Double-logarithmic stability for a parabolic inverse problem on CTA manifolds
Abstract
We establish a double-logarithmic stability estimate for the recovery of a time-dependent potential in the parabolic equation (c−1∂t−Δg+q)u=0 on a conformally transversally anisotropic manifold. The partial input--output map associates an initial state and lateral Dirichlet data supported in a prescribed open set with the terminal state and the Neumann trace on another open set. These sets contain the positive and negative boundary parts determined by a limiting Carleman weight, respectively. The measurement discrepancy is the operator norm of the difference of two such maps, whose range has improved Sobolev regularity. We prove stability under a uniform stability assumption for the attenuated geodesic ray transform on a family of non-tangential geodesics. This assumption is verified when the transversal manifold is simple and for regular families on certain non-simple manifolds. The proof uses Gaussian beam quasimodes with a uniform O(h1/2) concentration estimate, geometric optics solutions obtained from boundary Carleman estimates, and stability of the geodesic ray transform for small attenuation. Quantitative analytic continuation for Hilbert-space-valued Fourier transforms and Sobolev interpolation yield the double-logarithmic modulus.
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BibTeX
@misc{zheng2026doublelogarithmic,
title = {{Double-logarithmic stability for a parabolic inverse problem on CTA manifolds}},
author = {Yujian Zheng and Zhiwen Duan and Shiqi Jing},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16759},
eprint = {2609.16759},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16759},
}
RIS
TY - UNPB
TI - Double-logarithmic stability for a parabolic inverse problem on CTA manifolds
AU - Yujian Zheng
AU - Zhiwen Duan
AU - Shiqi Jing
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16759
UR - https://arxiv.org/abs/2609.16759
AB - We establish a double-logarithmic stability estimate for the recovery of a time-dependent potential in the parabolic equation (c^−1∂_t−Δ_g+q)u=0 on a conformally transversally anisotropic manifold. The partial input--output map associates an initial state and lateral Dirichlet data supported in a prescribed open set with the terminal state and the Neumann trace on another open set. These sets contain the positive and negative boundary parts determined by a limiting Carleman weight, respectively. The measurement discrepancy is the operator norm of the difference of two such maps, whose range has improved Sobolev regularity. We prove stability under a uniform stability assumption for the attenuated geodesic ray transform on a family of non-tangential geodesics. This assumption is verified when the transversal manifold is simple and for regular families on certain non-simple manifolds. The proof uses Gaussian beam quasimodes with a uniform O(h^1/2) concentration estimate, geometric optics solutions obtained from boundary Carleman estimates, and stability of the geodesic ray transform for small attenuation. Quantitative analytic continuation for Hilbert-space-valued Fourier transforms and Sobolev interpolation yield the double-logarithmic modulus.
ER -