Tensor-Train-based Algorithms for Aggregate State Estimation of Swarms with Interacting Agents

In this paper, we develop an efficient implementation of the gas-kinetic (GK) Probability Hypothesis Density (PHD) filter for aggregate swarm state estimation with interacting agents. We borrow a kinetic/mesoscopic partial differential equation (PDE) model of a swarm of interacting agents from biology moving in a plane with a heading state, which requires the computation of integrals up to five dimensions. In the context of the GK-PHD, we propagate this model by computing in a compressed format called the Tensor Train (TT) format, yielding better memory and runtime properties than a grid-based approach. Under certain assumptions, we prove that TT-GK-PHD has a time complexity of an order of magnitude better than the grid-based approach. Finally, we showcase the usefulness of our algorithm on a scenario which cannot be solved via the grid-based approach due to hardware memory constraints. Then in a computational experiment we demonstrate the better runtime and memory of TT-GK-PHD over the grid-based approach.