TY - JOUR

T1 - Stability of boundary distance representation and reconstruction of riemannian manifolds

AU - Katsuda, Atsushi

AU - Kurylev, Yaroslav

AU - Lassas, Matti

N1 - Funding Information:
The authors want to thank Y.D. Burago, A.P. Katchalov, N. Kossovskii, and T. Sakai for useful consultations and discussions. The research was supported by Finnish Centre of Excellence in Inverse Problems Research (Academy of Finland CoE–project 213476), EPSRC, UK and a Grant in Aid for Scientific Research(C)(2).

PY - 2007

Y1 - 2007

N2 - A boundary distance representation of a Riemannian manifold with boundary (M, g, ∂M) is the set of functions {rx ∈ C(∂M): x ∈ M}, where rx are the distance functions to the boundary, rx(z) = d(x, z), z ∈ ∂M. In this paper we study the question whether this representation determines the Riemannian manifold in a stable way if this manifold satisfies some a priori geometric bounds. The answer is affermative, moreover, given a discrete set of approximate boundary distance functions, we construct a finite metric space that approximates the manifold (M, g) in the Gromov-Hausdorff topology. In applications, the boundary distance representation appears in many inverse problems, where measurements are made on the boundary of the object under investigation. As an example, for the heat equation with an unknown heat conductivity the boundary measurements determine the boundary distance representation of the Riemannian metric which corresponds to this conductivity.

AB - A boundary distance representation of a Riemannian manifold with boundary (M, g, ∂M) is the set of functions {rx ∈ C(∂M): x ∈ M}, where rx are the distance functions to the boundary, rx(z) = d(x, z), z ∈ ∂M. In this paper we study the question whether this representation determines the Riemannian manifold in a stable way if this manifold satisfies some a priori geometric bounds. The answer is affermative, moreover, given a discrete set of approximate boundary distance functions, we construct a finite metric space that approximates the manifold (M, g) in the Gromov-Hausdorff topology. In applications, the boundary distance representation appears in many inverse problems, where measurements are made on the boundary of the object under investigation. As an example, for the heat equation with an unknown heat conductivity the boundary measurements determine the boundary distance representation of the Riemannian metric which corresponds to this conductivity.

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U2 - 10.3934/ipi.2007.1.135

DO - 10.3934/ipi.2007.1.135

M3 - Article

AN - SCOPUS:77951519381

SN - 1930-8337

VL - 1

SP - 135

EP - 157

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

IS - 1

ER -