This webinar will showcase some of the game-changing research supported by our Catalyst Grants program.
This event was recorded and will be on the UM Youtube channel shortly.
There are several different approaches to the numerical solution of partial differential equations. For example, finite-difference methods and finite-element methods discretize either the strong form or the weak form of the equation in real space, while spectral methods discretize the equation in reciprocal space. This project employs an alternative method which converts the differential equation into an integral equation by convolution with the Green’s function, followed by discretization and linear solution; the hope is that this approach is more amenable to adaptive refinement and parallelization than other methods. In the past, integral equation based methods were hindered by the difficulty of discretizing singular integrals and the cost of computing dense matrix-vector products, but these obstacles are being brought under control. We present our recent work in this area including (1) a GPU-accelerated barycentric treecode for long-range particle interactions, (2) applications in electrostatics, electronic structure, and vortex dynamics.
High-fidelity long-time scale simulations have been a challenge in a wide range of areas, including time-dependent electronic structure calculations and molecular dynamics. In particular, time-dependent density functional theory (TDDFT) calculations are limited to time-scales of the order of hundred femtoseconds, and MD simulations (even those based on interatomic potentials) are routinely limited to time-scales of the order of nanoseconds. However, there is very rich material phenomena, both at the quantum and atomistic scale, that occurs at time-scales that are orders of magnitude larger than the currently accessible range. In this talk, I will present the ideas we have been exploring as part of the MICDE catalyst grant to enable long time-scale simulations on a class of time-dependent problems. In particular, we investigate the use of exponential time-propagators as an alternative to the finite-difference based time-discretization of the PDEs. The ideas will be presented for time-dependent density functional theory and elastodynamics—as a prototypical problem for molecular dynamics—along with numerical results demonstrating the viability and computational efficiency of the proposed ideas.
This is joint work with Bikash Kanungo and Paavai Pari.
The real-time phase-resolved prediction of ocean waves is crucial for the safety of offshore operations. With the development of the remote sensing technology, it is now possible to reconstruct the phase-resolved ocean surface from radar measurements in real time. Using the reconstructed ocean surface as initial condition, nonlinear wave models such as the high-order spectral (HOS) method can be applied to predict the evolution of the ocean waves. However, the computations reply heavily on large CPU clusters which are usually not available in the offshore onboard environment, and the prediction can deviate quickly from the true wave evolution due to the chaotic nature of the nonlinear wave equations. To address these problems, we develop a novel GPU-accelerated computational framework, which features the coupling of HOS and an ensemble Kalman filter (EnKF) to reduce the uncertainties in the prediction. The new framework algorithm is tested and validated using both synthetic and real wave data, and is shown promising in fundamentally improving the real-time prediction capability of ocean waves.