arXiv subjects
Publications and source records attributed to Yaroslav Kurylev.
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.
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 $Ein(g)=T$, $T=T(g,ϕ)+F_1$, and $\square_g ϕ=F_2+S(g,ϕ,F_1,F_2)$. 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 $(g,ϕ)$ 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.
We study the spectral theory and inverse problem on asymptotically hyperbolic manifolds. The main subjects are as follows: (1)Location of the essential spectrum. (2)Absence of eigenvalues embedded in the continuous spectrum. (3)Limiting absorption principle for the resolvent and the absolute continuity of the continuous spectrum. (4)Construction of the generalized Fourier transform. (5)symptotic completeness of time-dependent wave operators. (6)Characterization of the space of scattering solutions to the Helmhotz equation in terms of the generalized Fourier transform. (7)Asymptotic expansion of scattering solutions to the Helmholtz equation and the S-matrix. (8)Representation of the fundamental solution to the wave equation in the upper-half space model. (9)Radon transform and the propagation of singularities for the wave equation. Finally, we shall discuss the inverse problem. Namely (10)Identification of the Riemannian metric from the scattering matrix.
We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is then non-compact, and has a finite number of cusps and elliptic singular points, which is regarded as a hyperbolic orbifold. We introduce a class of Riemannian surfaces with conical singularities on its finite part, having cusps and regular ends at infinity, whose metric is asymptotically hyperbolic. By observing solutions of the Helmholtz equation at the cusp, we define a generalized S-matrix. We then show that this generalized S-matrix determines the Riemannian metric and the structure of conical singularities.
The advent of transformation optics and metamaterials has made possible devices producing extreme effects on wave propagation. Here we give theoretical designs for devices, Schrödinger hats, acting as invisible concentrators of waves. These exist for any wave phenomenon modeled by either the Helmholtz or Schrödinger equations, e.g., polarized waves in EM, pressure waves in acoustics and matter waves in QM, and occupy one part of a parameter space continuum of wave-manipulating structures which also contains standard transformation optics based cloaks, resonant cloaks and cloaked sensors. For EM and acoustic Schrödinger hats, the resulting centralized wave is a localized excitation. In QM, the result is a new charged quasiparticle, a \emph{quasmon}, which causes conditional probabilistic illusions. We discuss possible solid state implementations.
It is generally believed that transformation optics based cloaking, besides rendering the cloaked region invisible to detection by scattering of incident waves, also shields the region from those same waves. We demonstrate a coupling between the cloaked and uncloaked regions, exposing a difference between cloaking for rays and waves. Interior resonances allow this coupling to be amplified, and careful choice of parameters leads to effective cloaks with degraded shielding. As one application, we describe how to use transformation optics to hide sensors in the cloaked region and yet enable the sensors to efficiently measure waves incident on the exterior of the cloak, an effect similar to the plasmon based approach of Alu' and Engheta.
We consider an invariant formulation of the system of Maxwell's equations for an anisotropic medium on a compact orientable Riemannian 3-manifold $(M,g)$ with nonempty boundary. The system can be completed to a Dirac type first order system on the manifold. We show that the Betti numbers of the manifold can be recovered from the dynamical response operator for the Dirac system given on a part of the boundary. In the case of the original physical Maxwell system, assuming that the entire boundary is known, all Betti numbers of the manifold can also be determined from the dynamical response operator given on a part of the boundary. Physically, this operator maps the tangential component of the electric field into the tangential component of the magnetic field on the boundary.
We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties : On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form $(dy)^2 + h(x,dx)$, $h(x,dx)$ being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to $(dy)^2 + h(x,dx)$. Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energy, we show that these two manifolds are isometric.
At any energy E > 0, we construct a sequence of bounded potentials $V^E_{n}, n\in\N$, supported in an annular region $B_{out}\setminus B_{inn}$ in three-space, which act as approximate cloaks for solutions of Schrödinger's equation: For any potential $V_0\in L^\infty(B_{inn})$ such that E is not a Neumann eigenvalue of $-Δ+V_0$ in $B_{inn}$, the scattering amplitudes $a_{V_0+V_n^E}(E,θ,ω)\to 0$ as $n\to\infty$. The $V^E_{n}$ thus not only form a family of approximately transparent potentials, but also function as approximate invisibility cloaks in quantum mechanics. On the other hand, for $E$ close to interior eigenvalues, resonances develop and there exist {\it almost trapped states} concentrated in $B_{inn}$. We derive the $V_n^E$ from singular, anisotropic transformation optics-based cloaks by a de-anisotropization procedure, which we call \emph{isotropic transformation optics}. This technique uses truncation, inverse homogenization and spectral theory to produce nonsingular, isotropic approximate cloaks. As an intermediate step, we also obtain approximate cloaking for a general class of equations including the acoustic equation.
This survey of recent developments in cloaking and transformation optics is an expanded version of the lecture by Gunther Uhlmann at the 2008 Annual Meeting of the American Mathematical Society.
