Ensemble Data Assimilation for Particle-based Methods

TL;DR

This work addresses applying ensemble data assimilation to particle-based, Lagrangian simulations by introducing two dedicated schemes: Remesh-EnKF, which consolidates all ensemble members onto a common particle grid for standard EnKF updates, and Part-EnKF, which updates intensities directly on each member’s native particle discretization. The methods are evaluated on a 1D advection–diffusion model and a 2D incompressible flow solved with a Vortex-In-Cell method, with grid-based EnKF as a benchmark. Key findings show that Remesh-EnKF generally offers robust performance with modest sensitivity to particle size, while Part-EnKF can approach grid-based results given sufficiently large discretization supports; however, Part-EnKF is sensitive to coverage and interpolation errors when the analyzed field extends beyond a member’s particle support. The results demonstrate the feasibility of applying EnKF corrections to mesh-free simulations and point to future improvements such as joint optimization of particle positions (displacement corrections) to further strengthen robustness and accuracy in highly dynamic Lagrangian systems.

Abstract

This study presents a novel approach to applying data assimilation techniques for particle-based simulations using the Ensemble Kalman Filter. While data assimilation methods have been effectively applied to Eulerian simulations, their application in Lagrangian solution discretizations has not been properly explored. We introduce two specific methodologies to address this gap. The first methodology employs an intermediary Eulerian transformation that combines a projection with a remeshing process. The second is a purely Lagrangian scheme designed for situations where remeshing is not appropriate. The second is a purely Lagrangian scheme that is applicable when remeshing is not adapted. These methods are evaluated using a one-dimensional advection-diffusion model with periodic boundaries. Performance benchmarks for the one-dimensional scenario are conducted against a grid-based assimilation filter Subsequently, assimilation schemes are applied to a non-linear two-dimensional incompressible flow problem, solved via the Vortex-In-Cell method. The results demonstrate the feasibility of applying these methods in more complex scenarios, highlighting their effectiveness in both the one-dimensional and two-dimensional contexts.

Figures (24)

• Figure 1: Illustration of the Remesh-EnKF. (\ref{['fig:remesh_enkf_1']}) The particles of the members are projected on a fixed Eulerian grid. (\ref{['fig:remesh_enkf_2']}) EnKF matrix operations are applied to nodal values. (\ref{['fig:remesh_enkf_3']}) New uniform sets of particles are generated with thresholding to interpolate the analyzed states.
• Figure 2: One-dimensional illustration of the Part-EnKF method. (\ref{['fig:part_enkf_1']}) A set of three particles discretizes a member forecast function. (\ref{['fig:remesh_enkf_2']}) The analyzed solution is determined thanks to equation \ref{['eq:analysed_field_F']}. (\ref{['fig:part_enkf_3']}) The forecast particle intensities are updated to fit the analyzed solution.
• Figure 3: The exact solution of the advection-diffusion problem at different times: $t=0$, $t= \frac{\pi}{v}$, and $t= \frac{2\pi}{v}$.
• Figure 4: On the left, the initial parameter sample with $v$ on the abscissa and $D$ on the ordinate. On the right, the initial ensemble states.
• Figure 5: Analysed solution at different assimilation steps for the Remesh-EnKF filter.