My research interests overlap with applied analysis, numerical analysis, numerical linear algebra, and most recently, data science. A common
theme to all my work is that it is typically centers around nonlinear PDEs that govern the evolution of fluid flows. Here are some specifics, for the
interested graduate student:
Data Assimilation:
Most models of evolutionary systems include an initial condition and boundary conditions. However, we typically do not know these conditions
exactly, and the small amounts of error present in them can cause poor solutions if simulations are run for more than just a short time (think
weather prediction). I am working on methods to incorporate measurement data, at any/all times, into the simulations to get better longer time accuracy.
Nonlinear Solvers:
Solving nonlinear systems equations shows up in most fields of mathematics, and I work on design of iterative procedures to solve such systems faster than current
state of the art methods. My main recent interest in this area is in spliting methods for nonlinear pde problems in fluids, and in using Anderson acceleration for general
nonlinear iterative solvers.
Numerical Method Development and Analysis:
For most equations of practical interest, many numerical methods already exist for approximating solutions. However, with fluid problems, simulations can run for days or weeks (or longer), and so
smart discretizations that can give more accurate solutions more efficiently are always needed. Determining accuracy can be a deep problem involving signficant functional analysis.
Model Reduction:
Many evolutionary systems have dominant recurrent modes/structures that drive them. If these modes can be extracted, then in some cases very small models can be created
which give very accurate solutions - in seconds, instead of hours or days for a regular simulation. Creating the very small system can be expensive, but this can be very
worthwhile in settings where many simulations of the same or similar problem needs done, such as in control, uncertainty quantification, or in parameter studies.
Applied Analysis:
My work in this area is about proving well-posedness for various nonlinear PDEs. Whether solutions exist to a particular equation, whether they are unique, how smooth they are, and how
they behave with respect to parameters are all very important things to know before approximating solutions numerically. Hence before one spends lots of
time developing algorithms and coding, one should make sure the PDE system is well-posed, and how smooth solutions are can help determine how to best do numerical approximations.