Browsing by Subject "INVERSE PROBLEM"

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  • Blåsten, Emilia; Zouari, Fedi; Louati, Moez; Ghidaoui, Mohamed S. (2019)
    In this note we present a reconstructive algorithm for solving the cross-sectional pipe area from boundary measurements in a tree network with one inaccessible end. This is equivalent to reconstructing the first order perturbation to a wave equation on a quantum graph from boundary measurements at all network ends except one. The method presented here is based on a time reversal boundary control method originally presented by Sondhi and Gopinath for one dimensional problems and later by Oksanen to higher dimensional manifolds. The algorithm is local, so is applicable to complicated networks if we are interested only in a part isomorphic to a tree. Moreover the numerical implementation requires only one matrix inversion or least squares minimization per discretization point in the physical network. We present a theoretical solution existence proof, a step-by-step algorithm, and a numerical implementation applied to two numerical experiments.
  • Helin, T.; Lassas, M.; Oksanen, L.; Saksala, T. (2018)
    Passive imaging refers to problems where waves generated by unknown sources are recorded and used to image the medium through which they travel. The sources are typically modelled as a random variable and it is assumed that some statistical information is available. In this paper we study the stochastic wave equation partial derivative(2)(t)u- Delta(g)u = chi W, where W is a random variable with the white noise statistics on R1+n, n >= 3, chi is a smooth function vanishing for negative times and outside a compact set in space, and Delta(g) is the Laplace Beltrami operator associated to a smooth non-trapping Riemannian metric tensor g on R-n. The metric tensor g models the medium to be imaged, and we assume that it coincides with the Euclidean metric outside a compact set. We consider the empirical correlations on an open set chi subset of R-n, C-T(t(1), x(1), t(2), x(2)) = 1/T integral(T)(0) u(t(1) s, x(1))u(t(2) s, x(2))ds, t(1), t(2) > 0, x(1), x(2) is an element of chi, for T > 0. Supposing that chi is non-zero on chi and constant in time after t > 1, we show that in the limit T -> infinity, the data C-T becomes statistically stable, that is, independent of the realization of W. Our main result is that, with probability one, this limit determines the Riemannian manifold (R-n, g) up to an isometry. (C) 2018 Elsevier Masson SAS. All rights reserved.
  • Lassas, Matti; Saksala, Teemu (2019)
    Let (N, g) be a Riemannian manifold with the distance function d(x, y) and an open subset M subset of N. For x is an element of M we denote by D-x the distance difference function D-x:F x F -> R, given by D-x(z(1), z(2)) = d(x, z(1)) - d(x, z(2)), z(1), z(2) is an element of F = N \ M. We consider the inverse problem of determining the topological and the differentiable structure of the manifold M and the metric g vertical bar M on it when we are given the distance difference data, that is, the set F, the metric g vertical bar F, and the collection D(M) = {D-x; x is an element of M}. Moreover, we consider the embedded image D(M) of the manifold M, in the vector space C(F x F), as a representation of manifold M. The inverse problem of determining (M, g) from D(M) arises e.g. in the study of the wave equation on R x N when we observe in F the waves produced by spontaneous point sources at unknown points (t, x) is an element of R x M. Then D-x (z(1), z(2)) is the difference of the times when one observes at points z(1) and z(2) the wave produced by a point source at x that goes off at an unknown time. The problem has applications in hybrid inverse problems and in geophysical imaging.
  • Stefanov, Plamen; Uhlmann, Gunther; Vasy, Andras (2018)
    We study the isotropic elastic wave equation in a bounded domain with boundary. We show that local knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the p-wave locally if there is a strictly convex foliation with respect to it, and similarly for the s-wave speed.
  • Rundell, William; Zhang, Zhidong (2020)
    We consider the recovery of a source term f (x, t) = p(x)q(t) for the nonhomogeneous heat equation in Omega x (0, infinity) where Omega is a bounded domain in R-2 with smooth boundary partial derivative Omega from overposed lateral data on a sparse subset of partial derivative Omega x (0, infinity). Specifically, we shall require a small finite number N of measurement points on partial derivative Omega and prove a uniqueness result, namely, the recovery of the pair (p, q) within a given class, by a judicious choice of N = 2 points. Naturally, with this paucity of overposed data, the problem is severely ill-posed. Nevertheless we shall show that, provided the data noise level is low, effective numerical reconstructions may be obtained.
  • Hauptmann, Andreas; Ikehata, Masaru; Itou, Hiromichi; Siltanen, Samuli (2019)
    An algorithm is introduced for using electrical surface measurements to detect and monitor cracks inside a two-dimensional conductive body. The technique is based on transforming the probing functions of the classical enclosure method by the Kelvin transform. The transform makes it possible to use virtual discs for probing the interior of the body using electric measurements performed on a flat surface. Theoretical results are presented to enable probing of the full domain to create a profile indicating cracks in the domain. Feasibility of the method is demonstrated with a simulated model of attaching metal sheets together by resistance spot welding.
  • Bosi, Roberta; Kurylev, Yaroslav; Lassas, Matti (2018)
    In 1995, Tataru proved a Carleman-type estimate for linear operators with partially analytic coefficients that is generally used to prove the unique continuation of those operators. In this paper, we use this inequality to study the stability of the unique continuation in the case of the wave equation with coefficients independent of time. We prove a logarithmic estimate in a ball whose radius has an explicit dependence on the C (1)-norm of the coefficients and on the other geometric properties of the operator.
  • Lassas, Matti; Oksanen, Lauri; Stefanov, Plamen; Uhlmann, Gunther (2020)
    We study the weighted light ray transform L of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze L as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities of a function f from its the weighted light ray transform Lf by a suitable filtered back-projection.
  • Ferreira, David Dos Santos; Kurylev, Yaroslav; Lassas, Matti; Liimatainen, Tony; Salo, Mikko (2020)
    In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an Fourier-Bros-Iagolnitzer-type transform at certain points in the transversal manifold. This leads to proving a uniqueness result for transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform, which has limited earlier results on this problem.