Research

Overview of Research

There is a growing need in the scientific community for rigorous uncertainty quantification. The problems studied in UQ are fundamental to all engineering and life sciences where uncertainty is ubiquitous. Forward UQ problems involve the propagation of uncertainties in model parameters and input data (e.g. initial conditions and boundary values) through solution operators to observable data. The inverse UQ problem involves the propagation of uncertainties in noisy or incomplete observable data to model parameters and input data. Any rigorous UQ analysis accounts for various sources of uncertainties, including those induced by numerical errors. Developing methods to estimate numerical error and propagate it into larger uncertainty assessments is a step towards the goal of building a rigorous UQ into the next generation of simulation codes.

My work with interdisciplinary groups has helped me to apply theoretical mathematics to many practical applications. Through funding from the DOE, DOD, and NSF, I have worked on applications ranging from subsurface contaminant transport modeling a chromium plume near the Los Alamos National Laboratory to storm surge resulting from hurricanes and tropical cyclones in the Gulf of Mexico.

Scientific Software

Now more than ever, research in computational and applied mathematics requires the ability to write professionally documented, organized, and open-source software. In my tenure as Director for the Center for Computational Mathematics, I co-developed, with Dr. Varis Carey, a short course on an Introduction to Scientific Programming with Python, which use Jupyter Notebooks to help novices and even more advanced programmers self assess their progress in situ.

BET

The BET software package is an open-source package for sample-based measure-theoretic inversion. This software package is based upon much of my published research in measure-theoretic inversion, and the major contributors to this package include Lindley Graham, Steve Mattis, Scott Walsh, and myself. The code is well documented and available online. The code includes many examples, several of which are taken from my published results as part of the general push towards scientific reproducibility efforts. The documentation for the examples also includes links to several relevant papers.

ADCIRC

ADCIRC is a system of computer programs for solving time dependent, free surface circulation and transport problems in two and three dimensions. These programs utilize the finite element method in space allowing the use of highly flexible, unstructured grids. Typical ADCIRC applications have included: (i) modeling tides and wind driven circulation, (ii) analysis of hurricane storm surge and flooding, (iii) dredging feasibility and material disposal studies, (iv) larval transport studies, (v) near shore marine operations.

I have implemented data assimiation libraries into the ADCIRC framework to improve storm surge forecasting in the Gulf of Mexico.