Showing papers by "Yaroslav Kurylev published in 2013"
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TL;DR: In this article, the authors considered the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions, and showed that the boundary measurements uniquely determine a mixed Fourier transform / attenuated geodesic ray transform (or integral against a more general semiclassical limit measure) of an unknown coefficient.
Abstract: We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Fourier transform / attenuated geodesic ray transform (or integral against a more general semiclassical limit measure) of an unknown coefficient. In particular, one obtains uniqueness results whenever the geodesic ray transform on the transversal manifold is injective. The second result shows that the boundary measurements in an infinite cylinder uniquely determine the transversal metric. The first result is proved by using complex geometrical optics solutions involving Gaussian beam quasimodes, and the second result follows from a connection between the Calderon problem and Gel'fand's inverse problem for the wave equation and the boundary control method.
66 citations
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TL;DR: It is shown that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvectors of a graph Laplace operator of a proximity graph on an epsilon-net.
Abstract: We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
6 citations
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TL;DR: In this article, the authors considered inverse problems for the Einstein equation with a time-depending metric on a 4-dimensional globally hyperbolic Lorentzian manifold and formulated the concept of active measurements for relativistic models.
Abstract: We consider inverse problems for the Einstein equation with a time-depending metric on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We formulate the concept of active measurements for relativistic models. We do this by coupling the Einstein equation with equations for scalar fields and study the system $Ein(g)=T$, $T=T(g,\phi)+F_1$, and $\square_g \phi=F_2+S(g,\phi,F_1,F_2)$. Here $F=(F_1,F_2)$ correspond to the perturbations of the physical fields which we control and $S$ is a secondary source corresponding to the adaptation of the system to the perturbation so that the conservation law $div_g(T)=0$ will be satisfied. The inverse problem we study is the question, do the observation of the solutions $(g,\phi)$ in an open subset $U\subset M$ of the space-time corresponding to sources $F$ supported in $U$ determine the properties of the metric in a larger domain $W\subset M$ containing $U$. To study this problem we define the concept of light observation sets and show that these sets determine the conformal class of the metric. This corresponds to passive observations from a distant area of the space which is filled by light sources (e.g. we see light from stars varying in time). One can apply the obtained result to solve inverse problems encountered in general relativity and in various practical imaging problems.
5 citations
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TL;DR: In this paper, a brief survey of the recent development of inverse scattering theory on non-compact Riemannian manifolds is given, where the main theme is the reconstruction of the manifold and the metric from the scattering matrix.
Abstract: We give a brief survey for the recent development of inverse scattering theory on non-compact Riemannian manifolds. The main theme is the reconstruction of the manifold and the metric from the scattering matrix.
5 citations
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TL;DR: In this article, an inverse problem associated with asymptotically hyperbolic orbifolds with a finite number of cusps and regular ends was considered, and a generalized $S$-matrix was introduced to determine the manifolds with Riemannian metric and the orbifold structure.
Abstract: We consider an inverse problem associated with $n$-dimensional asymptotically hyperbolic orbifolds $(n \geq 2)$ having a finite number of cusps and regular ends. By observing solutions of the Helmholtz equation at the cusp, we introduce a generalized $S$-matrix, and then show that it determines the manifolds with its Riemannian metric and the orbifold structure.
3 citations
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TL;DR: Greenleaf et al. as mentioned in this paper established optimal conditions under which the G-convergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps.
Abstract: We establish optimal conditions under which the G-convergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps. As an application we show that the approximate cloaking isotropic materials from [ Greenleaf, A.; Kurylev, Y.; Lassas, M.; Uhlmann, G. Approximate quantum and acoustic cloaking. J. Spectr. Theory (2011), no. 1, 27--80.] are independent of the source.
1 citations