62 research outputs found
Dynamic inverse problem in a weakly laterally inhomogeneous medium
An inverse problem of wave propagation into a weakly laterally inhomogeneous
medium occupying a half-space is considered in the acoustic approximation. The
half-space consists of an upper layer and a semi-infinite bottom separated with
an interface. An assumption of a weak lateral inhomogeneity means that the
velocity of wave propagation and the shape of the interface depend weakly on
the horizontal coordinates, , in comparison with the strong
dependence on the vertical coordinate, , giving rise to a small parameter
\e <<1. Expanding the velocity in power series with respect to \e, we
obtain a recurrent system of 1D inverse problems. We provide algorithms to
solve these problems for the zero and first-order approximations. In the
zero-order approximation, the corresponding 1D inverse problem is reduced to a
system of non-linear Volterra-type integral equations. In the first-order
approximation, the corresponding 1D inverse problem is reduced to a system of
coupled linear Volterra integral equations. These equations are used for the
numerical reconstruction of the velocity in both layers and the interface up to
O(\e^2).Comment: 12 figure
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 âł = Îââ2 associated with a Fuchsian group of the first kind Î containing parabolic elements. The surface âł 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
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
Reconstruction and stability in Gel'fand's inverse interior spectral problem
Assume that 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 on as well as the corresponding eigenfunctions restricted on an open set in . We then construct a stable approximation to the manifold . Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from 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
Spectral stability of metric-measure Laplacians
We consider a "convolution mm-Laplacian" operator on metric-measure spaces and study its spectral properties. The definition is based on averaging over small metric balls. For reasonably nice metric-measure spaces we prove stability of convolution Laplacian's spectrum with respect to metric-measure perturbations and obtain Weyl-type estimates on the number of eigenvalues
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
Reconstruction and interpolation of manifolds I: The geometric Whitney problem
We study the geometric Whitney problem on how a Riemannian manifold can be constructed to approximate a metric space . This problem is closely related to manifold interpolation (or manifold learning) where a smooth -dimensional surface , needs to be constructed to approximate a point cloud in . 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 and interpolated to a smooth surface. We also give algorithms that solve the problems for finite data
G-convergence, Dirichlet to Neumann maps and invisibility
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 [19] are independent of the source
Inverse problems for the connection Laplacian
We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a corollary we derive an elliptic analogue of the main result which solves a CalderĂłn problem for connections on a cylinder
Full-wave invisibility of active devices at all frequencies
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 . We give two basic constructions for cloaking a region
contained in a domain from measurements of Cauchy data of waves at \p
\Omega; we pay particular attention to cloaking not just a passive object, but
an active device within , interpreted as a collection of sources and sinks
or an internal current.Comment: Final revision; to appear in Commun. in Math. Physic
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