An inverse spectral problem for the wave equation with a varying Robin parameter at the boundary
Abstract
An inverse spectral problem for a (1+1)-dimensional wave equation with wave speed c(x) is considered. The spectral data consist of the eigenvalues of the Neumann--Dirichlet problem for the operator −c2(x)∂x2 together with their derivatives with respect to a Robin parameter at the boundary. We propose a two-step recovery procedure. In the first step, the Neumann-to-Dirichlet map Λ is reconstructed from the spectral data using the results of arXiv/0406582. In the second step, the reconstructed Λ is used within the regularization strategy for the boundary control method presented in arXiv:1509.04478. As an improvement, the regularization of the control problems employs a weighted combination of L2 and H1 penalty terms, and we prove a uniform convergence result for the family of optimal controls, thereby allowing the regularization parameters to be tuned simultaneously. We also prove that the forward spectral map, which assigns to a wave speed its spectral data, is Lipschitz continuous, and that the inverse spectral map underlying the reconstruction procedure is Hölder continuous on a subclass of wave speeds. The method is implemented numerically and validated on both smooth and piecewise smooth wave speed profiles. Numerical experiments demonstrate that the recovery of the wave speed exhibits high accuracy and robustness.
Related
- Statistical comparison of reconstruction methods for the inverse boundary problem of the one-dimensional wave equation
- On an inverse boundary problem in elastodynamics
- The inverse elastic scattering problem in the cavity with impedance boundary
- A Schiffer-type inverse obstacle problem for the planar Schrödinger equation with a single measurement
- An inverse free boundary problem
BibTeX
@misc{hannukainen2026inverse,
title = {{An inverse spectral problem for the wave equation with a varying Robin parameter at the boundary}},
author = {Antti Hannukainen and Petr Kulikov and Matti Lassas and Lauri Oksanen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13629},
eprint = {2610.13629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13629},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An inverse spectral problem for the wave equation with a varying Robin parameter at the boundary
AU - Antti Hannukainen
AU - Petr Kulikov
AU - Matti Lassas
AU - Lauri Oksanen
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13629
UR - https://arxiv.org/abs/2610.13629
AB - An inverse spectral problem for a (1+1)-dimensional wave equation with wave speed c(x) is considered. The spectral data consist of the eigenvalues of the Neumann--Dirichlet problem for the operator −c^2(x)∂_x^2 together with their derivatives with respect to a Robin parameter at the boundary. We propose a two-step recovery procedure. In the first step, the Neumann-to-Dirichlet map Λ is reconstructed from the spectral data using the results of arXiv/0406582. In the second step, the reconstructed Λ is used within the regularization strategy for the boundary control method presented in arXiv:1509.04478. As an improvement, the regularization of the control problems employs a weighted combination of L^2 and H^1 penalty terms, and we prove a uniform convergence result for the family of optimal controls, thereby allowing the regularization parameters to be tuned simultaneously. We also prove that the forward spectral map, which assigns to a wave speed its spectral data, is Lipschitz continuous, and that the inverse spectral map underlying the reconstruction procedure is Hölder continuous on a subclass of wave speeds. The method is implemented numerically and validated on both smooth and piecewise smooth wave speed profiles. Numerical experiments demonstrate that the recovery of the wave speed exhibits high accuracy and robustness.
ER -