# Research

## Applied and Computational Mathematics

**Uncertainty Quantification**

We develop and deploy efficient methods for quantification of predictive uncertainty. This uncertainty stems from the approximate nature of any mathematical model, model parameters. These stem from data sparsity, incomplete knowledge, etc. We adopt a probabilistic framework for uncertainty quantification and deal with stochastic differential equations (equations with random coefficients).

## Environmental Fluid Mechanics

**Physics-informed machine learning**

*Dynamic Mode Decomposition (DMD)*

DMD is a method for dynamical system analysis and prediction from high-dimensional data. Based on the power of Singular Value Decomposition (SVD), DMD is able to extract the low-rank structure from the data as well as separating temporal and spatial features. The close connections between DMD and the Koopman operator provide theory to interpret the capability of this equation-free, data-driven method. In our research, we are taking advantage of the DMD approach to accelerate multiscale simulations. The accuracy and efficiency of DMD in learning dynamics are explored both analytically and numerically.

## Biomedical Modeling

**Multiscale and multiphysics modeling**

**Propagation of noise/fluctuations across scales and model components**

Construction and analysis of a tightly-coupled domain decomposition for highly nonlinear hydrogen diffusion through a three-layer membrane subject to a stochastic boundary concentration. Analysis of noise propagation in a hybrid atomistic-continuum solver for highly nonlinear systems, with the one-dimensional stochastic Ginzburg-Landau equation acting as a testbed.