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Lipschitz Stability of Travel Time Data

Joonas Ilmavirta, Antti Kykkänen, Matti Lassas, Teemu Saksala, Andrew Shedlock

Abstract

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel'fand's inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.

Lipschitz Stability of Travel Time Data

Abstract

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel'fand's inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.

Paper Structure

This paper contains 21 sections, 15 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Travel Time Data and the Main Results
  3. Some related inverse problems
  4. Organization of this paper
  5. The scope of this paper: When is a compact length space is BE or BE?
  6. Each FLIE-space is a BLIE-space after possible choosing a smaller epsilon
  7. Examples of sufficient or necessary conditions for BLIE spaces
  8. When is the travel time map a global isometry?
  9. Truncated Compact Length Spaces
  10. The travel time map of AE-space preserves the length structure
  11. Properties of the truncation
  12. Verification of the BE-property from the travel time data of a AE-space
  13. Proofs of the Main Theorems
  14. Examples of FLIEe-manifolds
  15. Examples of FLIEe domains in Rn
  16. ...and 6 more sections

Key Result

Theorem 6

Let $\varepsilon, D > 0$. Let $(X_1,d_1)$ and $(X_2,d_2)$ be two compact length spaces with closed measurement sets $S_1 \subset X_1$ and $S_2 \subset X_2$. Suppose that the diameters of $X_1$ and $X_2$ are less than $D$, and both spaces are $\mathrm{BLIE}_{\varepsilon}$. If there is a homeomorphism If the travel time data of spaces $X_1$ and $X_2$ coincide, then these metric spaces are isometric.

Figures (1)

  • Figure 1: Two illustrations of our example of a Herglotz manifold.

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Theorem 6: Stability of Travel Time Data: First Version
  • Definition 7
  • Definition 8
  • Theorem 9: Stability of Travel Time Data: Second Version
  • Corollary 10
  • ...and 32 more