# 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).

*The method of distributions*

Rather than solving the same model thousands of times (Monte Carlo simulations), we derive a single deterministic equation for the joint probability density function (PDF) or cumulative distribution function (CDF) of the random solutions of the original stochastic system. The resulting PDF/CDF equations provide a systematic way to study the effect of random parameters/inputs on key quantities of interest. (Recent application areas: initiation events in energetic materials, flow and reactive transport in heterogeneous media, detection of leaks and blockages in pipes, vehicular traffic)

*Data assimilation and PDF methods*

Data can be used via Bayesian updating to reduce the uncertainty on the parameters and the inputs of a stochastic system described by the PDF equation. We are investigating how data (in the form of measurements, observations, or synthetic Monte Carlo realizations) can help to inform closures/corrections for inaccurate stochastic models, and reduce the uncertainty on the predictions.

*Numerical Methods for High Dimensional PDEs*

The Boltzmann Transport Equation has important applications in both non-continuum fluid mechanics and semiconductor research. It is an example of a high-dimensional partial-differential equation (PDE) and describes the evolution of a joint probability function. Another example is PDF/CDF equations. Due to the curse of dimensionality these kinds of equations are notoriously hard to solve. I work on parallel tensor methods to solve high-dimensional PDEs. *Global sensitivity analysis*