Numerical reconstruction of the kinetic chemotaxis kernel from macroscopic measurement, wellposedness and illposedness
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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.
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