Electromagnetic wormholes and virtual magnetic monopoles

We describe new configurations of electromagnetic (EM) material parameters, the electric permittivity $\epsilon$ and magnetic permeability $\mu$, 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.Comment: 4 pages, 3 figure

Hyperbolic inverse problem with data on disjoint sets

We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and the Neumann traces are restricted on another subset. We show that the restricted Dirichlet-to-Neumann map determines the geometry and the lower order terms in the wave equation, up the natural gauge invariances, along a convex foliation of the manifold. The main novelty is the recovery of the lower order terms when the supports of the Dirichlet traces are disjoint from the set on which the Neumann traces are restricted. We allow the lower order terms to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy

Rigidity of broken geodesic flow and inverse problems

Consider broken geodesics alpha([0, 1]) on a compact Riemannian manifold (M, g) with boundary of dimension n >= 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 alpha([0, 1]) starting at and ending to the boundary partial derivative M we know the starting point and direction (alpha(0), alpha'(0)), the end point and direction (alpha(1), alpha'(1)), and the length 1. 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 nonconstant 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

Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces

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 ℳ = Γ∖ℍ2 associated with a Fuchsian group of the first kind Γ containing parabolic elements. The surface 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

Gel'fand inverse problem for a quadratic operator pencil

In this paper we consider the inverse boundary value problem for the operator pencil A(lambda) = a(x, D)-i lambda b(0)(x)-lambda(2) where a(x, D) is an elliptic second-order operator on a differentiable manifold M with boundary. The manifold M can be interpreted as a Riemannian manifold (M, g) where g is the metric generated by a(x, D). We assume that the Gel'fand data on the boundary is given; i.e., we know the boundary partial derivative M and the boundary values of the fundamental solution of A(lambda), namely, R-lambda(x, y), x, y is an element of partial derivative M, lambda is an element of C. We show that if (M, g) satisfies some geometric condition then the Gel'fand data determine the manifold M, the metric g, the coefficient b(0)(s) uniquely and also the equivalence class of a(x, D) with respect to the group of generalized gauge transformations

Reconstruction and stability in Gel'fand's inverse interior spectral problem

Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian $\Delta$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit logarithmic stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gel'fand's inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric Boundary Control metho

Dynamic inverse problem for hyperbolic equation and continuation of boundary data

We consider an inverse problem for a second order hyperbolic initial boundary value problem on a compact Riemannian manifold M with boundary

Inverse problem for wave equation with sources and observations on disjoint sets

We consider an inverse problem for a hyperbolic partial differential equation on a compact Riemannian manifold. Assuming that $\Gamma_1$ and $\Gamma_2$ are two disjoint open subsets of the boundary of the manifold we define the restricted Dirichlet-to-Neumann operator $\Lambda_{\Gamma_1,\Gamma_2}$. This operator corresponds the boundary measurements when we have smooth sources supported on $\Gamma_1$ and the fields produced by these sources are observed on $\Gamma_2$. We show that when $\Gamma_1$ and $\Gamma_2$ are disjoint but their closures intersect at least at one point, then the restricted Dirichlet-to-Neumann operator $\Lambda_{\Gamma_1,\Gamma_2}$ determines the Riemannian manifold and the metric on it up to an isometry. In the Euclidian space, the result yields that an anisotropic wave speed inside a compact body is determined, up to a natural coordinate transformations, by measurements on the boundary of the body even when wave sources are kept away from receivers. Moreover, we show that if we have three arbitrary non-empty open subsets $\Gamma_1,\Gamma_2$, and $\Gamma_3$ of the boundary, then the restricted Dirichlet-to-Neumann operators $\Lambda_{\Gamma_j,\Gamma_k}$ for $1\leq j<k\leq 3$ determine the Riemannian manifold to an isometry. Similar result is proven also for the finite-time boundary measurements when the hyperbolic equation satisfies an exact controllability condition

Reconstruction and interpolation of manifolds I: The geometric Whitney problem

We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold interpolation (or manifold learning) where a smooth $n$-dimensional surface $S\subset {\mathbb R}^m$, $m>n$ needs to be constructed to approximate a point cloud in ${\mathbb R}^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric. We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary. Moreover, we characterise the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius. The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalisation of the Whitney embedding construction where approximative coordinate charts are embedded in ${\mathbb R}^m$ and interpolated to a smooth surface. We also give algorithms that solve the problems for finite data