Upcoming PhD Defenses

Departmental PhD Thesis Exams

Tomas Dominguez Chiozza

Monday, February 12th, 2024
10:00 a.m. (sharp)
BA6183/ Zoom Web Conference

PhD Candidate: Tomas Dominguez Chiozza
Supervisor: Dmitry Panchenko
Thesis title: PDF iconA Hamilton-Jacobi approach to the stochastic block model

PDF iconComplete copy of Tomas' thesis

This thesis addresses the problem of recovering the community structure in the stochastic block model with two communities. The stochastic block model is a random graph model with planted clusters widely employed as the canonical model to study clustering and community detection. The focus is on the fundamental limits of community detection, quantified by the asymptotic mutual information between the observed network and the actual community structure. This mutual information is studied using the Hamilton-Jacobi approach, pioneered by Jean-Christophe Mourrat.

The first contribution of this thesis is a detailed description of the Hamilton-Jacobi approach, and its application to computing the limit of the mutual information in the dense stochastic block model, where the average degree of a node diverges with the total number of nodes. The main novelty is a wellposedness theory for Hamilton-Jacobi equations on positive half-space that leverages the monotonicity of the non-linearity to circumvent the imposition of an artificial boundary condition as previously done in the literature.

The second contribution of this thesis is a novel well-posedness theory for an infinite-dimensional Hamilton-Jacobi equation posed on the set of non-negative measures and with a monotonic non-linearity.
Such an infinite-dimensional Hamilton-Jacobi equation appears naturally when applying the Hamilton- Jacobi approach to the sparse stochastic block model, where the total number of nodes diverges while the average degree of a node remains bounded. The solution to the infinite-dimensional Hamilton-Jacobi equation is defined as the limit of the solutions to an approximating family of finite-dimensional Hamilton- Jacobi equations on positive half-space. In the special setting of a convex non-linearity, a Hopf-Lax variational representation of the solution is also established.

The third contribution of this thesis is a conjecture for the limit of the mutual information in the sparse stochastic block model, and a proof that this conjectured limit provides a lower bound for the asymptotic mutual information. In the case when links across communities are more likely than links within communities, the asymptotic mutual information is known to be given by a variational formula. It is also shown that the conjectured limit coincides with this formula in this case.

Past PhD Defenses

Kevin (Min Seong) Park

Wednesday, December 13, 2023
3:00 p.m.
BA6183/ Zoom Web Conference

PhD Candidate: Kevin Min Seong Park
Supervisor: Adam Stinchcombe
Thesis title: PDF iconTemporal Difference Learning for viscous incompressible flow

This thesis presents a stochastic numerical method for computing viscous incompressible flow. By Itô’s lemma, the solution to a linear parabolic PDE is a martingale over an appropriate probability measure induced by Brownian motion. Given an initial boundary value problem, a functional corresponding to the martingale condition is minimized numerically through deep reinforcement learning. This methodology is well-suited for high dimensional PDEs over irregular domains, as it is mesh-free and sampling techniques can avoid the curse of dimensionality.

The extension to computing viscous incompressible flow is done by first formulating a martingale condition for the viscous Burgers’ equation. Its solution is obtained by a fixed point iteration for which a proof of convergence in L2 is provided. The constrained minimization problem subject to divergence-free vector fields is designed for the incompressible Navier-Stokes equations. The velocity is determined without the pressure gradient. The stochastic numerical method avoids difficulties arising from coupling of velocity and pressure terms by globally maintaining incompressibility. Furthermore, pressure can be recovered from the computed velocity in a post-processing step.

The numerical implementation details are provided, including errors from statistical sampling. Simulations of various flow scenarios are showcased, including those with analytical solutions such as Stokes’ flow in a revolving ball, Poiseuille flow, and the Taylor-Green vortex. Additional validation is acquired from comparing against numerical solutions for cavity flow, and flow past a disk. Analysis is undertaken to determine bounds on the statistical and numerical error. A number of improvements in deep learning and generalizations to broader classes of PDES are proposed as possible avenues of future research. Software is available at github.com/mskpark/DRLPDE