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

*Co-mentor: Houman Owhadi*

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 include partial differential equations and fluid dynamics, with practical emphases in data assimilation, optimization, high performance computing, and numerical analysis. I find the intersection of theoretical and applied mathematics to be more fruitful than studying either subject in isolation.

## Postdoctoral Scholars

*Co-mentor: Tapio Schneider*

Eviatar's interests lie in developing methods to improve prediction of high-dimensional, chaotic, and sparsely observed systems, with a focus on the Earth system. In particular, he has worked on methods that use tools from ensemble-based data assimilation to combine dynamical model forecasts with machine learning. He is now working on ensemble-based inversion methods to correct the statistical properties of dynamical systems. He is also interested in the application of concepts from dynamical systems theory to understand the dynamics and predictability of climate.

My research addresses issues at the heart of computational engineering, sciences, and medicine: to understand, enhance, and control the quality, validity, and reliability of simulation-based predictions of complex physical systems. I have worked on various uncertainty quantification and optimization problems associated with models governed by parametric partial differential equations. I have extensively worked on application problems associated with computational polymer science. I am currently working on constitutive modeling for solid mechanics.

*Primary mentor: Tapio Schneider*

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.

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.

I am primarily interested in developing efficient Bayesian inference techniques for probabilistic models such as Gaussian processes and stochastic (partial) differential equations, with the goal of making them applicable in real-world applications. Methods I have considered include variational inference, inclusion of physical or geometric/topological inductive biases, and deep learning methodologies. Prior to this, I have also done research at the intersection of geometric and statistical mechanics, as well as analysis of stochastic fluid PDEs.

*Primary mentor: Houman Owhadi*

My research interests are in numerical algorithms for PDEs, mean field games, and gaussian processes.

## Graduate Students

*Secondary advisor: Tapio Schneider*

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.

*Secondary advisor: Franca Hoffmann*

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.

*Primary advisor: Jean-Phillipe Avouac*

My research delves into the area of high-dimensional chaotic dynamical systems. I am particularly interested in extreme events, data assimilation, uncertainty quantification, and the application of machine learning to uncover constitutive laws. Currently, my focus is on studying the predictability of earthquakes and Slow Slip Events (SSEs). In the intersection of mathematics, geophysics, and data science, my work aims to shed light on complex, high-dimensional systems that have the potential to improve our understanding and ability to predict natural phenomena.

*Primary advisor: Tapio Schneider*

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 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.

*Primary advisor: Kaushik Bhattacharya*

I am interested in problems at the intersection of mechanics, data-driven modelling, statistics and machine learning. Currently, I am working on multiscale modelling problems and learning thermodynamically stable material constitutive laws from data.

*Primary advisor: Anima Anandkumar*

My research interests lie at the intersection of computer science and applied mathematics. I work on data-driven methods for computational problems in natural science.