### Wonjun Lee

#### Publications and Preprints

Computational Mean-Field Information Dynamics Associated With Reaction Diffusion Equations

• Wuchen Li, Wonjun Lee, Stanley Osher
• Submitted, 2021.
We formulate and compute a class of mean-field information dynamics based on reaction diffusion equations. Given a class of nonlinear reaction diffusion and entropy type Lyapunov functionals, we study their gradient flow formulations. We write the “mean-field” metric space formalisms and derive Hamiltonian flows therein. These Hamiltonian flows follow saddle point systems of the proposed mean-field control problems. We apply primal-dual hybrid-gradient algorithms to compute the mean field information dynamics. Several numerical examples are provided.

Tropical Optimal Transport and Wasserstein Distances in Phylogenetic Tree Space

We study the problem of optimal transport on phylogenetic tree space from the perspective of tropical geometry, and thus define the Wasserstein-p distances for probability measures in this continuous metric measure space setting. With respect to the tropical metric -- a combinatorial metric on the space of phylogenetic trees -- the cases of \(p = 1, 2)\are treated in detail, which give an efficient way to compute geodesics and also provide theoretical foundations for geometric insight a statistical framework on tree space. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances, and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport on sets of phylogenetic trees. Several numerical examples are provided.

Mean Field Control Problems For Vaccine Distribution

• Wonjun Lee, Siting Liu, Wuchen Li, Stanley Osher
• Submitted, Apr, 2021.
With the invention of the COVID-19 vaccine, shipping and distributing are crucial in controlling the pandemic. In this paper, we build a mean-field variational problem in a spatial domain, which controls the propagation of pandemic by the optimal transportation strategy of vaccine distribution. Here we integrate the vaccine distribution into the mean-field SIR model designed in [arXiv:2006.01249]. Numerical examples demonstrate that the proposed model provides practical strategies in vaccine distribution on a spatial domain.

The Back-And-Forth Method For Wasserstein Gradient Flows

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced in [arXiv:1905.12154] to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large scale gradient flows simulations for a large class of internal energies including singular and non-convex energies.

Controlling Propagation of epidemics via mean-field games

• Wonjun Lee, Siting Liu, Hamidou Tembine, Wuchen Li, Stanley Osher
• SIAM Journal on Applied Math, Dec, 2020.
The coronavirus disease 2019 (COVID-19) pandemic is changing and impacting lives on a global scale. In this paper, we introduce a mean-field control modelin controlling the propagation of epidemics on a spatial domain. The control variable,the spatial velocity, is first introduced for the classical disease models, such as the SIRmodel. For this proposed model, we provide fast numerical algorithms based on proximal primal-dual methods. Numerical experiments demonstrate that the proposed modelillustrates how to separate infected patients in a spatial domain effectively.

Joint Task Assignment and Trajectory Optimization for a Mobile Robot Swarm by Mean-Field Game

• Yuhan Kang, Siting Liu, Wonjun Lee, Hongliang Zhang, Wuchen Li, Zhu Han
• IEEE GLOBECOM, Dec, 2020.
In recent years, there has been a growing interest in utilizing mobile robot swarm to execute several tasks at the same time. However, how to assign tasks to the swarm and optimize the trajectory of the robots scientifically to minimize energy consumption is still a big challenge. In this paper, we consider a mobile robot swarm system where a large number of robots are deployed by a centralized controller to execute a series of tasks, such as target detection tasks, cooperatively. The controller controls the velocity strategy of each robot, and makes corresponding task assignment decisions to minimize the overall cost of the robot swarm. Since the number of involving robots is large, it will be extremely difficult to consider the interaction between them. In this regard, we adopt the concept of mean-field term to approximate the behaviors and states of the robots, and formulate the joint task assignment and trajectory optimization problem as a mean-field game. To solve the problem efficiently, a primal-dual hybrid gradient algorithm is proposed to find the optimal trajectory and corresponding task assignment decisions for each robot. The numerical simulation results show the effectiveness of the proposed algorithm.

Energy-efficient Velocity Control for Massive Rotary-Wing UAVs: A Mean Field Game Approach

• H. Gao, Wonjun Lee, Wuchen Li, Zhu Han, Stanley Osher, Vincent Poor
• IEEE GLOBECOM, Dec, 2020.
When a disaster happens in the metropolitan area, the wireless communication systems are highly affected, degrading the efficiency of the search and rescue (SAR) mission. An emergent wireless network must be deployed quickly and efficiently to preserve human lives. Teams of low-altitude rotary-wing unmanned aerial vehicle (UAVs) is preferable to be utilized as an on-demand temporal wireless network because they are generally faster to deploy, flexible to reconfigure, and able to provide better communication services with short line-of-sight links. However, rotary-wing UAVs’ limited on-board batteries require that they need to recharge and reconfigure frequently during a mission. Therefore, we formulate the velocity control problem for massive rotary-wing UAVs as a Schrödinger bridge problem which can describe the frequent reconfiguration of UAVs. Then we transform it into a mean field game and solve it with the G-prox primal dual hybrid gradient (PDHG) method. Finally, we show the efficiency of our algorithm and analyze the influence of wind dynamics with numerical results.

Generalized unnormalized optimal transport and its fast algorithms

• Wonjun Lee, Rongjie Lai, Wuchen Li, Stanley Osher
• Journal of Computational Physics, Nov, 2020.
We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the $L^p$ optimal transport with $L^p$ distance. For $p=1$, we derive the corresponding $L^1$ generalized unnormalized Kantorovich formula. We further show that the problem becomes a simple $L^1$ minimization which is solved efficiently by a primal-dual algorithm. For $p=2$, we derive the $L^2$ generalized unnormalized Kantorovich formula, a new unnormalized Monge problem and the corresponding Monge-Amp\`ere equation. Furthermore, we introduce a new unconstrained optimization formulation of the problem. The associated gradient flow is essentially related to an elliptic equation which can be solved efficiently. Here the proposed gradient descent procedure together with the Nesterov acceleration involves the Hamilton-Jacobi equation which arises from the KKT conditions. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithms.