CMG Collaborative Research: Fast and Efficient Radial Basis Function Algorithms for Geophysical Modeling on Arbitrary Geometries

Numerical method innovations in the geosciences have not paralleled the

explosion in computer hardware development over the last decades. Yet, for

scientific computing to advance, it is crucial that novel numerical

approaches are developed that improve the simplicity, flexibility, and

accuracy of algorithms while taking advantage of this revolution in hardware

technology. The current project is aimed at exactly that objective: to

develop fast, efficient and parallelizable radial basis function (RBF)

methodologies, enhanced by novel graphics processing unit (GPU) technology,

for applications in computational geosciences. A collaborative team of

computational mathematicians and geophysicists has been established for this

purpose. The RBF approach is very attractive in that it achieves high-order

accuracy for arbitrary geometries in n-dimensional space, naturally permits

local refinement, is mesh-free (no grids), algorithmic complexity does not

increase with dimension and, generally offers higher accuracy with longer

time steps than traditional spectral methods for a given number of nodes.

However, RBFs are still in an early developmental stage; key issues to be

addressed are: 1) development of localized high-order RBF-finite difference

(RBF-FD) stencils on irregular node layouts in n-dimensional space for

arbitrary geometries with stable time-stepping, 2) adaptive local node

refinement schemes for RBF-FD, 3) development of hybrid spectral RBF and FD

schemes to maximize advantages of RBFs and while minimizing their

computational cost, and 4) implementation of the method on hardware

accelerators, such as GPUs. Scientific targets for application include 3D

mantle convection with varying properties, models of the geodynamo in an

elliptic geometry, and tsunami modeling with irregular coastline topography.

By advancing the frontier of computational mathematics that will take

advantage of today?s booming technology industry, society can be offered a

more in depth understanding of geophysical processes related to mantle

convection, the geodynamo, and tsunamis. These phenomena play a key role in

continental movement, polarity reversal of the earth?s magnetic field,

earthquakes, and coastal flooding. The proposed innovative mathematicalcomputational approach may hold the key to unlocking long-standing questions of fundamental importance in such geophysics areas. An interdisciplinary collaborative team of computational mathematicians and geophysicists has been assembled from a national lab and four universities (NCAR-Colorado, Boise State Univ.-Idaho, Univ. of Minnesota, Florida State Univ., and the Univ. of California?Davis) to develop new methods in computational mathematics for addressing these critical geophysical problems. The project supports collaborative participation among researchers at different stages in their careers as well as Ph.D, Masters, and undergraduate students in 4 states and in a myriad of scientific and mathematical disciplines. These students will have the opportunity to participate as members of an interdisciplinary team composed of senior personnel with a demonstrated commitment to education and research. NCAR will serve not only as an integrating hub for scientific endeavors but provide a medium for students from across the country to work together on multiple facets of the proposal. Through the SciPARCS 10 week internship program at NCAR, students will have the opportunity to engage in joint collaborative research, preparing them for a career in computational geosciences for the 21st century.