Professor Stuart's research is focused on the development of foundational mathematical and algorithmic frameworks for the seamless integration of models with data. He works in the Bayesian formulation of inverse problems for differential equations, and in data assimilation for dynamical systems.

# People

## Faculty

## Staff

## von Karman Instructors

My research is on developing scalable computational methods for uncertainty quantification in complex physical systems. In particular, my work leverages measure transport and dimension reduction to design principled techniques for probabilistic modeling and Bayesian inference. Broadly, I am motivated by using these algorithms to improve predictions and gain insights on systems in engineering, geophysics, and medicine.

My research interests are motivated by the need for scalable computational methods to enable and enhance decision-making in engineering, scientific, and medical applications. In particular, my work focuses on the development of principled low-dimensional model approximations through model reduction and scientific machine learning, and on designing multi-fidelity formulations for optimization and uncertainty quantification that embed these approximations in decision-making settings.

## Postdoctoral Scholars

Research Interests in Applied analysis and simulation: Partial differential equations, optimization, deterministic modelling, inverse problems, numerical analysis and implementation; Uncertainty quantification and probabilistic techniques: Bayesian inverse problems, optimal design, scalability of algorithms, machine learning.

Michelle Feng is a postdoctoral researcher working on developing topological tools for analyzing complex social systems, especially social systems that are informed by spatial patterns (e.g. housing, voting). Her other research interests include studying network structure (especially higher order structures in networks), and the intersection of topology and machine learning. Outside of mathematics, she is interested in advocacy, literature, and crafting.

My research interest lies in advancing fundamental understanding and predictive modeling for real world engineering applications and important natural phenomena. Previously, I focus on developing mathematical models and advanced computational algorithms (e.g. embedded boundary method and high order methods). Recently, I start to explore data-driven approaches to improve these models and quantify uncertainties (e.g. neural networks and Bayesian inversion).

I am interested in the development of efficient methods for operator approximation and their application to Bayesian data assimilation for fluid flows. Part of my research focuses on emerging operator learning architectures, aiming to contribute to the theoretical underpinnings of these methods, and seeking to improve their efficiency and reliability. My other research focuses on fluid flows, their ill-posedness and implications for their reliable approximation by numerical methods.

## Graduate Students

I come from a numerical PDEs background, having worked on finite difference, finite element and pseudo-spectral methods for Schroedinger and Navier--Stokes equations. My current research is developing data assimilation tools for global climate models, in particular, the focus is on experimental design questions and ODE/PDE averaging. I am also interested in Koopman analysis, diffusion maps, and parameter estimation.

My work lies at the intersection of data assimilation, stochastic analysis, dynamical systems, computational statistics and machine learning. In general, I am interested in using theoretical insights from analysis to develop novel computationally efficient numerical algorithms.

I am broadly interested in theoretical and computational math problems arising at the intersection of physics, computer, and information science. Specifically, I’ve been doing research in multiscale analysis, numerical analysis, machine learning and statistics.

Ervik’s research interests broadly encompass stochastic models for the climate. He has a background in applied mathematics and is currently working on developing stochastic closures for subgrid-scale cloud models. The goal of this research is to develop a model that more faithfully incorporates uncertainty. As part of this, he also works on methods for robust parameter estimation and uncertainty quantification.

I am interested in developing novel methods within the intersections of dynamical systems, machine learning, and data assimilation, and have most often applied these methods to biomedical contexts, including modeling and prediction of the glucose-insulin system.

I have research interests in theory and algorithms for high-dimensional scientific and data-driven computation. My current work is centered on operator regression, with application to efficient surrogates for forward and inverse problems arising from models of physical systems. To this end, I develop and utilize tools from machine learning, model reduction, and numerical/statistical analysis.