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23
Mathematics of thermoacoustic tomography
 European Journal Applied Mathematics
"... The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography. 1 ..."
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Cited by 29 (6 self)
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The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography. 1
Boundary Regularity for the Ricci Equation, Geometric Convergence, and Gel'fand's Inverse Boundary Problem
"... Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The secon ..."
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Cited by 14 (13 self)
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Abstract This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts. 1.
Thermoacoustic tomography with variable sound speed
 Inverse Problems
, 2009
"... Abstract. We study the mathematical model of thermoacoustic tomography in media with a variable speed for a fixed time interval [0, T] so that all signals issued from the domain leave it after time T. In case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann s ..."
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Cited by 10 (0 self)
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Abstract. We study the mathematical model of thermoacoustic tomography in media with a variable speed for a fixed time interval [0, T] so that all signals issued from the domain leave it after time T. In case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann series expansion. We give almost necessary and sufficient conditions for uniqueness and stability when the measurements are taken on a part of the boundary. 1.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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FORWARD AND INVERSE SCATTERING ON MANIFOLDS WITH ASYMPTOTICALLY CYLINDRICAL ENDS
, 905
"... Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codi ..."
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Cited by 2 (0 self)
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Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange 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. 1.
Acoustic TimeReversal Mirrors in the Framework of OneWay Wave Theories
"... We investigate the implications of directional wavefield decomposition with a view to time reversibility. In particular, we discuss how wavefield decomposition preserves the reciprocity theorem of timeconvolution type but looses the reciprocity theorem of timecorrelation type. As a consequence, a ..."
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Cited by 1 (1 self)
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We investigate the implications of directional wavefield decomposition with a view to time reversibility. In particular, we discuss how wavefield decomposition preserves the reciprocity theorem of timeconvolution type but looses the reciprocity theorem of timecorrelation type. As a consequence, a perfect `timereversal mirror' in the framework of oneway wave theory does not exist: We find that on the wavefront set (`classical limit') a timereversal mirror can retrofocus the wavefield to its originating source, but that nonperfectly retrofocusing lowerorder distributions contribute to the process as well. These distributions can be attributed to `evanescent' wave constituents but are not negligible; we will study them explicitly. As a peculiarity, we discuss how a Schrodingerlike equation can be obtained out of the (exact) frequencydomain oneway wave equation. This involves an approximation  known in ocean acoustics and exploration seismology as the `parabolic equation' approximation  that restores timereversibility.
SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS
, 2006
"... Abstract. We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structur ..."
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Cited by 1 (0 self)
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Abstract. We study scattering and inverse scattering theories for asymptotically complex hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic family of operators in the Heisenberg calculus on the boundary, which is a contact manifold with a pseudohermitian structure. Then we define radiation fields as in the real asymptotically hyperbolic case, and reconstruct the scattering operator from those fields. As an application we show that the manifold, including its topology and the metric, are determined up to invariants by the scattering matrix at all energies.
On the local structure of the KleinGordon field on curved spacetimes
, 2000
"... This paper investigates waveequations on spacetimes with a metric which is locally analytic in the time. We use recent results in the theory of the noncharacteristic Cauchy problem to show that a solution to a waveequation vanishing in an open set vanishes in the “envelope” of this set, which may ..."
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This paper investigates waveequations on spacetimes with a metric which is locally analytic in the time. We use recent results in the theory of the noncharacteristic Cauchy problem to show that a solution to a waveequation vanishing in an open set vanishes in the “envelope” of this set, which may be considerably larger and in the case of timelike tubes may even coincide with the spacetime itself. We apply this result to the real scalar field on a globally hyperbolic spacetime and show that the field algebra of an open set and its envelope coincide. As an example there holds an analog of Borchers ’ timelike tube theorem for such scalar fields and hence, algebras associated with world lines can be explicitly given. Our result applies to cosmologically relevant spacetimes.
Boundary Control and Dynamical Reconstruction of Vector Fields (The BCMethod )
, 1998
"... An approach to the dynamical inverse problems (IP's) based upon their relations to the Boundary Control Theory ( the socalled BCmethod ) is developed. The method is applied to the problem of reconstruction of a vector field given on a Riemannian manifold via the response operator (the dynamical Di ..."
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An approach to the dynamical inverse problems (IP's) based upon their relations to the Boundary Control Theory ( the socalled BCmethod ) is developed. The method is applied to the problem of reconstruction of a vector field given on a Riemannian manifold via the response operator (the dynamical DirichlettoNeumann map ). A peculiarity of the case under consideration is that the operator which governs an evolution of the corresponding dynamical system is nonselfadjoint. The paper announces the results and gives a brief description of technique of the BCmethod. 2 ESAIM: Proc., Vol. 4, 1998, 16 1 Introduction An approach to the dynamical inverse problems (IP's) based upon their relations to the Boundary Control Theory ( the socalled BCmethod ) is developed. The method is applied to the problem of reconstruction of a vector field given on a Riemannian manifold via the response operator (the dynamical DirichlettoNeumann map ). A peculiarity of the case under consideration is that...
Identification of
"... : We consider the problem of recovering the coefficient q(x) in the equation u t = \Delta\Delta \Deltau \Gamma qu from boundary observations. Uniqueness of q based on knowledge of the `Neumann 7! Dirichlet response operator' is shown as an implication of (known) corresponding results concerning th ..."
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: We consider the problem of recovering the coefficient q(x) in the equation u t = \Delta\Delta \Deltau \Gamma qu from boundary observations. Uniqueness of q based on knowledge of the `Neumann 7! Dirichlet response operator' is shown as an implication of (known) corresponding results concerning the inverse problem for the corresponding hyperbolic equation w tt = \Delta\Delta \Delta w \Gamma qw. This is then reduced to use of the response to a single input with some consideration of computational approximation. Key Words: identification, parabolic, partial differential equation, uniqueness, approximation. 1 This research has been partially supported by the U.S. National Academy of Science under the NAS/NRC Project Development Program. 1. Introduction We consider the problem of identifying the (unknown) coefficient q = q(x) in the parabolic partial differential equation u t = \Delta\Delta \Deltau \Gamma qu on Q := (0; T ) \Theta\Omega ; (1.1) assuming input/output access only a...