The overall direction of my research is in the area of optimal design and control of distributed parameter systems. This is a multidisciplinary area and I perform a majority of my research at the Interdisciplinary Center for Applied Mathematics (ICAM). My main area of study is in system level and computational aspects of these problems covering research areas:
computational science/scientific computing, control theory, numerical analysis, numerical methods for PDEs, optimization, reduced-order modeling and sensitivity analysis.
More details from some of these research areas are included below.
My interest in sensitivity analysis goes back to my Masters research. The current line of research was initially motivated by the role of sensitivity analysis in computing derivatives for optimal design of a forebody simulator. An overview of sensitivity analysis as well as a number of practical applications are described in more detail in the following links: overview and applications.
The availability of System X and tera-scale computing capability on campus has led Traian Iliescu and I to develop a research code (ViTLES - the Virginia Tech Large-Eddy Simulator) for developing and testing models for high Reynolds number flows. This code makes use of parallelism through domain decomposition and the use of PETSc. A website with more details on ViTLES is here.
ViTLES was developed on equipment obtained from SCHEV ETF funds and an NSF SCREMS grant.
Reduced-order modeling refers to the construction of low-dimensional models for high-dimensional or infinite dimensional systems. The typical method for construction of reduced-order models is based on finding a suitable low-dimensional basis (by eg. proper orthogonal decomposition) and forming a model by projecting the high dimensional model onto this basis (by eg. Galerkin projection).
These models are important for either fast model evaluations in applications such as forecasting, data assimilation and optimization, or for building low-dimensional models for in applications such as control.
Along with Chris Beattie, Serkan Gugercin and Traian Iliescu, I am working on developing methods to build reduced-order models for complex fluid problems. The need for parallel methods and better strategies for model construction are well known.
A working page for our research on reduced-order modeling is here.
I am very grateful to have received support for this research from the Air Force Office of Scientific Research (AFOSR), the National Research Council (NRC), and the National Science Foundation (NSF):
I was fortunate to spend extended research stays at the Air Vehicles Directorate at AFRL, SCS at Florida State University, and Genie Mecanique at Ecole Polytechnique de Montreal in recent years.