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.

The purpose of this article is to define the radiation fields on asymptotically hyperbolic manifolds and to use them to study scattering theory. The radiation fields on R n and on asymptotically Euclidean manifolds were introduced by F.G. Friedlander in a series of papers starting in the early 1960’s [10, 11, 12, 13, 14]. His program of using the radiation fields to obtain the scattering matrix in that general setting was

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.

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 time-convolution type but looses the reciprocity theorem of time-correlation type. As a consequence, a perfect `time-reversal mirror' in the framework of one-way wave theory does not exist: We find that on the wavefront set (`classical limit') a time-reversal mirror can retrofocus the wavefield to its originating source, but that non-perfectly retrofocusing lower-order 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 Schrodinger-like equation can be obtained out of the (exact) frequency-domain oneway wave equation. This involves an approximation -- known in ocean acoustics and exploration seismology as the `parabolic equation' approximation -- that restores timereversibility.

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.

An approach to the dynamical inverse problems (IP's) based upon their relations to the Boundary Control Theory ( the so-called BC-method ) 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 Dirichlet-to-Neumann 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 BC-method. 2 ESAIM: Proc., Vol. 4, 1998, 1-6 1 Introduction An approach to the dynamical inverse problems (IP's) based upon their relations to the Boundary Control Theory ( the so-called BC-method ) 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 Dirichlet-toNeumann map ). A peculiarity of the case under consideration is that...

: 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...

Abstract. We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the Laplace-Beltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in stable way when manifold is a priori known to satisfy natural geometrical conditions related to curvature and other invariant quantities. I. Introduction In this paper we consider the questions of stability and approximate reconstruction in the inverse boundary spectral problem which is also called the generalized Gelfand inverse problem [13] for Riemannian manifolds. To formulate the problem and the main results, we need to introduce some basic notations. We will denote by (M,g) an unknown,

Abstract: The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell’s equations written for differential forms over a 3-manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle over the manifold. The second part of the article deals with the dynamical inverse boundary value problem of determining the electromagnetic material parameters from boundary measurements. By using the boundary control method, it is proved that the dynamical boundary data determines the electromagnetic travel time metric as well as the scalar wave impedance on the manifold. This invariant result leads also to a complete characterization of the non-uniqueness of the corresponding inverse problem in bounded domains of R 3.

problem for a Dirac-type equation on a