Razvan C. Fetecau

Hannah Potgieter, Razvan C. Fetecau, Steven J. Ruuth

We approximate the first Dirichlet eigenpair of the $p$-Laplace operator for $2 \leq p < \infty$ on both Euclidean and surface domains. We emphasize large $p$ values and discuss how the $p \to \infty$ limit connects to the underlying geometry of our domain. Working with large $p$ values introduces significant numerical challenges. We present a surface finite element numerical scheme that combines a Newton inverse-power iteration with a new domain rescaling strategy, which enables stable computations for large $p$. Numerical experiments in $1$D, planar domains, and surfaces embedded in $\mathbb{R}^3$ demonstrate the accuracy and robustness of our approach and show convergence towards the $p \to \infty$ limiting behavior.

Xinyu Zhao, Razvan C. Fetecau, Mo Chen

Collaborative autonomous multi-agent systems covering a specified area have many potential applications, such as UAV search and rescue, forest fire fighting, and real-time high-resolution monitoring. Traditional approaches for such coverage problems involve designing a model-based control policy based on sensor data. However, designing model-based controllers is challenging, and the state-of-the-art classical control policy still exhibits a large degree of sub-optimality. In this paper, we present a reinforcement learning (RL) approach for the multi-agent efficient domain coverage problem involving agents with second-order dynamics. Our approach is based on the Multi-Agent Proximal Policy Optimization Algorithm (MAPPO). Our proposed network architecture includes the incorporation of LSTM and self-attention, which allows the trained policy to adapt to a variable number of agents. Our trained policy significantly outperforms the state-of-the-art classical control policy. We demonstrate our proposed method in a variety of simulated experiments.

Razvan C. Fetecau, Hui Huang, Daniel Messenger et al.

We establish the zero-diffusion limit for both continuous and discrete aggregation models over convex and bounded domains. Compared with a similar zero-diffusion limit derived in [44], our approach is different and relies on a coupling method connecting PDEs with their underlying SDEs. Moreover, our result relaxes the regularity assumptions on the interaction and external potentials and improves the convergence rate (in terms of the diffusion coefficient). The particular rate we derive is shown to be consistent with numerical computations.