Increasing stability for the inverse potential problem of the fractional Helmholtz equation
Abstract
This paper is concerned with the inverse potential scattering problem for the three-dimensional fractional Helmholtz equation. For every 0<α<1, we establish an increasing stability of determining the potential function from multifrequency exterior measurements of the scattered field. The stability consists of a Lipschitz data discrepancy and a logarithmic remainder that decreases when the upper measured wavenumber increases. The key ingredient in the analysis is to construct appropriate geometric optics solutions, which relate the X-ray transform of the potential to the measured fields. As a consequence, the stability is derived by applying analytic continuation. Two-dimensional numerical examples with noisy near-field data illustrate the benefit of enlarging the measured frequency band for three representative fractional orders.
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BibTeX
@misc{qiu2026increasing,
title = {{Increasing stability for the inverse potential problem of the fractional Helmholtz equation}},
author = {Dong Qiu and Xiang Xu and Yue Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04749},
eprint = {2611.04749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04749},
}
RIS
TY - UNPB
TI - Increasing stability for the inverse potential problem of the fractional Helmholtz equation
AU - Dong Qiu
AU - Xiang Xu
AU - Yue Zhao
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04749
UR - https://arxiv.org/abs/2611.04749
AB - This paper is concerned with the inverse potential scattering problem for the three-dimensional fractional Helmholtz equation. For every 0<α<1, we establish an increasing stability of determining the potential function from multifrequency exterior measurements of the scattered field. The stability consists of a Lipschitz data discrepancy and a logarithmic remainder that decreases when the upper measured wavenumber increases. The key ingredient in the analysis is to construct appropriate geometric optics solutions, which relate the X-ray transform of the potential to the measured fields. As a consequence, the stability is derived by applying analytic continuation. Two-dimensional numerical examples with noisy near-field data illustrate the benefit of enlarging the measured frequency band for three representative fractional orders.
ER -