  Bacterial motion is guided by external stimuli (chemotaxis), and the motion
described on the mesoscopic scale is uniquely determined by a parameter $K$
that models velocity change response from the bacteria. This parameter is
termed chemotaxis kernel. In a practical setting, experimental data was
collected to infer this kernel. In this article, a PDE-constrained optimization
framework is deployed to perform this reconstruction using velocity-averaged,
localized data taken in the interior of the domain. The problem can be
well-posed or ill-posed depending on the data preparation and the experimental
setup. In particular, we propose one specific design that guarantees numerical
reconstructability and local convergence. This design is adapted to the
discretization of $K$ in space and decouples the reconstruction of local values
of $K$ into smaller cell problems, opening up parallelization opportunities.
Numerical evidences support the theoretical findings.
