1. ODE and PDE Models for Traffic Flow

Project mentors: Dr. Wen Shen and Dr. Tianyou Zhang

We consider the first-order, following- the-leader model, for an infinitely long highway with $$N$$ identical cars, and possible traffic lights at certain locations. The unknown variable $$x_{i}(t)$$ denotes the position of the $$i^{\mbox{th}}$$ car at time $$t$$, with $$x_{i+1}(t)-x_{i}(t)\geq L$$ for all $$t$$ and $$i$$, where $$L$$ is the vehicle length. Initially, the $$i^{\mbox{th}}$$ car is at position $$\bar{x_{i}}$$. During time intervals when the lights are green, we consider the following microscopic model:

$\dot{x_{N}}(t)=V,\;\;\;\;\dot{x_{i}}(t)=\mathcal{V}((x_{i+1}-x_{i})(t))-e^{-t/\epsilon},\;\;\;\; i=1,\dots,N-1.\label{eq:Traffic1}$

The foremost car $$x_{N}$$ has the empty highway ahead of it, thus it travels with a fixed speed $$V$$. All other cars travel with a speed that depends only on the distance between the current car and the one ahead, described by the monotone decreasing function $$\mathcal{V}\in\mathbf{C}^{0,1}([L,+\infty\;[0,V])$$ satisfying suitable assumptions. Here the parameter $$\epsilon>0$$ may be treated as a relaxation parameter.

With $$\epsilon=0$$, as $$N\rightarrow+\infty$$, eq. ([eq:Traffic1]) converges to the macroscopic Lighthill-Whitham model

$\rho_{t}+(v(\rho)\rho)_{x}=0,\;\;\;\;\rho(0,x)=\bar{\rho}(x).\label{eq:Traffic2}$

Here $$\rho(t,x)$$ is the density of cars, and $$v(\rho)$$ is the velocity of cars, assumed to be a decreasing function of the density. Equation ([eq:Traffic2]) is simply the conservation of cars. One observes that the two models are related through particle paths, i.e. the trajectories of single cars in eq. ([eq:Traffic2]), $$\dot{x}=(v(\rho)\rho)$$ with $$x(0)=\bar{x}$$. The students will conduct numerical simulations for model ([eq:Traffic1]) using ODE solvers, with the option of adding traffic lights, and model ([eq:Traffic2]) using the Lax-Friedrichs method or front tracking. They will observe the behavior of ([eq:Traffic1]) for large $$N$$ , with $$\varepsilon=0$$ and $$\varepsilon>0,$$ and observe the convergence to ([eq:Traffic2]) and to a relaxation model. In order to reduce congestion on the highway, our goal is to maximize the flux $$\rho v$$ at a chosen point $$X$$, by varying the parameters in the model. The students will run numerical simulations with different values of $$L$$, $$V$$ , $$\epsilon$$, and different functions $$\mathcal{V}$$. Students may also include other factors into the model, then run simulations and evaluate the performance of these new models.

2. A Harvesting Model for a Fishery.

Project mentors: Dr. Wen Shen and Dr. Tien Khai Nguyen

In this project, we study optimal harvesting of marine resource. The analysis proceeds in two steps: understanding first a control problem, then a differential game.

An optimal control problem.

Denote by $$x(t)$$ the size of a fish population at time $$t$$, subject to harvesting. This evolves according to $\dot{x}=\alpha x(M-x)-xu,\;\;\;\; u\in[0,u_{\mbox{max}}].$ Here the control $$u(t)$$ accounts for the fishing effort, while the product $$x(t)u(t)$$ is the actual catch at time $$t$$. The constants $$M$$, $$\alpha$$ describe the maximum sustainable population and a reproduction rate, respectively.

We consider the optimal harvesting problem in infinite time horizon, exponentially discounted: maximize $\int_{0}^{\infty}e^{-\gamma t}\left[x(t)u(t)-cu(t)\right]\, dt\;\;\;\;\mbox{over}\;\;\;\; u\in[0,u_{\mbox{max}}],$ where $$c$$ is a unitary cost of the harvesting effort, and $$\gamma$$ is the discount rate.

• Derive a Hamilton-Jacobi (H-J) equation for the value function $$V(y)$$=[maximum payoff that can be achieved when the initial population is $$x(0)=y$$]. Since $$x\in\mathbb{R}$$, this takes the form of an implicit ODE.

• Design a numerical algorithm to compute solutions of the implicit ODE.

• Compute the value function, and use it to find the optimal feedback control $$u=U(x)$$.

• Compute the asymptotic value of $$x(t)$$ as $$t\rightarrow\infty$$, and describe how this depends on the discount rate $$\gamma$$ and the harvesting cost $$c$$.

A differential game.

Next, consider the case of $$n$$ equal fishermen, modeled as a noncooperative game. Calling $$u_{i}$$ the harvesting effort of the $$i^{\mbox{th}}$$ fisherman, the fish population now evolves according to $\dot{x}=\alpha x(M-x)-x\cdot\sum_{i=1}^{n}u_{i}(t),\;\;\;\; u_{i}(t)=[0,u_{\mbox{max}}].$

The payoff of the $$i^{\mbox{th}}$$ player is $J_{i}=\int_{0}^{\infty}e^{-\gamma t}\left[x(t)-c\right]u_{i}(t)\, dt.$

We say that an $$n$$-tuple of feedback strategies $$\left(u_{1}^{*}(x),\dots,u_{n}^{*}(x)\right)$$ is a non-cooperative Nash equilibrium if for each $$i=1,\dots,n$$ the feedback control $$u_{i}^{*}$$ is optimal for the control problem: $\mbox{maximize }J_{i},\;\;\mbox{subject to }\;\;\dot{x}=\alpha x(M-x)-xu_{i}^{*}(t)-\sum_{j=1\,(j\neq i)}^{n}xu_{i}^{*}(x),\;\; u_{i}(t)=[0,u_{\mbox{max}}].$

• Assuming that all players adopt the same strategy: $$u_{1}^{*}(x)=u_{2}^{*}(x)=\dots=u_{n}^{*}(x)$$, write a H-J equation describing the value functions $$V_{i}(y)=V(y)$$, which are the same for all players.

