Applied Math

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…

The advent of generative AI has turbocharged the development of a myriad of commercial applications, and it has slowly started to permeate to scientific computing. In this talk we discussed how recasting the formulation of old and new problems within a probabilistic approach opens the door to…

In this talk, we shall discuss our recent work which shows that in the periodic homogenization of viscous HJ equations in any spatial dimension the effective Hamiltonian does not necessarily inherit the quasiconvexity property (in the momentum variables) of the original Hamiltonian. Moreover,…

This talk will discuss some nontrivial but often pleasant effects of large learning rates, which are commonly used in machine learning practice for improved empirical performances, but defy traditional theoretical analyses. I will first quantify how large learning rates can help gradient descent…

This presentation first discusses the derivation of radiative transfer equations for acoustic waves propagating in a randomly fluctuating half-space in the weak-scattering regime, and the study of boundary effects through an asymptotic analysis of the Wigner transform of the…

Abstract