195 research outputs found

    Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

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    In this article we study the fan-beam Radon transform Dm{\cal D}_m of symmetrical solenoidal 2D tensor fields of arbitrary rank mm in a unit disc D\mathbb D as the operator, acting from the object space L2(D;Sm){\mathbf L}_{2}(\mathbb D; {\bf S}_m) to the data space L2([0,2π)×[0,2π)).L_2([0,2\pi)\times[0,2\pi)). The orthogonal polynomial basis sn,k(±m){\bf s}^{(\pm m)}_{n,k} of solenoidal tensor fields on the disc D\mathbb D was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator Dm{\cal D}_m was obtained. The inversion formula for the fan-beam tensor transform Dm{\cal D}_m follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.Comment: LaTeX, 37 pages with 5 figure

    Simultaneous Identification of the Diffusion Coefficient and the Potential for the Schr\"odinger Operator with only one Observation

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    This article is devoted to prove a stability result for two independent coefficients for a Schr\"odinger operator in an unbounded strip. The result is obtained with only one observation on an unbounded subset of the boundary and the data of the solution at a fixed time on the whole domain

    Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions

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    We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.Comment: 43 page

    Global stability for the multi-channel Gel'fand-Calder\'on inverse problem in two dimensions

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    We prove a global logarithmic stability estimate for the multi-channel Gel'fand-Calder\'on inverse problem on a two-dimensional bounded domain, i.e. the inverse boundary value problem for the equation Δψ+vψ=0-\Delta \psi + v\, \psi = 0 on DD, where vv is a smooth matrix-valued potential defined on a bounded planar domain DD

    Increasing stability for the inverse problem for the Schr\"odinger equation

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    In this article, we study the increasing stability property for the determination of the potential in the Schr\"odinger equation from partial data. We shall assume that the inaccessible part of the boundary is flat and homogeneous boundary condition is prescribed on this part. In contrast to earlier works, we are able to deal with the case when potentials have some Sobolev regularity and also need not be compactly supported inside the domain

    CALDER 3N'S INVERSE PROBLEM with A FINITE NUMBER of MEASUREMENTS

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    We prove that an L 1e potential in the Schr\uf6dinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace W. As a corollary, we obtain a similar result for Calder\uf3n's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces W, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of W

    A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

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    We consider a transmission wave equation in two embedded domains in R2R^2, where the speed is a1>0a1 > 0 in the inner domain and a2>0a2 > 0 in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strictly convex and a1>a2a1 > a2 . As a consequence of this inequality, uniqueness and Lip- schitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement

    Inverse problems with partial data for a magnetic Schr\"odinger operator in an infinite slab and on a bounded domain

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    In this paper we study inverse boundary value problems with partial data for the magnetic Schr\"odinger operator. In the case of an infinite slab in RnR^n, n3n\ge 3, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same hyperplane. This is a generalization of the results of [41], obtained for the Schr\"odinger operator without magnetic potentials. In the case of a bounded domain in RnR^n, n3n\ge 3, extending the results of [2], we show the unique determination of the magnetic field and electric potential from the Dirichlet and Neumann data, given on two arbitrary open subsets of the boundary, provided that the magnetic and electric potentials are known in a neighborhood of the boundary. Generalizing the results of [31], we also obtain uniqueness results for the magnetic Schr\"odinger operator, when the Dirichlet and Neumann data are known on the same part of the boundary, assuming that the inaccessible part of the boundary is a part of a hyperplane

    Inverse scattering at fixed energy on surfaces with Euclidean ends

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    On a fixed Riemann surface (M0,g0)(M_0,g_0) with NN Euclidean ends and genus gg, we show that, under a topological condition, the scattering matrix S_V(\la) at frequency \la > 0 for the operator Δ+V\Delta+V determines the potential VV if VC1,α(M0)eγd(,z0)jL(M0)V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0) for all γ>0\gamma>0 and for some j{1,2}j\in\{1,2\}, where d(z,z0)d(z,z_0) denotes the distance from zz to a fixed point z0M0z_0\in M_0. The topological condition is given by Nmax(2g+1,2)N\geq\max(2g+1,2) for j=1j=1 and by Ng+1N\geq g+1 if j=2j=2. In \rr^2 this implies that the operator S_V(\la) determines any C1,αC^{1,\alpha} potential VV such that V(z)=O(eγz2)V(z)=O(e^{-\gamma|z|^2}) for all γ>0\gamma>0.Comment: 21 page

    Determination of the characteristic directions of lossless linear optical elements

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    We show that the problem of finding the primary and secondary characteristic directions of a linear lossless optical element can be reformulated in terms of an eigenvalue problem related to the unimodular factor of the transfer matrix of the optical device. This formulation makes any actual computation of the characteristic directions amenable to pre-implemented numerical routines, thereby facilitating the decomposition of the transfer matrix into equivalent linear retarders and rotators according to the related Poincare equivalence theorem. The method is expected to be useful whenever the inverse problem of reconstruction of the internal state of a transparent medium from optical data obtained by tomographical methods is an issue.Comment: Replaced with extended version as published in JM
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