Parallel Solvers for Eigenproblems in Quantum Mechanics
Creators: name: University Of California Office Of The President
Publisher: name: University of California Office of the President
publisherIdentifier: https://ror.org/00dmfq477
publisherIdentifierScheme: ROR
Year: 2024

description: Large-scale quantum-mechanical (ab initio) materials calculations are a cornerstone of modern materials research and, as such, comprise a substantial portion of all large-scale calculations around the world today. At LLNL alone, up to 40% of the massively parallel BlueGene/L supercomputer has been devoted to such calculations. The importance of these calculations stems from their underlying quantum-mechanical (QM) nature, yielding insights inaccessible to experiment and robust, predictive power unattainable by more approximate, empirical schemes. However, because such ab initio calculations are so computationally intensive, a vast range of real materials problems of utmost importance to the Laboratory and wider scientific community remain inaccessible by such rigorous, quantum mechanical means. Indeed, even with the computational platforms available today, key calculations such as transition-metal melt curves can require in excess of a year of massively parallel computation using existing planewave based solution methods. To address this critical deficiency, there have been substantial efforts in recent years to develop new real space QM methods, in which computations can be better localized, allowing much more natural and efficient parallel implementation. In the vast majority of such ab initio calculations, the dominant computational step is the repeated solution of the large, eigenproblems produced by the discretization of the underlying (typically Kohn-Sham) QM problem in the chosen finite-dimensional basis. This proposal aims squarely at addressing that key computational bottleneck affecting all such calculations. The generalized eigenproblem is of particular importance as it is the variety produced by the majority of basis sets in current use. Furthermore, it has assumed increasing importance in recent years with the advent of modern projector-augmented-wave methods. In this proposal, we gather leading experts at UC Davis and LLNL to address this key computational problem: the development and implementation of efficient, parallel solvers and preconditions for the large, sparse, generalized eigenproblems produced by modern real space quantum mechanical materials methods. The developments will be of potentially wide impact: indeed such ``nonlinear eigenproblems'' occur in a broad range of applications areas from electronic-structure calculations to the design of MEMS to the simulation of electromagnetic fields in particle accelerators. In order to maximize impact, we will focus in particular on the eigenproblems produced by the new partition-of-unity finite-element (PUFE) based ab initio methods under current development at LLNL. By virtue of their highly efficient orbital-polynomial bases, these new methods produce eigenproblems of a quite different nature than those of other methods in current use. The efficient, parallel solution of these eigenproblems will require the development of algorithms which exploit the unique nature of both the larger self-consistent electronic-structure problem, in which thousands of eigenvectors may be required with excellent initial approximations available, and the particular system matrices themselves, in order to construct efficient preconditioners. A successful outcome will advance the new PUFE ab initio method substantially, enabling key applications goals immediately, and potentially benefit all other such methods in current development and use today. A successful implementation in the LLNL PUFE code in particular stands to change the way the largest, most complex quantum mechanical calculations are done, affording an order-of-magnitude advance over the current state of the art. This project will bring together the leading expertise of Dr. Pask at LLNL in real-space methods, Prof. Sukumar at UC Davis in PUFE techniques, and Prof. Bai at UC Davis in large-scale eigenproblem solution methods. It will undertake the mentoring and development of a graduate student and postdoc at UC Davis, foster the existing collaboration, and begin new ones. In so doing, it will initiate a key collaboration of mathematicians, computer scientists, and engineers at the University with key applications scientists at the Laboratory, leveraging existing developments, expertise, and computational resources, to advance this critical area of modern materials research.