Lipschitz Stability of Travel Time Data

TL;DR

This paper establishes Lipschitz stability for reconstructing certain length spaces from travel time data on a subset, extending classical inverse boundary problems to more general spaces.

Contribution

It proves Lipschitz stability for reconstructing length spaces from travel time data, including non-simple Riemannian manifolds and metric trees.

Findings

Lipschitz stability is achieved for a broad class of length spaces.

Includes examples like non-simple Riemannian manifolds and metric trees.

Extends classical inverse boundary problems to more general settings.

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.