PBDW state estimation: noisy observations; configuration-adaptive background spaces; physical interpretations (Q2786519)

A numerical simulations of a Vlasov-type kinetic description of flocking models NEWLINE\[NEWLINE \frac{\partial f}{\partial t}+v\cdot \nabla_xf+\nabla_v\cdot Q(f)=0\;\;\;\forall x\in\Omega\subset\mathbb{R}^d,\;\;\;v\subset\mathbb{R}^d,NEWLINE\]NEWLINE NEWLINE\[NEWLINEf(0,x,v)=f_0(x,v), NEWLINE\]NEWLINE is considered. The distribution function \(f=f(t,x,v)\) describes the probability of finding an individual at time \(t>0\) at the position of the phase space \((x,v)\). The integral operator \(Q,\) the flocking operator, characterizes the nonlocal interactions. The authors propose a velocity scaling method to solve the considered equations, with a new choice of scaling factor which captures the exact scaling of the system. A numerical scheme is also designed to efficiently solve the coupled evolutions of the kinetic equation together with the rescaling function.