Stability of boundary distance representation and reconstruction of riemannian manifolds

Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)135-157
Number of pages23
JournalInverse Problems and Imaging
Volume1
Issue number1
DOIs
Publication statusPublished - Jan 1 2007
Externally publishedYes

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All Science Journal Classification (ASJC) codes

  • Analysis
  • Modelling and Simulation
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

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