IPI
On the range of the attenuated magnetic ray transform for connections and Higgs fields
Gareth Ainsworth Yernat M. Assylbekov
Inverse Problems & Imaging 2015, 9(2): 317-335 doi: 10.3934/ipi.2015.9.317
For a two-dimensional simple magnetic system, we study the attenuated magnetic ray transform $I_{A,\Phi}$, with attenuation given by a unitary connection $A$ and a skew-Hermitian Higgs field $\Phi$. We give a description for the range of $I_{A,\Phi}$ acting on $\mathbb{C}^n$-valued tensor fields.
keywords: inverse problems Ray transforms tensor tomography magnetic geodesics.
IPI
Boundary and scattering rigidity problems in the presence of a magnetic field and a potential
Yernat M. Assylbekov Hanming Zhou
Inverse Problems & Imaging 2015, 9(4): 935-950 doi: 10.3934/ipi.2015.9.935
In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$ and a potential $U$. For brevity, this type of systems are called $\mathcal{MP}$-systems. On simple $\mathcal{MP}$-systems, we consider both the boundary rigidity problem and scattering rigidity problem. Unlike the cases of geodesic or magnetic systems, knowing boundary action functions or scattering relations for only one energy level is insufficient to uniquely determine a simple $\mathcal{MP}$-system up to natural obstructions, even under the assumption that the boundary restriction of the system is given, and we provide some counterexamples. By reducing an $\mathcal{MP}$-system to the corresponding magnetic system and applying the results of [6] on simple magnetic systems, we prove rigidity results for metrics in a given conformal class, for simple real analytic $\mathcal{MP}$-systems and for simple two-dimensional $\mathcal{MP}$-systems.
keywords: magnetic field action. gauge invariance Boundary rigidity potential
IPI
Reconstruction in the partial data Calderón problem on admissible manifolds
Yernat M. Assylbekov
Inverse Problems & Imaging 2017, 11(3): 455-476 doi: 10.3934/ipi.2017021

We consider the problem of developing a method to reconstruct a potential $q$ from the partial data Dirichlet-to-Neumann map for the Schrödinger equation $(-Δ_g+q)u=0$ on a fixed admissible manifold $(M,g)$. If the part of the boundary that is inaccessible for measurements satisfies a flatness condition in one direction, then we reconstruct the local attenuated geodesic ray transform of the one-dimensional Fourier transform of the potential $q$. This allows us to reconstruct $q$ locally, if the local (unattenuated) geodesic ray transform is constructively invertible. We also reconstruct $q$ globally, if $M$ satisfies certain concavity condition and if the global geodesic ray transform can be inverted constructively. These are reconstruction procedures for the corresponding uniqueness results given by Kenig and Salo [7]. Moreover, the global reconstruction extends and improves the constructive proof of Nachman and Street [14] in the Euclidean setting. We derive a certain boundary integral equation which involves the given partial data and describes the traces of complex geometrical optics solutions. For construction of complex geometrical optics solutions, following [14] and improving their arguments, we use a certain family of Green's functions for the Laplace-Beltrami operator and the corresponding single layer potentials. The constructive inversion problem for local or global geodesic ray transforms is one of the major topics of interest in integral geometry.

keywords: Inverse problems Calderón's problem Dirichlet-to-Neumann map

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