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February  2007, 1(1): 135-157. doi: 10.3934/ipi.2007.1.135

Stability of boundary distance representation and reconstruction of Riemannian manifolds

1. 

Department of Mathematics, Okayama University, Tsushima-naka, Okayama, 700-8530, Japan

2. 

Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, United Kingdom

3. 

Department of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Finland

Received  August 2006 Revised  August 2006 Published  January 2007

A boundary distance representation of a Riemannian manifold with boundary $(M,g,$∂$\M)$ is the set of functions $\{r_x\in C $ (∂$\M$) $:\ x\in M\}$, where $r_x$ are the distance functions to the boundary, $r_x(z)=d(x, z)$, $z\in$∂$M$. 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.
Citation: Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems & Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135
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