the author focuses on using analytic methods in the study of some fundamental theorems in Riemmannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, Lyusternik and Fet theorem and the existence of harmonic mappings.
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A global uniqueness theorem for an inverse boundary value problem
John Sylvester,Gunther Uhlmann +1 more
TL;DR: In this paper, the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n? 3.
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