Bulletin of the American Physical Society
2005 APS March Meeting
Monday–Friday, March 21–25, 2005; Los Angeles, CA
Session H32: Novel Computational Algorithms II |
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Sponsoring Units: DCOMP Chair: Ron Mickens, Clark-Atlanta Room: LACC 507 |
Tuesday, March 22, 2005 8:00AM - 8:36AM |
H32.00001: Simple but effective finite difference methods for simulating shock phenomena arising in continuum mechanics Invited Speaker: In this talk I will discuss two relatively simple finite difference schemes that are extremely effective in capturing the finite-time blow-up exhibited by nonlinear acceleration waves under certain conditions. We will test these schemes in the context of an initial-boundary value problem that involves a sinusoidal input signal. The first considers transverse propagation in a nonlinear soft tissue model while in the second finite-amplitude acoustic waves in Darcy-type porous media are studied. With these schemes, which are implemented on a desktop PC using the software package {\sc Mathematica 5.0}, we are able to capture over 95\% of the \lq \lq shocking-up" process, as well as illustrate the acceleration wave's other possible evolutionary paths. Finally, all numerical results will be supported by analytical work. [Preview Abstract] |
Tuesday, March 22, 2005 8:36AM - 8:48AM |
H32.00002: Model A Dynamics in Potts Models and Pure Lattice Gauge Theory Bernd Berg, Alexei Bazavov, Alexander Velytsky Model A Dynamics in Potts Models and Pure Lattice Gauge Theory (A. Bazavov and B.A. Berg, Florida State University, A. Velytsky, UCLA) We consider model A dynamics for a quench from the disordered (confined) to the ordered (deconfined) QCD phase. The linear theory of spinodal decomposition is compared with MC data and the critical mode of the linear approximation is determined, which is related to the Debye screening mass. The quench leads to competing vacuum domains, which are difficult to equilibrate. Structure functions show pronounced peaks. We study their finite size behavior as well as the gluonic energy density in the presence of such peaks. [Preview Abstract] |
Tuesday, March 22, 2005 8:48AM - 9:00AM |
H32.00003: Improved Wang-Landau algorithm for the joint density of states of continuous models Chenggang Zhou, T.C. Schulthess, D.P. Landau The joint density of states of a statistical physical system is the key to calculating thermodynamic observables at all temperatures and external fields. For example, \textit{$\rho $}($M$, $E)$ of a Heisenberg ferromagnet is a generalization of \textit{$\rho $}($E)$, from which magnetization and susceptibility at all temperatures can be obtained. A well-known method to calculate \textit{$\rho $}($E)$ is the Wang-Landau algorithm [1], which can be in principle extended to the joint density of states. Unfortunately, a straightforward application of the Wang-Landau algorithm to this type of problem turns out to be inefficient. We thus adopt a number of strategies to accelerate the simulation and to increase the performance of the algorithm in low-density regimes. In particular, we replace the conventional binning scheme with kernel density estimation, so that the algorithm is intrinsically suitable for continuous systems. This version of the Wang-Landau algorithm is also generally applicable to classical statistical physics models, including discrete models with large size. We also discuss other promising applications to magnetic nano-particles and in biophysics. [1] F. Wang and D. P. Landau, Phys. Rev. Lett. \textbf{86}, 2050 (2001). *This research is supported by the Department of Energy through the Laboratory Technology Research Program of OASCR and the Computational Materials Science Network of BES under Contract No. DE-AC05-00OR22725 with UT-Battelle LLC, and by NSF DMR-0341874. [Preview Abstract] |
Tuesday, March 22, 2005 9:00AM - 9:12AM |
H32.00004: Genetic-Algorithm first-principles prediction of novel ground state Alex Zunger, Volker Blum$^{2,}$ Binary A$_{1-x}$B$_x$ alloys can exist in any of the $2^N$ possible configurations on a lattice of $N$ points; finding the true ground-state thus requires mapping the LDA total-energies on an Ising-like Hamiltonian with generally numerous pair and many-body interactions. Finding which many-body interactions best reproduce the LDA energies is tedious, and often involves subjectives choices. Here we use artificial (but objective) intelligence to make such choice. We combine DFT calculated total energies of $O(50)$ configurations for each binary alloy in the Nb, Ta, Mo, W system with a ``Mixed Basis Cluster Expansion'' whose interaction types are chosen by a {\em genetic algorithm} search.[1] We thus derive the energy for {\em any} bcc configuration (in practice, $\sim 3,000,000$ structures). This (Ising-like) functional is then searched for $2^N$ configurations to find $T=0$ ground state structures, and to compute (via Monte Carlo) finite T thermodynamics and short- range order. We find rather surprising ground state -- very different from those suspected by the approach of ``rounding up the usual suspects''\\ $[1]$ G.Hart, V.Blum and A.Zunger (submitted) [Preview Abstract] |
Tuesday, March 22, 2005 9:12AM - 9:24AM |
