| Abstract: |
| On the mesoscopic level, motion of individual particles can be modeled by a kinetic transport equation for the population density $f(t,x,v)$ as a function of time $t$, space $x$ and velocity $v \in V$. A relaxation term on the right hand side accounts for scattering due to self-induced velocity changes and typically involves a parameter $K(x,v,v`)$ encoding the probability of changing from velocity $v`$ to $v$ at location $x$:
\begin{equation} \nonumber
\partial_t f(t,x,v) + v \cdot \nabla f(t,x,v) = \int K(x,v,v`) f(t,x,v`) - K(x,v`,v)f(t,x,v) dv` .
\end{equation}
This hyperbolic model is widely used to model bacterial motion, called chemotaxis.
We study the inverse parameter reconstruction problem whose aim is to recover the scattering parameter $K$ and that has to be solved when fitting the model to a real situation. We restrict ourselves to macroscopic, i.e. velocity averaged data $\rho = \int f dv$ as a basis of our reconstruction. This introduces additional difficulties, which can be overcome by the use of short time interior domain data. In this way, we can establish theoretical existence and uniqueness of the reconstruction, study its macroscopic limiting behavior and numerically conduct the inversion under suitable data generating experimental designs.
This work based on a collaboration with Kathrin Hellmuth (W\urzburg, Germany), Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China). |
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