Research interestsMy primary research activities are centered on the development and numerical analysis of high-order accurate and spectral methods for solving partial differential equations. During the last few years I have focused on discontinuous Galerkin methods, nodal elements, methods for complex geometries, and time-advancement methods, but I have also worked on high-order finite difference methods, spectral and spectral element methods, particle based methods. Recently I have taken a substantial interest in reduced basis methods and high-dimensional problems arising in uncertainty quantification. Throughout my career I have worked on many aspects of high-performance and parallel computing. Applications to CFD, electromagnetics, plasma physics, general relativity.
Discontinuous Galerkin methods and applications
All aspects of discontinuous Galerkin methods and their application are being considered. Emphasis is on nodal techniques and approached that enable efficient and accurate implementations. Recent activities include preconditioning, efficient time-stepping methods, and limiting in problems with shocks while applications include water waves, electromagnetics, plasma physics, general relativity.
Computational electromagnetics and plasma physics
Discontinuous Galerkin methods continue to be developed to enable the accurate and efficient solution of large scale problems in time-domain electromagnetics and kinetic plasma physics. The emphasis is on maturing such techniques to enable to solution of large scale geometrically complex applications with a strong interaction between geometries, materials, fields and particles. This is an ongoing effort involving collaborators throughout the world, including both academic and industrial partners.
Certified reduced basis methods for wave problems
While computational techniques continue to improve, the challenges associated the rapid solution of parameterized partial differential equations remains a major computational bottleneck. However, such techniques are essential for the practical solution of many types of problems found in optimization, design, and real time control. In contrast to more traditional reduced basis methods, we focus in this project on techniques for which a rigorous and tight error can be computed during the online computation, hence resulting in a true certified reduced basis methods. We focus on wave dominated problems and in particular on Maxwell’s equations, expressed both on differential and integral form. This effort is the result of a close collaboration between MIT, Brown and University of Paris VI.
Numerical relativity and black hole dynamics
The goal in this project is to develop robust and accurate computational techniques suitable for the modeling of black hole dynamics, binary black hole inspiral and plunge, extreme mass ratio star black-hole orbiting etc. Such models enable the computation of radiated space-time waveforms, supporting large international experimental facilities such as LIGO in the detection of gravity waves. The computational tools being analysed and developed are based on discontinuous Galerkin methods and spectral penalty multi-domain methods. The effort is a collaboration between Brown, Cornell and Caltech and also has as an aim to model neutron-star, black-hole dynamics.
High-dimensional problems and uncertainty quantification
As computational techniques for solving deterministic problems, the uncertainty and randomness associated with large scale realistic problems continue to present limits to how accurate such problems can be solved. In this effort we focus on the development of efficient computational techniques to deal with such situations, leading to high-dimensional problems. We consider chaos expansions, AVOVA based techniques for parameter compression as well as other methods from statistics and probability. Applications are drawn from electromagnetic scattering, EMC/EMI problems as well as general parametric sensitivity studies of systems of differential equations.
GPU accelerated computing
The use of Graphics Processing Units (GPU’s) for large scale scientific computing problems has the potential to truly transform the types of problems we consider and the algorithmic choices must suitable for such highly parallel architectures. We are developing GPU accelerated discontinuous Galerkin methods for general partial differential equations, often seeing performance increased of 20-40 times over pure CPU based computations. This impressive acceleration is due partly to the excellent mapping of this highly local algorithm onto this particular architecture -- a feature which is found to be most advantageous for moderate to high orders of approximation.
Geophysical flows are characterized by being highly complex multi-physics, multi-scale phenomena, requiring the use of advanced computational techniques to enable the accurate modeling of such problems. In this effort we focus on the development and analysis of high-order accurate methods for problems including magma flow in the mantel and the formation of volcanos and global circulation models for climate prediction. This is done in a ongoing collaboration with the Geophysics Department at Brown.