Reconstructing the kinetic chemotaxis kernel using macroscopic data: well-posedness and ill-posedness

TL;DR

This paper investigates the reconstruction of the chemotaxis kernel in bacterial motion models using PDE-constrained optimization, identifying conditions for well-posedness and proposing a design for guaranteed numerical reconstructability.

Contribution

It introduces a specific experimental design and computational framework that ensure the stable reconstruction of the chemotaxis kernel from interior data, with parallelizable computation.

Findings

The proposed design guarantees numerical reconstructability.

The reconstruction problem can be well-posed or ill-posed depending on data setup.

Numerical experiments support theoretical results.

Abstract

Bacterial motion is steered 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 called chemotaxis kernel. In a practical setting, it is inferred by experimental data. We deploy a PDE-constrained optimization framework 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.