Analysis of Stochastic Linear Algebra Methods with Nicholas Marshall

Dr. Nicholas Marshall is interested in problems involving an interplay between analysis, geometry, and probability (especially those motivated by data science).

Dr. Marshall's project this summer will focus on stochastic linear algebra methods. In the field of machine learning, stochastic optimization methods, such as stochastic gradient descent, have become the standard method for a wide range of problems. These stochastic methods exhibit many advantageous qualities which are not well understood mathematically. This project aims to advance the mathematical understanding of these stochastic methods in the linear algebra setting. Linear algebra provides a setting where developing a precise theoretical understanding of special cases of these stochastic methods is feasible.

Investigating connections between stochastic processes and analysis of Partial Differential Equations with Radu Dascaliuc

Dr. Radu Dascaliuc's research focuses on qualitative properties and long-time behavior of nonlinear evolution equations, using both analytic and probabilistic approaches.

Dr. Dascaliuc's project this summer will focus on Investigating connections between stochastic processes and analysis of Partial Differential Equations.

Many deterministic evolution Partial Differential Equations (PDE) can be naturally connected to stochastic structures that shed light on the behavior of the solutions. Perhaps the most famous example of such a connection is the heat equation and Brownian motion. For the incompressible Navier-Stokes equations (NSE), the nonlinearity, which is responsible for mixing fluids on various scales, leads to a branching stochastic process, with an associated random binary tree – a stochastic cascade. Studying the probabilistic properties of these cascades can potentially help understand connections between the statistical nature of the empirical laws of turbulence and determinism of the underlying equations of motion – such as NSE. This project will study stochastic cascades for various simplified turbulence models.

Random Models for Electromagnetics with Nathan Gibson

Dr. Nathan Gibson investigates problems in the general areas of Applied and Computational Mathematics with specific interests in Numerical Analysis, Partial Differential Equations, Inverse Problems and Uncertainty Quantification.

Dr. Gibson's project this summer is related to electromagnetic signals, including pulses, which are used in numerous high-impact applications. The dynamics of the propagation of these waves is modeled using Maxwell's equations. Additional models are necessary for representing a material's response to the electromagnetic fields. Natural variability at the microscale can be captured by endowing dielectric parameters with probability distributions. Inverse problems in this context include material identification from scattered field data or optimal design of metamaterials. This project will explore optimization under uncertainty problems involving computational electromagnetics.

Past years

The proceedings from the past few years give an idea of the variety and scope of previous projects.