Research

Overview:

Moore's Law and an enormous multi-decade national investment in massively parallel computational resources have enabled numerical simulation to have a transformational impact on fields as diverse as aircraft development, bioinformatics and weather simulation. However, many problems of critical importance remain, in any meaningful sense, computationally intractable. A common theme throughout the hardest problems is the tight coupling of different underlying physical phenomena over a broad range of scales. To truly capture the promise of numerical simulation, there is an outstanding need to develop a new generation of scalable algorithms to efficiently simulate this type of complex, multiscale, problem.

There are three categories of applications our group considers. The first is related to modeling complex multi physics problems in plasma science. The second area we work on is the development of methods that are aimed at solving interface problems in polymer membranes; think fuel cells, solar cells, and batteries. The third area we work on is the development of ultra-fast methods (sub-linear methods) for identification of sparse signals.

The application-oriented work is facilitated by our focus on the development of high order methods (here high order can be thought of as high accuracy) for general classes of problems, that we then tweak for key applications. Our work on high order methods for hyperbolic conservation laws is a good example of this kind of work, where we developed general methods that we then extend to fluid models of plasmas.

​More than half of our work centers on new method development, with the goal of making an impact on science through developing methods that enable us to explore problems that are beyond the scope of the previous generation of methods. The work is driven by advances in hardware, such as general-purpose graphics processing units (GPGPUs), the intel multi-core Phi technology and the heterogeneous computing platforms that have resulted from the synergy of these accelerators in new massively parallel computing platforms, such as the TITAN machine down at Oak Ridge National Laboratory. To be able to leverage these new computing platforms effectively, one needs to rethink our previous generation of algorithms in the context of these multi-core giants, where the number of floating-point operations is increasing at a rate far faster than the amount of memory one has in these computers.

​Plasma Science

So, what is a plasma? If you start with a solid and add energy to it, the solid melts and becomes a liquid. If you add even more energy to this system, eventually the liquid evaporates into a gas. Finally, if you add even more energy into this system, the atoms will eventually breakdown into their constituent elements, electrons, and ions. It is this fourth state of matter that is known as a plasma.

Membrane Morphology

​Membrane morphology for separator membranes is a subject of interest in the study of polymer physics. Membranes do work by allowing ions to move in one direction but not electrons, forcing the electrons to complete a circuit by flowing through an external connection. The morphology of the membrane is key because it is a strong indicator of how well the membrane will perform in devices such as batteries, solar cells, and fuel cells.

Sub-Linear Transforms

​Sub-linear transforms are a class of methods that leverage knowledge of a signal one is looking for to develop a method that is faster at identifying the signal than just applying the standard methods. In this case, we are interested in signals that are sparse, and it is the sparsity of the signal that we are using as a tool in rapidly identifying the signal. This could be thought of like the auto tuner in your car, when it finds all available stations for you, except in this case the bandwidth over which we are looking is large, maybe 4-Terahertz, and we are only looking for around 100 to 1,000 signals in this wide spectrum. These are a class of methods that can play an important role in data science where big data does not mean lots of meaningful information.  In fact, often the data one is seeking can be thought of as sparse. ​

​High Order Methods

In general, we look to develop high order methods for a range of general problems in science and engineering. Many of these methods are explicitly designed with the notion of using new hardware or leveraging hardware we have not been fully utilizing. For example, we have done work on parallel time stepping methods for advancing models both parallel in time as well as in space. This can help us utilize the potential of a multi-core computer more efficiently. We have also done work on new generic hyperbolic solvers (fluid solvers) which make it easier to ensure things like positivity of the pressure while maintaining high order solutions.

Reduced Order Models

Many physical systems of interest are too computationally costly to tackle directly, and in many engineering instances we want to optimize features of these systems. Reduced order modeling is a method to allow for computationally cheaper models that maintain physical properties of the system.