# Particle Density Estimation with Grid-Projected Adaptive Kernels

## TL;DR

This paper introduces a grid-projected adaptive kernel density estimation method that combines binning and KDE for efficient, boundary-aware particle density estimation in fluid transport simulations.

## Contribution

It develops a hybrid KDE approach with local bandwidth optimization, improving efficiency and boundary handling over traditional KDE methods.

## Key findings

- Enhanced computational efficiency demonstrated in examples
- Effective boundary condition handling achieved
- Adaptive bandwidth improves density estimation accuracy

## Abstract

The reconstruction of smooth density fields from scattered data points is a procedure that has multiple applications in a variety of disciplines, including Lagrangian (particle-based) models of solute transport in fluids. In random walk particle tracking (RWPT) simulations, particle density is directly linked to solute concentrations, which is normally the main variable of interest, not just for visualization and post-processing of the results, but also for the computation of non-linear processes, such as chemical reactions. Previous works have shown the superiority of kernel density estimation (KDE) over other methods such as binning, in terms of its ability to accurately estimate the "true" particle density relying on a limited amount of information. Here, we develop a grid-projected KDE methodology to determine particle densities by applying kernel smoothing on a pilot binning; this may be seen as a "hybrid" approach between binning and KDE. The kernel bandwidth is optimized locally. Through simple implementation examples, we elucidate several appealing aspects of the proposed approach, including its computational efficiency and the possibility to account for typical boundary conditions, which would otherwise be cumbersome in conventional KDE.