Georgia Institute of Technology College of Sciences

Finding Cheeger cuts of graphs is an NP-hard problem, and one often resorts to approximate solutions. In the literature, spectral graph theory provides the most popular approaches for obtaining such approximate solutions. Recently, K.C. Chang introduced a novel nonlinear spectral graph theory and proved that the seek of Cheeger cuts is equivalent to solving a constrained optimization problem. However, this resulting optimization problem is also very challenging as it involves a non-differentiable function over a non-convex set that is composed of simplex cells of different dimensions. In this talk, we will discuss an ADMM algorithm for solving this optimization problem and provide some convergence analysis. Experimental results will be presented for typical graphs, including Petersen's graph and Cockroach graphs, the well-known Zachary karate club graph, and some preliminary applications in material sciences.

In this talk, I will give an introduction to the implicit boundary integral method based on the co-area formula and it provides a simple quadrature rule for boundary integral on general surfaces.  Then, I will focus on the application of solving the linearized Poisson Boltzmann equation, which is used to model the electric potential of protein molecules in a solvent. Near the singularity, I will briefly discuss the choices of regularization/correction and illustrate the effect of both cases. In the end, I will show the numerical analysis for the error estimate. 

In this talk, we discuss generative modeling algorithms motivated by the time reversal and reflection properties of diffusion processes. Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. We develop SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. Besides the quest for generating images at ever higher resolution, our primary motivation is to create a well-posed infinite-dimensional learning problem so that we can discretize it consistently at multiple resolution levels. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting by ensuring the well-posedness of forward and reverse processes, and derive the convergence of the approximation of multilevel training. We illustrate that approximating the score function with an operator network is beneficial for multilevel training.

In the second part of this talk, we propose the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and demonstrate its scalability in constrained generative modeling.

Diffusion models, particularly score-based generative models (SGMs), have emerged as powerful tools in diverse machine learning applications, spanning from computer vision to modern language processing. In the first part of this talk, we delve into the generalization theory of SGMs, exploring their capacity for learning high-dimensional distributions. Our analysis show that SGMs achieve a dimension-free generation error bound when applied to a class of sub-Gaussian distributions characterized by certain low-complexity structures.  In the second part of the talk, we consider the application of diffusion models in solving partial differential equations (PDEs). Specifically, we present the development of a physics-guided diffusion model designed for reconstructing high-fidelity solutions from their low-fidelity counterparts. This application showcases the adaptability of diffusion models and their potential to scientific computation.