In a recent Letter, Cummer et al. give a description of material parameters for acoustic wave propagation giving rise to a 3D spherical cloak, and verify the cloaking phenomenon on the level of scattering coefficients. A similar configuration has been given by Chen and Chan. In this Comment, we show that these theoretical constructions follow directly from our earlier work http://arxiv.org/abs/math/0611185 on full wave analysis of cloaking for the Helmholtz equation with respect to Riemannian metrics. Furthermore, the analysis there covers the case of acoustically radiating objects being enclosed in the cloaked region.
In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $Σ$ into two components and we know the eigenvalues $λ_j$ of the Laplace operator on $(M,g)$ and also the Cauchy data, on $Σ$, of the corresponding eigenfunctions $ϕ_j$, i.e. $ϕ_j|_Σ,\partial_νϕ_j|_Σ$, where $ν$ is the normal to $Σ$. We prove that these data determine $(M,g)$ uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, $λ_j$ and $ϕ_j|_Σ$ only. However, if $Σ$ consists of at least two components, $Σ_1, Σ_2$, we are still able to determine $(M,g)$ assuming some conditions on $M$ and $Σ$. These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting $M$ along $Σ_i$, $i=1,2$, and are of a generic nature. We consider also some other inverse problems on $M$ related to the above with data which is easier to obtain from measurements than the spectral data described.
We analyze, both analytically and numerically, the effectiveness of cloaking an infinite cylinder from observations by electromagnetic waves in three dimensions. We show that, as truncated approximations of the ideal permittivity and permeability tensors tend towards the singular ideal cloaking fields, so that the anisotropy ratio tends to infinity, the $D$ and $B$ fields blow up near the cloaking surface. We also consider cloaking with and without the SHS (soft-and-hard surface) lining. We demonstrate numerically that cloaking is significantly improved by the SHS lining, with both the far field of the scattered wave significantly reduced and the blow up of $D$ and $B$ prevented.
Cloaking devices are prescriptions of electrostatic, optical or electromagnetic parameter fields (conductivity $σ(x)$, index of refraction $n(x)$, or electric permittivity $ε(x)$ and magnetic permeability $μ(x)$) which are piecewise smooth on $\mathbb R^3$ and singular on a hypersurface $Σ$, and such that objects in the region enclosed by $Σ$ are not detectable to external observation by waves. Here, we give related constructions of invisible tunnels, which allow electromagnetic waves to pass between possibly distant points, but with only the ends of the tunnels visible to electromagnetic imaging. Effectively, these change the topology of space with respect to solutions of Maxwell's equations, corresponding to attaching a handlebody to $\mathbb R^3$. The resulting devices thus function as electromagnetic wormholes.
We describe new configurations of electromagnetic (EM) material parameters, the electric permittivity $ε$ and magnetic permeability $μ$, that allow one to construct from metamaterials objects that function as invisible tunnels. These allow EM wave propagation between two points, but the tunnels and the regions they enclose are not detectable to EM observations. Such devices function as wormholes with respect to Maxwell's equations and effectively change the topology of space vis-a-vis EM wave propagation. We suggest several applications, including devices behaving as virtual magnetic monopoles.
Consider a broken geodesics $α([0,l])$ on a compact Riemannian manifold $(M,g)$ with boundary of dimension $n\geq 3$. The broken geodesics are unions of two geodesics with the property that they have a common end point. Assume that for every broken geodesic $α([0,l])$ starting at and ending to the boundary $\partial M$ we know the starting point and direction $(α(0),α'(0))$, the end point and direction $(α(l),α'(l))$, and the length $l$. We show that this data determines uniquely, up to an isometry, the manifold $(M,g)$. This result has applications in inverse problems on very heterogeneous media for situations where there are many scattering points in the medium, and arises in several applications including geophysics and medical imaging. As an example we consider the inverse problem for the radiative transfer equation (or the linear transport equation) with a non-constant wave speed. Assuming that the scattering kernel is everywhere positive, we show that the boundary measurements determine the wave speed inside the domain up to an isometry.
There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or "cloaking") from observation by electromagnetic (EM) waves. Here, we prove invisibility, with respect to solutions of the Helmholtz and Maxwell's equations, for several constructions of cloaking devices. Previous results have either been on the level of ray tracing [Le,PSS] or at zero frequency [GLU2,GLU3], but recent numerical [CPSSP] and experimental [SMJCPSS] work has provided evidence for invisibility at frequency $k\ne 0$. We give two basic constructions for cloaking a region $D$ contained in a domain $Ω$ from measurements of Cauchy data of waves at $\p Ω$; we pay particular attention to cloaking not just a passive object, but an active device within $D$, interpreted as a collection of sources and sinks or an internal current.
We consider an admissible Riemannian polyhedron with piece-wise smooth boundary. The associated Laplace defines the boundary spectral data as the set of eigenvalues and restrictions to the boundary of the corresponding eigenfunctions. In this paper we prove that the boundary spectral data prescribed on an open subset of the polyhedron boundary determine the admissible Riemannian polyhedron uniquely.