Directed bacterial motion due to external stimuli (chemotaxis) can, on the mesoscopic phase space, be described by a velocity change parameter $K$. The numerical reconstruction for $K$ from experimental data provides useful insights and plays a crucial role in model fitting, verification and prediction. In this article, the PDE-constrained optimization framework is deployed to perform the reconstruction of $K$ from velocity-averaged, localized data taken in the interior of a 1D domain. Depending on the data preparation and experimental setup, this problem can either be well- or ill-posed. We analyze these situations, and propose a very specific design that guarantees local convergence. The design is adapted to the discretization of $K$ and decouples the reconstruction of local values into smaller cell problem, opening up opportunities for parallelization. We further provide numerical evidence as a showcase for the theoretical results.

翻译：由外部刺激（趋化性）引起的定向细菌运动，在介观相空间中可通过速度变化参数 $K$ 进行描述。基于实验数据对 $K$ 进行数值重建，可为模型拟合、验证与预测提供关键见解并发挥重要作用。本文采用偏微分方程约束优化框架，基于一维区域内部的速度平均局部数据进行 $K$ 的重建。根据数据制备方法与实验设置，该问题可能呈现适定性或不适定性特征。我们分析了这两种情形，并提出一种可保证局部收敛性的特殊设计方案。该方案适配 $K$ 的离散化格式，将局部值的重建解耦为更小规模的胞元问题，从而为并行化计算创造条件。此外，我们提供数值实验作为理论结果的验证案例。