随机分数阶亥姆霍兹方程反源问题的稳定性
Stability for the inverse source problem of the stochastic fractional Helmholtz equation
- Zhejiang University（浙江大学）
- Central China Normal University（华中师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究白噪声驱动的随机分数阶亥姆霍兹方程反源问题，证明单频数据唯一确定源方差，并利用多频数据建立递增稳定性估计，稳定性随频率带宽上界增加而改善。
AI中文摘要：
本文关注由白噪声驱动的随机分数阶亥姆霍兹方程的反源问题。对于每个分数阶$0<α<1$，我们证明了直接问题的出射分布解的存在性和唯一性，并给出了高频下的预解估计，同时建立了其随机表示。对于反问题，我们证明了在单一频率下，源的方差可以由相关随机外部数据唯一确定。我们进一步利用多频相关外部数据建立了反问题的递增稳定性估计。我们的稳定性结果表明，随着所用频率带宽上界的增加，稳定性也会改善。分析采用了几何光学解的构造，将相关数据与方差的X射线变换联系起来。
英文摘要：
This paper is concerned with the inverse source problem for the stochastic fractional Helmholtz equation driven by white noise. For every fractional order $0<α<1$, we prove the existence and uniqueness of the outgoing distributional solution to the direct problem with resolvent estimates at high frequencies, and establish its stochastic representation. For the inverse problem, we demonstrate that the variance of the source can be uniquely determined by the correlated random exterior data at a single frequency. We further establish an increasing stability estimate for the inverse problem by using the multifrequency correlated exterior data. Our stability result shows that as the upper bound of the bandwidth of the utilized frequency increases, the stability will also improve. The analysis employs the construction of geometric optics solutions, which connects the correlated data to the X-ray transform of the variance.
