Applied Math

A longstanding challenge in data science is to effectively quantify systems of interest by integrating information from heterogeneous datasets, a problem known as multiview learning. In this talk, I will present recent advancements in this direction, focusing on novel algorithms based on…

Manifold optimization has found wide applications across various scientific and engineering domains. In this talk, I will present our recently developed algorithms for large-scale decentralized and federated manifold optimization. In addition, I will present a retraction-free and penalty…

In this talk, we explore two classical image processing tasks motivated by cryo-electron microscopy imaging: tomographic image denoising and rigid image registration. Both tasks inherently involve operations of 2D rotations, where leveraging specific transforms can significantly enhance the…

The Muskat problem on the half-plane models motion of an interface between two fluids of distinct densities in a porous medium that sits atop an impermeable layer, such as oil and water in an aquifer above bedrock.  We develop a local well-posedness theory for this model in the stable…

In many situations, the combined effect of advection anddiffusion enhances dissipation. I will talk about this in two contexts: The first is for a randomly shifted alternating shear flows where we show that dissipation enhancement occurs on time scale O(\ln κ), where κ is the molecular…

In this talk, we will present a martingale based neural network, SOC-MartNet, for solving high-dimensional Hamilton-Jacobi-Bellman (HJB) equations where no explicit expression is needed for the Hamiltonian $\inf_{u \in U} H(t,x,u, z,p)$, and stochastic optimal control problems (SOCP) with…

High-resolution imaging in complex media, such as turbulent air, underwater environments, or biological tissues, faces challenges due to wavefront distortion caused by scattering from inhomogeneities. I will describe an approach for imaging point-like sources in scattering media when large and…

Perhaps the most elegant mathematical definition of privacy of data is called "differential privacy". I will describe a somewhat more general framework, which leads to some fun questions at the interface of probability and metric geometry. This talk is based on joint work with March Boedihardjo…

I will discuss the problem of solving a system of equations F(x)=0,for x a d-dimensional unit vectors and D a non-linear map from R^d to R^n whose components are independent, rotationally invariant Gaussian processes. We studied this problem under the proportional asymptotics in which n and…