非自治演化方程中基于非局部观测的空间依赖系数的线性化唯一性
Linearized uniqueness of space dependent coefficients in a non-autonomous evolution equation from non-local observations
AI总结：
本文针对含双线性控制项的非自治时变偏微分方程，证明了基于含空间域加权积分的非局部时间迹线观测的空间依赖系数线性化唯一性，并将其应用于扩散方程势函数识别及Bloch-Torrey方程的定量磁共振成像参数重建。
AI中文摘要：
本文研究含双线性控制项的非自治时变偏微分方程中空间依赖系数的识别问题。我们证明了基于时间迹线观测的线性化唯一性，尤其考虑了以状态在空间域上的加权积分形式呈现的非局部观测。该结果在抽象框架下得到推导，并首先应用于扩散方程中势函数的识别作为示例，其次应用于基于Bloch-Torrey方程的定量磁共振成像中，将平衡磁化强度、弛豫率和场不均匀性作为空间依赖量进行重建，这是本研究的实际应用动机。
英文摘要：
In this paper, we consider identification of space dependent coefficients in a time dependent PDE, which is non-autonomous, due to a bilinear control term. We prove linearized uniqueness from time trace observations, in particular also considering non-local observations in the form of weighted integrals of the state over he spatial domain. The result is obtained in an abstract setting and applied, first of all to the identification of a potential in a diffusion equation for illustration purposes, second to the reconstruction of equilibrium magnetization, relaxation rates and field inhomogeneity as space dependent quantities in model based qunatitative magnetic resonance imaging with the Bloch-Torrey equation, which constitutes the real world application motivating this study.
