  This work studies how to estimate the mean-field density of large-scale
systems in a distributed manner. Such problems are motivated by the recent
swarm control technique that uses mean-field approximations to represent the
collective effect of the swarm, wherein the mean-field density (especially its
gradient) is usually used in feedback control design. In the first part, we
formulate the density estimation problem as a filtering problem of the
associated mean-field partial differential equation (PDE), for which we employ
kernel density estimation (KDE) to construct noisy observations and use
filtering theory of PDE systems to design an optimal (centralized) density
filter. It turns out that the covariance operator of observation noise depends
on the unknown density. Hence, we use approximations for the covariance
operator to obtain a suboptimal density filter, and prove that both the density
estimates and their gradient are convergent and remain close to the optimal one
using the notion of input-to-state stability (ISS). In the second part, we
continue to study how to decentralize the density filter such that each agent
can estimate the mean-field density based on only its own position and local
information exchange with neighbors. We prove that the local density filter is
also convergent and remains close to the centralized one in the sense of ISS.
Simulation results suggest that the centralized suboptimal density filter is
able to generate convergent density estimates, and the local density filter is
able to converge and remain close to the centralized filter.