H32.00005: Optimization the localized orbitals in nearly O(N) electronic structure methods Qingzhong Zhao, Wenchang Lu, Jerry Bernholc By using localized orbitals that are \textit{variationally optimized }for each system, it is possible to evaluate the DFT total energy $O(N) $steps. However, the convergence of such methods is often impractically slow. We found it advantageous [1] to use a small, but larger than the minimum, number of non-orthogonal orbitals. These orbitals vanish beyond a fixed but fairly large localization radius around each atom. Parallelized diagonalization is used to accelerate convergence and the scaling remains nearly linear for up to 2000-3000 atoms. However, the localization approximation resulted in the total energy somewhat above the DFT limit. We show that by adding supplementary short-range orbitals, the true DFT limit is reached and the convergence properties improve significantly. Furthermore, the low-lying conduction states become sufficiently accurate for the description of Green's functions entering non-linear calculations of electron transport, which utilize this compact basis to reach very large system sizes. We will discuss both scalability issues and example applications during the talk. 1. J.-L. Fattebert and J. Bernholc, Phys. Rev. B \underline {62}, 1713 (2000). [Preview Abstract] |
Tuesday, March 22, 2005 9:24AM - 9:36AM |
H32.00006: Direct enumeration investigation of bandgaps and effective masses of semiconductor alloys Peter A. Graf, Kwiseon Kim, Wesley B. Jones, Gus L. W. Hart We present and apply an approach to directly enumerate the bandgaps and effective masses of all possible configurations of a given alloy whose unit cell contains up to a specified number of atoms. This method allows us to map the space of bandgaps and effective masses versus alloy composition and atomic configuration. We demonstrate that a large range of bandgaps and masses are available for a given composition for AlGaAs and GaInP alloys. Furthermore, the maxima and minima occur for structures we can identity. By decomposing the space of possible structures into categories based on superlattice structure, patterns emerge. For example, bandgap maxima typically occur in [$0$ $h$ $k$] superlattices with $h\neq k$, and minima typically occur in $[1 1 1]$ superlattices. Finally, we discuss convergence of the method with respect to the unit cell size, and the effects of uniaxial strain by modeling growth on a substrate. [Preview Abstract] |
Tuesday, March 22, 2005 9:36AM - 9:48AM |
H32.00007: Atomistic material design by optimization Kwiseon Kim, Peter A. Graf, Wesley B. Jones We cast the problem of discovering atomic configurations with desired properties as a constrained global optimization problem. Here the free variables are the location and identity of every atom in a material and the objective function is built from the desired electronic properties. For example, we can minimize the bandgap or we can optimize for a target of combined bandgap and effective mass. We present two evolutionary optimization methods (a genetic algorithm [1] and a scatter search algorithm [2]), and two applications (semiconductor alloys and quantum dots). We describe the application specific mutation and crossover operation necessary, as well as the constraints and how they are maintained during the search of the space of atomic configurations. We highlight past successes, current challenges, and future prospects for this novel method. [1] D. Levine, PGAPack: Parallel Genetic Algorithm Library (1998). [2] M. Laguna and R. Marti, Scatter Search, Methodology and Implementation in C, Kluwer, Boston (2003). [Preview Abstract] |
Tuesday, March 22, 2005 9:48AM - 10:00AM |
H32.00008: Physically-motivated dynamical algorithms for the graph isomorphism problem Shiueyuan Shiau, Robert Joynt, S.N. Coppersmith The graph isomorphism problem (GI) plays a central role in the theory of computational complexity and has importance in physics and chemistry as well. No polynomial-time algorithm for solving GI is known. We investigate quantum physics-based polynomial- time algorithms for solving the graph isomorphism problem in which the graph structure is reflected in the behavior of a dynamical system. The method is to construct graph invariants based on the two-particle Green's function on the graph. The algorithm has been tested on strongly regular graphs - graphs that are known to be very difficult to distinguish by conventional means - up N=35, where N is the number of vertices. The GF for non-interacting fermions successfully distinguishes all pairs of non-isomorphic graphs for N $<$ 35 while that for hard-core bosons works for all graphs tested. [Preview Abstract] |
Tuesday, March 22, 2005 10:00AM - 10:12AM |
H32.00009: Object-oriented Development of an All-electron Gaussian Basis DFT Code for Periodic Systems John Alford, Samuel Trickey We report on the construction of an all-electron Gaussian-basis DFT code for systems periodic in one, two, and three dimensions. This is in part a reimplementation of algorithms in the serial code, GTOFF, which has been successfully applied to the study of crystalline solids, surfaces, and ultra-thin films. The current development is being carried out in an object-oriented parallel framework using C++ and MPI. Some rather special aspects of this code are the use of density fitting methodologies and the implementation of a generalized Ewald technique to do lattice summations of Coulomb integrals, which is typically more accurate than multipole methods. Important modules that have already been created will be described, for example, a flexible input parser and storage class that can parse and store generically tagged data (e.g. XML), an easy to use processor communication mechanism, and the integrals package. Though C++ is generally inferior to F77 in terms of optimization, we show that careful redesigning has allowed us to make up the run-time performance difference in the new code. Timing comparisons and scalability features will be presented. The purpose of this reconstruction is to facilitate the inclusion of new physics. Our goal is to study orbital currents using modified gaussian bases and external magnetic field effects in the weak and ultra-strong ( $\sim 10^{5}$ T) field regimes. This work is supported by NSF-ITR DMR-0218957. [Preview Abstract] |
