Kinetic equations describe various particle-based physical phenomena on the mesoscopic level through the time (t) evolution of the particle density f(t,x,v) in the space (x) and velocity (v). One example is bacterial motion in response to an external stimulus, termed chemotaxis. In a setting with fixed chemical concentration, we study a parameter reconstruction problem for this model and seek for the scattering kernel K(x,v,v’) that accounts for instantaneous velocity jumps in the microscopic run-and-tumble particle model and uniquely determines the motion. Experimental restrictions suggest the use of data on the macroscopic bacteria density $\rho(t,x) = \int f(t,x,v) dv$. Given the discrepancy in scale of parameter and data, we study its suitability for our reconstruction task in two frameworks and show that: - the function K can be recovered point wise from specifically designed experiments that leverage the scale discrepancy through short time interior domain data. - in a numerical setting where K is piece wise constant, it can be reconstructed through cost minimization, if a suitable experimental design is chosen. We hope that this insight helps practitioners to design their experiments.
This is joint work with Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China).