Dynamic kernel density estimation in the chaotic advection problem
The aim of the project is to develop numerical strategies for obtaining accurate estimations of the tracer concentration fields from the ensemble of stochastic trajectory solutions of the Lagrangian particle dispersion model. Specifically, more information can be obtained in a stochastic problem than just a static density. We can go beyond standard kernel density estimation (KDE) methods by exploiting the fact that we are solving a dynamic problem. Hence we can introduce a new class of methods, which we will call the dynamic KDE (DKDE). The starting point of DKDE is the Green's function representation of the solution to the advection-diffusion equation (or Fokker-Planck equation). The Green's function for a short-time interval is then approximated by considering the leading order WKBJ series solution of the advection-diffusion equation, under the assumption that the diffusivity parameter is small.
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