• Describe the common optimal feedback strategy $$u_{1}^{*}(x)$$ implemented by every player.

• Compute $$\lim_{t\rightarrow\infty}x(t)$$, and describe how it depends on $$\alpha$$, $$\gamma$$, and $$n$$.

• Compare the results with the optimal control case. Describe how the payoff changes if a maximum harvesting quota for each fisherman is imposed. What happens if $$u_{\mbox{max}}$$ is reduced?

3. Geometry and combinatorics of convex codes and feedforward neural networks

The goal of this project is understanding the relationship of the coding properties of feedforward neural networks on one hand and arrangements of convex sets on another. A binary code is a set of binary patterns (i.e. the strings of 0s and 1s) of length n. A convex code is a set of binary patterns that can be possibly obtained as the patterns of intersection of n convex open sets. Feedforward neural networks generate binary codes that are convex.

It is known that one can construct a network with a prescribed set of maximal patterns (or a prescribed simplicial complex). However it remains an open question if any convex code may be encoded on a neural feedforward network. This project will explore possible approaches to (i) finding a convex code that is not realizable on feedforward networks or (ii) finding a construction for encoding every convex code by a feedforward network. The tools to be used for the project are a combination of elementary combinatorics, topology and computational approaches.

4. Multiple pendulums

Project mentor: Dr. Shuhao Cao

The most notable example of multiple pendulum is the double pendulum. Double pendula normally consist two linkages connecting two rods. One is free to rotate with respect to a point, the other is free to rotate with respect to a point on the first rod. The double pendulum gives us a glimpse of how chaotic the behavior can be for such a simple physical system. The derivation of the equation of motion for a double pendulum can be facilitated by introducing generalized coordinates and calculus of variations, which is the gateway to the Lagrangian mechanics.

The two variables (degrees of freedom) to model the double pendulum are $$\theta$$ and $$\phi$$, the angles between each rod and the vertical axis. If we have the rods with the same mass $$m$$ and the same length $$l$$, the Lagrangian is $\mathcal{L} = ml^2\dot{\theta}^2 + \tfrac{1}{2} m l^2 \dot{\phi}^2 + ml^2 \dot{\theta} \dot{\phi} \cos(\theta ??? \phi) + 2 m gl \cos\theta + mgl \cos \phi.$ The equation of motion, analogous to $$F=ma$$, can be derived from above Lagrangian. For certain initial energy (initial positions or velocities), it is known that the trajectories these rods exhibit are “chaotic”.

• Derive the Euler-Lagrange equation for the double pendulum with different shapes and masses, and for the triple pendulum.

• Implement a numerical integration to compute the trajectories of different pendula above.

• Investigating the possible stable orbits, and possible way to stabilize the system (external force, etc). Learn how to visualize the Poincarè sections for the pendula.

5. Electromagnetic wave propagation

Project mentor: Dr. Shuhao Cao

Maxwell’s equations describe the mechanism of how light propagates through space and time, and how magnetism and electricity are intertwined. On a straight line, the equations can be simplified to $\begin{cases} \partial_t (\varepsilon E) + \partial_x H =j, \\ \partial_t (\mu H) + \partial_x E =0. \end{cases}$ In this equation, $$E$$ stands for the electric field’s strength and direction, and $$H$$ stands for the same quantities for magnetic field. The $$\varepsilon$$ is the dielectric permittivity which describes how easy a material will be polarized by an external electric field. The modern engineering techniques introduce many nonlinear materials such that $$\varepsilon$$ can be complex and depends on $$E$$. When electromagnetic waves of a certain frequency propagates through such materials, new frequencies of waves will be created.

• Learn the modern numerical techniques for simulating the wave propagation like finite element and discontinuous Galerkin.

• Investigate the behavior of different frequencies of electromagnetic waves propagates through a material with quadratic nonlinearity $$\varepsilon = \alpha + \beta E^2$$.

6. Is policing an evolutionarily stable job in a community?

Project mentor: Dr. Tim Reluga

Today in America, there is an intense conversation on-going about the behavior and roles of our police forces, revolving around the tragic deaths of both citizens and police officers. Police play a vital role in our society - they are the only legal entity through which force is applied to enforce law and order. But the mis-application of this force due to error and bias does harm and creates resentment within the very communities police are serving.

While there is a rich and complex cultural dialog ongoing, this project aims at a more fundamental question – when does a community evolutionarily benefit from having some individuals play the role of police, and when is the role of police itself evolutionarily stable.

We’ll explore at least two possible modelling approaches to develop a theory of policing:

• Replicator dynamics. The most common evolutionary model used for populations with a small number of pure strategies is the replicator equation $\dot{x_i}=x_i\left(\left(Ax\right)_i-x^TAx\right) .$ We’ll explore extending Hawk-Dove and Rock-Paper-Scissors versions of this equation to include Police, and study the dynamic consequences.

• Wahl’s theory of task allocation. In 2002, Wahl proposed a unique group-selection equation in which survival was dependent on the efficient allocation of tasks within the group. $\dot{x_i} = x_i [ f_i(x) - \phi(x)], \quad \phi(x) = \sum_{j=1}^{n}{x_j f_j(x)}$ We’ll use this to explore competition between tribes with and without police.

My conjecture is that catastrophic collapse of police-strategies can be a natural consequence of increasing resource pressure on a population, independent of other cultural factors.