Inverse heat source problem from terminal data on a set of low Hausdorff dimension
Abstract
We study the recovery of a spatial source in the heat equation from terminal observations on a compact interior set E with positive β dimensional Hausdorff content for some β>d−1. The prescribed temporal forcing vanishes on a nontrivial interval immediately preceding the observation time. Under a spectral Sobolev prior and a quantitative noncancellation condition on the terminal spectral multipliers, we establish conditional logarithmic stability. For scattered observations on such sets, we derive a sampling inequality under a quasi uniformity assumption. Under explicit sampling and independent noise assumptions, we prove a logarithmic convergence rate in probability for the corresponding Tikhonov reconstructions with a data driven parameter choice. Two numerical experiments using a finite candidate approximation of a fixed Hausdorff set assess discrete spectral sampling stability and illustrate the effects of the waiting interval and measurement noise on reconstruction.
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BibTeX
@misc{yu2026inverse,
title = {{Inverse heat source problem from terminal data on a set of low Hausdorff dimension}},
author = {Kai Yu and Zhiyuan Li and Zhidong Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35014},
eprint = {2610.35014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35014},
}
RIS
TY - UNPB
TI - Inverse heat source problem from terminal data on a set of low Hausdorff dimension
AU - Kai Yu
AU - Zhiyuan Li
AU - Zhidong Zhang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35014
UR - https://arxiv.org/abs/2610.35014
AB - We study the recovery of a spatial source in the heat equation from terminal observations on a compact interior set E with positive β dimensional Hausdorff content for some β>d−1. The prescribed temporal forcing vanishes on a nontrivial interval immediately preceding the observation time. Under a spectral Sobolev prior and a quantitative noncancellation condition on the terminal spectral multipliers, we establish conditional logarithmic stability. For scattered observations on such sets, we derive a sampling inequality under a quasi uniformity assumption. Under explicit sampling and independent noise assumptions, we prove a logarithmic convergence rate in probability for the corresponding Tikhonov reconstructions with a data driven parameter choice. Two numerical experiments using a finite candidate approximation of a fixed Hausdorff set assess discrete spectral sampling stability and illustrate the effects of the waiting interval and measurement noise on reconstruction.
ER -