My research interests are broadly in applied probability theory by leveraging probabilistic principles and incorporating them into applicable models. In particular, my undergraduate research has focused on rigorously studying and implementing interacting particle systems to study how the microscopic "random" behavior of individual particles can be organized to recover specific macroscopic behaviors of the ensemble. I outline some of my current projects below.
Particle-Based Optimization Methods
Doing optimization accurately and efficiently is a critical part of computational science and engineering. However, for non-convex functions (as in a), typical gradient-based methods fail. To get around these issues, swarm optimization models have become popular. In particular, particle swarm optimization (PSO) is a widely implemented swarm algorithm that was proposed to study the intelligent behavior of birds or fish.
We first write down the original PSO model as in 1 , with the physical constraint that ($c_1, c_2 > 0$). The matrices (R_1^n, R_2^n \in \mathbb{R}^{d\times d}) are diagonal matrices with uniformly distributed entries on the interval $[0,1]$, and are generated for each particle and iteration. The PSO update equations are:
\begin{equation} x^{n+1}_i = x^n_i + v^{n+1}_i \[ v^{n+1}_i = v^{n}_i + c_1R_1^n(y^n_i - x^n_i) + c_2R_2^n(\overline{y}^n - x^n_i) \] \end{equation}