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

## 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.

My research focuses on Uncertainty Quantification for complex simulators as the ones encountered in climate modeling and engineering applications. My research interests gravitate around Bayesian inverse problems, stochastic processes, numerical analysis, and both stochastic and deterministic partial differential equations. I am also interested in large scale applications of mathematical models, often encountered in numerical optimisation and pattern recognition tasks as in machine learning. As a professional, I have specialised in numerical methods, statistical modelling, data visualisation and applied probability models.

My research interests lie at the interface of model-driven and data-driven approaches, developing new tools for PDE and data analysis. My current interests are among others non-linear drift-diffusion equations, kinetic theory, many particle systems, gradient flows, entropy methods, optimal transport, functional inequalities, parabolic and hyperbolic scaling techniques, hypocoercivity and diffusive models in mathematical biology. Furthermore, I am interested in developing rigorous mathematical tools for data analysis in the context of inverse problems, Bayesian inference and machine learning, unsupervised and semi-supervised learning, focusing on data clustering and classification, graph Laplacians and their continuum counterparts, spectral analysis, uncertainty quantification and consistency analysis.

My research interests lie at the intersection of applied mathematics, probability and statistics. Broadly speaking, I work on the analysis, development and application of methods for estimating parameters and quantifying uncertainty. I am particularly interested in the Bayesian approach for solution of inverse problems. My interest in this field stems from industrial projects in atmospheric dispersion and focused ultrasound treatment.

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'm interested in data-driven manifold learning and dimensionality reduction methods specifically in the context of multiscale, stochastic, high-dimensional complex systems. My work focuses on developing algorithmic and theoretical aspects of optimal design of measurements and inputs for forecasting and control. This optimization relies on model reduction to obtain a low-rank representation of dynamics. I have applied this framework to sensor placements in fluid flows, ocean temperature data, aircraft shimming data, and more generally to optimal sensor and actuator placement for linear time invariant systems.

My research interests are primarily in the fields of modeling, estimation, control, and numerical optimization of systems governed by dynamical systems. I am also interested in machine learning, data assimilation, and statistics applied to the problems in the areas of mathematical biology and biomedicine. My current research is focused on modeling the glucose-insulin dynamics in humans.

My previous research mainly focuses on data-driven turbulence modeling by using Bayesian inference and machine learning techniques. More recently, I also started to explore generative learning techniques (e.g. generative adversarial networks) to emulate and predict PDE-governed systems. In general, my research interests lie in an interdisciplinary area of computational physics, applied mathematics and statistics.

## 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 research interests lie at the intersection of learning theory and inverse problems. I am keen on the development of the mathematical theory of learning and its implications for advancement of numerical algorithms. Further I am interested in the application of existing learning systems to physical problems arising in the sciences.

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.

Nicholas has research interests in computational science and mathematics, including numerical methods for partial differential equations and optimal control. Motivated by applications from the physical sciences and engineering, he works in the analysis and development of modern computational methods for inverse problems and data assimilation in dynamical systems. Nick is also interested in connections between the above areas and learning algorithms.