Lipschitz Stability of Travel Time Data (bibtex)

by Joonas Ilmavirta, Antti Kykkänen, Matti Lassas, Teemu Saksala and 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.

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Reference:

Lipschitz Stability of Travel Time Data (Joonas Ilmavirta, Antti Kykkänen, Matti Lassas, Teemu Saksala and Andrew Shedlock), 2024. [show abstract] [hide 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. [arXiv]

Bibtex Entry:

@unpublished{stable-bdf,
    author = {Joonas Ilmavirta and Antti Kykk\"{a}nen and Matti Lassas and Teemu Saksala and Andrew Shedlock},
    title = {{Lipschitz Stability of Travel Time Data}},
    month = oct,
    year = 2024,
    arxiv = {2410.16224},
    url = {http://users.jyu.fi/~jojapeil/pub/stable-bdf.pdf},
    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.}
}