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 and Sampling Optimization and sampling are quintessential tasks in applied mathematics, both of which arise naturally from solving inverse problems. In applications, $\mathcal{G}$ is expensive to evaluate (e.g., solving a PDE), and derivatives of $\mathcal{G}$ are unavailable.
These tasks arise from formulating the Bayesian inverse problem, where one wishes to recover a parameter $\theta \in \mathbb{R}^d$ from data $y \in \mathbb{R}^m$, with $\mathcal{G} : \mathbb{R}^d \to \mathbb{R}^m$ and $\eta \sim \mathcal{N}(0,\Gamma)$: $$ y = \mathcal{G}(\theta) + \eta. $$
For a Gaussian prior $\mathbb{P}(\theta)=\mathcal{N}(\theta_0,\Sigma)$, the posterior $\mathbb{P}(\theta \mid y)$ can be written $$ \mathbb{P}(\theta \mid y) \propto \exp!\left(-\tfrac12\,\lvert y-\mathcal{G}(\theta)\rvert_{\Gamma}^{2} - \tfrac12\,\lvert \theta-\theta_0\rvert_{\Sigma}^{2}\right) = \exp!\big(-\Phi(\theta)\big), $$ where $\lvert x\rvert_{A}^{2} := \langle x, A^{-1}x\rangle$.
Some natural questions are: 1) Optimization: Find the MAP estimate (the maximizer of $\mathbb{P}(\theta \mid y)$). 2) Uncertainty Quantification: Sample from the posterior $\mathbb{P}(\theta \mid y)$.
Interacting particle methods have become a popular alternative because they: (i) do not need gradient information ✓, (ii) allow for parallelization ✓, (iii) enjoy convergence guarantees through mean-field analysis* ✓
* under appropriate assumptions on the interaction and potential.