We have made some initial progress on this problem by developing a parallel adaptive multigrid eigenvalue solver, which integrates adaptive mesh refinement techniques with an iterative eigenvalue solver based on multigrid. We have applied our solver to several model problems with considerable success; for example, we have generated a highly accurate solution to the H2+ problem. While these model problems are not of direct materials interest, their solution does demonstrate the feasibility of our computational approach for larger systems, and the dramatic improvements possible through the use of adaptivity---a 100-fold savings in time and memory.
We are currently extending our adaptive multigrid solver to handle self-consistency, which arises in materials of technological significance, such as high temperature super-conductors.
This is a multidisciplinary investigation. The collaboration team includes:
Eric Bylaska (UCSD Chemistry and Biochemistry)
Scott Kohn (Center for Applied Scientific Computing, Lawrence Livermore National Laboratory))
Beth Ong (UCSD Mathematics)
John Weare (UCSD Chemistry and Biochemistry)
Alan Edelman (MIT Mathematics)
Parallel Software Abstractions for Structured Adaptive Mesh Methods, Scott R. Kohn and Scott B. Baden, Journal of Parallel and Distributed Computing, (To appear)
Software Abstractions and Computational Issues in Parallel Structured Adaptive Mesh Methods for Electronic Structure Calculations , Scott Kohn, John Weare, M. Elizabeth Ong, and Scott B. Baden. IMA Volumes in Mathematics and its Applications, Volume 117, Structured Adaptive Mesh Refinement (SAMR) Grid Methods, S. B. Baden, N.Chrisochoides, M. Norman, and D. Gannon, Eds., Springer-Verlag, 1999, pp. 75-95. (Proc. Workshop on Structured Adaptive Mesh Refinement Grid Methods, March 12-13, 1997, Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN. March 12-13, 1997).
Parallel Adaptive Mesh Refinement for Electronic Structure Calculations , Scott Kohn, John Weare, Elizabeth Ong, and Scott Baden. Proc. 8th SIAM Conf. on Parallel Proc. for Scientific Computing, March 1997, Minneapolis, MN.
"A Parallel Software Infrastructure for Dynamic Block-Irregular Scientific Calculations," Scott R. Kohn, UCSD CSE Dept. Tech. Rep. CS95-429, Jun. 1995. (PhD Dissertation, send email to email@example.com for hardcopy.) PDF ps.gz
Scalable Parallel Numerical Methods and Software Tools for Material Design. E. Bylaska, S. Kohn, S. Baden, M.E.G. Ong, J. Weare, A. Edelman, R. Kawai. PHYSICS COMPUTING '95 (Annual Meeting of the APS Division of Computational Physics), June 5-8 1995, Pittsburgh, PA.
A Parallel Software Infrastructure for Structured Adaptive Mesh Methods, (Also available in HTML form. ) Scott R. Kohn and Scott B. Baden, Proc. Supercomputing '95, San Diego, CA, December 1995.}
Structured adaptive mesh algorithms dynamically allocate computational resources to accurately resolve interesting portions of a numerical calculation. Such methods are difficult to implement and parallelize because they rely on dynamic, irregular data structures. We have developed an efficient, portable, parallel software infrastructure for adaptive mesh methods; our software provides computational scientists with high-level facilities that hide low-level details of parallelism and resource management. We have applied our software infrastructure to the solution of adaptive eigenvalue problems arising in materials design. We describe our software infrastructure and analyze its performance. We present computational results which indicate that the uniformity restrictions imposed by a data parallel Fortran implementation would significantly impact performance.
The Parallelization of an Adaptive Multigrid Eigenvalue Solver with LPARX, Scott R. Kohn and Scott B. Baden, Proc. 7th SIAM Conf. on Parallel Proc. for Scientific Computing, February 1995, San Francisco, CA. (Also CSE Tech. Rept. CS94-387, 9/94.)
We have developed a parallel adaptive eigenvalue solver and applied it to a model problem in theoretical materials science. Our method combines adaptive mesh refinement techniques with a novel multigrid eigenvalue algorithm. By exploiting adaptivity, we have reduced computation time and memory consumption by more than two orders of magnitude. We have implemented our solver using the LPARX parallel programming system, which considerably simplified the programming and enabled us to run the same code on a diversity of high performance parallel architectures.
Scalable Parallel Numerical Methods and Software Tools for Material Design, Eric J. Bylaska, Scott R. Kohn, Scott B. Baden, Alan Edelman, Ryiochi Kawai, Maria E. G. Ong, and John H. Weare, Proc. 7th SIAM Conf. on Parallel Proc. for Scientific Computing, February 1995, San Francisco, CA. (Also CSE Tech. Rept. CS94-388, 9/94.)
A new method of solution to the local spin density approximation to
the electronic Schroedinger equation is presented. The method is based
on an efficient, parallel, adaptive multigrid eigenvalue solver. It is
shown that adaptivity is both necessary and sufficient to accurately solve
the eigenvalue problem near the singularities at the atomic centers. While
preliminary, these results suggest that direct real space methods may provide
a much needed method for efficiently computing the forces in complex materials.