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Solving a Class of Nonlinear Eigenvalue Problems by Newton's Method

Description: We examine the possibility of using the standard Newton's method for solving a class of nonlinear eigenvalue problems arising from electronic structure calculation. We show that the Jacobian matrix associated with this nonlinear system has a special structure that can be exploited to reduce the computational complexity of the Newton's method. Preliminary numerical experiments indicate that the Newton's method can be more efficient for small problems in which a few smallest eigenpairs are needed.
Date: July 2, 2009
Creator: Gao, Weiguo; Yang, Chao & Meza, Juan C.
Partner: UNT Libraries Government Documents Department

Steepest Descent

Description: The steepest descent method has a rich history and is one of the simplest and best known methods for minimizing a function. While the method is not commonly used in practice due to its slow convergence rate, understanding the convergence properties of this method can lead to a better understanding of many of the more sophisticated optimization methods. Here, we give a short introduction and discuss some of the advantages and disadvantages of this method. Some recent results on modified versions of the steepest descent method are also discussed.
Date: February 12, 2010
Creator: Meza, Juan C.
Partner: UNT Libraries Government Documents Department

A Constrained Optimization Algorithm for Total Energy Minimizationin Electronic Structure Calculation

Description: A new direct constrained optimization algorithm forminimizing the Kohn-Sham (KS) total energy functional is presented inthis paper. The key ingredients of this algorithm involve projecting thetotal energy functional into a sequences of subspaces of small dimensionsand seeking the minimizer of total energy functional within eachsubspace. The minimizer of a subspace energy functional not only providesa search direction along which the KS total energy functional decreasesbut also gives an optimal "step-length" to move along this searchdirection. A numerical example is provided to demonstrate that this newdirect constrained optimization algorithm can be more efficient than theself-consistent field (SCF) iteration.
Date: July 26, 2005
Creator: Yang, Chao; Meza, Juan C. & Wang, Lin-Wang
Partner: UNT Libraries Government Documents Department

The Use of Trust Regions in Kohn-Sham Total EnergyMinimization

Description: The Self Consistent Field (SCF) iteration, widely used forcomputing the ground state energy and the corresponding single particlewave functions associated with a many-electronatomistic system, is viewedin this paper as an optimization procedure that minimizes the Kohn-Shamtotal energy indirectly by minimizing a sequence of quadratic surrogatefunctions. We point out the similarity and difference between the totalenergy and the surrogate, and show how the SCF iteration can fail whenthe minimizer of the surrogate produces an increase in the KS totalenergy. A trust region technique is introduced as a way to restrict theupdate of the wave functions within a small neighborhood of anapproximate solution at which the gradient of the total energy agreeswith that of the surrogate. The use of trust region in SCF is not new.However, it has been observed that directly applying a trust region basedSCF(TRSCF) to the Kohn-Sham total energy often leads to slowconvergence.We propose to use TRSCF within a direct constrainedminimization(DCM) algorithm we developed in \cite dcm. The keyingredients of theDCM algorithm involve projecting the total energyfunction into a sequence of subspaces of small dimensions and seeking theminimizerof the total energy function within each subspace. Theminimizer of a subspace energy function, which is computed by TRSCF, notonly provides a search direction along which the KS total energy functiondecreases but also gives an optimal "step-length" that yields asufficient decrease in total energy. A numerical example is provided todemonstrate that the combination of TRSCF and DCM is more efficient thanSCF.
Date: May 30, 2006
Creator: Yang, Chao; Meza, Juan C. & Wang, Lin-wang
Partner: UNT Libraries Government Documents Department

Derivative-free optimization methods for surface structuredetermination of nanosystems

Description: Many properties of nanostructures depend on the atomicconfiguration at the surface. One common technique used for determiningthis surface structure is based on the low energy electron diffraction(LEED) method, which uses a high-fidelity physics model to compareexperimental results with spectra computed via a computer simulation.While this approach is highly effective, the computational cost of thesimulations can be prohibitive for large systems. In this work, wepropose the use of a direct search method in conjunction with an additivesurrogate. This surrogate is constructed from a combination of asimplified physics model and an interpolation that is based on thedifferences between the simplified physics model and the full fidelitymodel.
Date: October 18, 2007
Creator: Meza, Juan C.; Garcia-Lekue, Arantzazu; Abramson, Mark A. & Dennis, John E.
Partner: UNT Libraries Government Documents Department

Science-Driven Computing: NERSC's Plan for 2006-2010

Description: NERSC has developed a five-year strategic plan focusing on three components: Science-Driven Systems, Science-Driven Services, and Science-Driven Analytics. (1) Science-Driven Systems: Balanced introduction of the best new technologies for complete computational systems--computing, storage, networking, visualization and analysis--coupled with the activities necessary to engage vendors in addressing the DOE computational science requirements in their future roadmaps. (2) Science-Driven Services: The entire range of support activities, from high-quality operations and user services to direct scientific support, that enable a broad range of scientists to effectively use NERSC systems in their research. NERSC will concentrate on resources needed to realize the promise of the new highly scalable architectures for scientific discovery in multidisciplinary computational science projects. (3) Science-Driven Analytics: The architectural and systems enhancements and services required to integrate NERSC's powerful computational and storage resources to provide scientists with new tools to effectively manipulate, visualize, and analyze the huge data sets derived from simulations and experiments.
Date: May 16, 2005
Creator: Simon, Horst D.; Kramer, William T.C.; Bailey, David H.; Banda,Michael J.; Bethel, E. Wes; Craw, James M. et al.
Partner: UNT Libraries Government Documents Department