Applied Mathematics
High-performance sparse-matrix algorithms
Sparse-matrix problems are at the heart of many scientific and engineering applications of importance to DOE and the nation. Some applications are fusion energy, accelerator physics, structural dynamics, computational chemistry, and groundwater simulations. Teranishi et al. studied the effect of data mapping on the performance of triangular solutions. They showed that a careful choice of data mapping, coupled with the use of selective inversion, can significantly reduce the amount of communication required in the solution of a sparse triangular linear system.
K. Teranishi, P. Raghavan, and E. G. Ng, “Reducing communication costs for parallel sparse triangular solution,” in Proceedings of the SC2002 Conference (2002). ASCR-MICS, SciDAC, NSF
Optimizing memory hierarchy
The BeBOP toolkit automatically tunes scientific codes by applying statistical techniques to produce highly tuned numerical subroutines. Vuduc et al. used the BeBOP tool SPARSITY to investigate memory hierarchy issues with such sparse matrix kernels as Ax (matrix-vector products) and AT Ax (normal equation products). They evaluated optimizations on a benchmark set of 44 matrices and four platforms, showing speedups of up to a factor of 4.2.
R. Vuduc, A. Gyulassy, J. Demmel, and K. Yelick. "Memory hierarchy optimizations and performance bounds for sparse AT Ax." U.C. Berkeley Technical Report UCB/CS-03-1232, February 2003. ASCR-MICS, SciDAC, NSF, Intel
Modeling the component structures of flames
Combustion in turbulent flows may take the form of a thin flame wrapped around vortical structures. Marzouk et al. used the flame-embedding approach to decouple computations of the “outer” nonreacting flow and the combustion zone by breaking the flame surface into a number of elemental flames, each incorporating the local impact of unsteady flow-flame interaction (Figure 2). Results show that using the flame leading-edge strain rate gives reasonable agreement in cases of low-strain-rate and weak-strain-rate gradient within the flame structure. This agreement deteriorates substantially when both are high. Agreement between the 1D model and the 2D simulation improves greatly when the actual strain rate at the reaction zone of the 1D flame is made to match that of the 2D flame.
Y. M. Marzouk, A. F. Ghoniem, and H. N. Najm, "Toward a flame embedding model for turbulent combustion simulation," AIAA Journal 41, 641 (2003). ASCR-MICS
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Improving the accuracy of front tracking in fluid mixing
Acceleration-driven fluid-mixing instabilities play important roles in inertially confined nuclear fusion, as well as in stockpile stewardship. Turbulent mixing is a difficult and centrally important issue for fluid dynamics that affects such questions as the rate of heat transfer by the Gulf Stream, resistance of pipes to fluid flow, combustion rates in automotive engines, and the late-time evolution of a supernova. To provide a better understanding of these instabilities, Glimm et al. have developed a new, fully conservative front tracking algorithm that results in one-order-of-accuracy improvement over most finite difference simulations and shows good agreement with experimental results.
J. Glimm, X.-L. Li, Y.-J. Liu, Z. L. Xu, and N. Zhao, “Conservative front tracking with improved accuracy,” SIAM J. Sci. Comp. (in press). ASCR-MICS, FES, ARO, NSF, LANL, NSFC
Predicting the energies of quantum well states
Understanding the electronic structures of quantum-well states (QWSs) in metallic thin films can lay the groundwork for designing new materials with predictable properties. An et al. compared the experimental electronic properties of QWSs in a copper film grown on a cobalt substrate with ab initio simulations. The calculations confirm the concept of the quantization condition inherent in the phase-accumulation model (PAM) to predict the energies of QWSs as a function of their thickness, and to provide new insight into their nature. The results also show that band structures and reflection phases obtained from either experiment or ab initio theory can quantitatively predict QWS energies within the PAM model.
J. M. An, D. Raczkowski, Y. Z. Wu, C. Y. Won, L. W. Wang, A. Canning, M. A. Van Hove, E. Rotenberg, and Z. Q. Qiu, “The quantization condition of quantum-well states in Cu/Co(001),” Phys. Rev. B 68, 045419 (2003). ASCR-MICS, NSF
Reduced-density matrices for electronic structure calculations
Accurate first-principles electronic structure calculations are of special importance for spectroscopy and for chemical-reaction dynamics. Reduced-density matrices (RDM) that have linear scaling for large N offer a first-principles alternative to density-functional and semi-empirical calculations for large systems. Zhao et al. reformulated the existing RDM method from a primal to a dual formulation, substantially reducing the computational cost. They computed the ground state energy and dipole moment of 47 small molecules and molecular ions, both open and closed shells, achieving results more accurate than those from other approximations.
Z. Zhao, B. Braams, M. Fukuda, M. Overton, and J. Percus, “The reduced density matrix method for electronic structure calculations and the role of three-index representability,” J. Chem. Phys. (submitted, 2003). ASCR-MICS, NSF