Tuesday, March 22, 2005 10:12AM - 10:24AM |
H32.00010: Finite Element Discrete Variable Method for the Solution of the Time-Dependent Schroedinger Equation Barry Schneider, Lee Collins We demonstrate how a discretization of the spatial Hamiltonian, using the finite element discrete variable representation, may be combined with the Lie-Trotter-Suzuki (LTS) approach to the time-propagation operator, to produce an extremely efficient algorithm for the solution of the time-dependent Schroedinger equation. The algorithm is explicit, unconditionally stable, scales linearly with the number of basis functions used for the spatial discretization and is easily parallelized. Calculations using a second and fourth order accurate version of LTS propagators will be compared on a few model problems for efficiency and accuracy. [Preview Abstract] |
Tuesday, March 22, 2005 10:24AM - 10:36AM |
H32.00011: Hybrid DFT/Thomas-Fermi-like simulations of solvated molecules in water Miroslav Hodak, Wenchang Lu, Jerry Bernholc We have developed a hybrid method for efficient and accurate treatment of systems that involve a chemically active part surrounded by a large number of solvent molecules. The chemically active region is treated with Density Functional Theory (DFT), while most of the solvent molecules are treated with a variant of Thomas-Fermi (TF) theory, in which the TF kinetic energy functional was replaced by one based on the generalized gradient approximation (GGA). These solvent molecules are also assumed to be rigid and have frozen electron densities. The exchange of solvent molecules between the two regions is allowed as is the overlap of densities. We show that the calculation in which small mumber of water molecules is treated with DFT, while the TF-like approach is used for most of the molecules, leads to a good description of the entire water system. This calculation takes only a fraction of computer time required for a full DFT treatment of the whole system. Initial applications to biological systems will also be discussed. [Preview Abstract] |
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H32.00012: Positivity Preserving Nonstandard Finite Difference Schemes for PDE's Having Cross-Diffusion Terms Ronald Mickens Many phenomena in the natural and engineering sciences can be modeled by coupled systems of nonlinear partial differential equations (PDE) in which cross-diffusion terms occur. Such terms correspond to expressions for which the diffusion coefficients depend on dependent variables other than the one represented in the first-order time derivative for a given equation. An example is $(vu_x)_x$, where $v$ and $u$ are two different dependent variables. Since such terms can appear as either a positive or negative sign, it becomes critical to construct numerical integration schemes that preserve a positivity condition. The positivity condition is the requirement that if $(v,u,\dots)$ are non-negative at $t=0$, then as they evolve in time, they remain non-negative. Since, for many systems the variables $(v,u,\dots)$ are densities or particle numbers, the significance of a positivity condition is obvious. Since many of the standard numerical integration methods do not strictly enforce this condition, they may give rise to numerical instabilities, i.e., solutions to the discrete equations not corresponding to any actual solutions of the PDE's. We demonstrate that the nonstandard finite difference procedures of Mickens [1] can provide positivity preserving schemes. We are also able to obtain relationships between the space and time step-sizes. \par\noindent [1] R. E. Mickens, Finite Difference Models of Differential Equations (World Scientific, 1994). [Preview Abstract] |
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H32.00013: Numerical classical and quantum mechanical simulations of charge density wave models Andrew Beckwith First, using a driven harmonic oscillator model by a numerical scheme as initially formulated by Littlewood, we present a computer simulation of charge density waves (CDW); next, we use this simulation to show how the dielectric model presented via this procedure leads to a blow up at the initialization of a threshold field E$_{T}$. Finding this approach highly unphysical, we initiated inquiry into alternative models. We investigate how to present the transport problem of CDW quantum mechanically, through a numerical simulation of the massive Schwinger model. We find that this single-chain quantum mechanical simulation used to formulate solutions to CDW transport is insufficient for transport of soliton-antisolitons (S-S') through a pinning gap model of CDW. We show that a model Hamiltonian with Peierls condensation energy used to couple adjacent chains (or transverse wave vectors) permits formation of S-S' that can be used to transport CDW through a potential barrier. This addition of the Peierls condensation energy term is essential for any quantum model of CDW to give a numerical simulation to tunneling behavior. [Preview Abstract] |
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