Particle Control in Phase Space by Global K-Means Clustering

TL;DR

This paper introduces a k-means clustering based iterative method for controlling particle populations in PIC simulations, enabling efficient merging and splitting while preserving physical properties like energy and momentum.

Contribution

The novel contribution is applying k-means clustering to optimally merge and split particles in phase space, maintaining accuracy and conservation laws in PIC codes.

Findings

High accuracy energy and momentum conservation at compression ratios up to 3.

Effective particle splitting using k-means for optimal phase space placement.

The method is general and applicable beyond Vlasov-Maxwell PIC codes.

Abstract

We devise and explore an iterative optimization procedure for controlling particle populations in particle-in-cell (PIC) codes via merging and splitting of computational macro-particles. Our approach, is to compute an optimal representation of the global particle phase space structure while decreasing or increasing the entire particle population, based on k-means clustering of the data. In essence the procedure amounts to merging or splitting particles by statistical means, throughout the entire simulation volume in question, while minimizing a 6-dimensional total distance measure to preserve the physics. Particle merging is by far the most demanding procedure when considering conservation laws of physics; it amounts to lossy compression of particle phase space data. We demonstrate that our k-means approach conserves energy and momentum to high accuracy, even for high compression ratios, $\mathcal{R} \approx 3$ --- \emph{i.e.}, $N_{f} \lesssim 0.33N_{i}$. Interestingly, we find that an accurate particle splitting step can be performed using k-means as well; this from an argument of symmetry. The split solution, using k-means, places splitted particles optimally, to obtain maximal spanning on the phase space manifold. Implementation and testing is done using an electromagnetic PIC code, the \ppcode. Nonetheless, the k-means framework is general; it is not limited to Vlasov-Maxwell type PIC codes. We discuss advantages and drawbacks of this optimal phase space reconstruction.