Large Scale Intrusive Optimization and Applications
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Today's need for rapid software development has generated a great interest in employing Commercial-Off-The-Shelf (COTS) software products as a way of managing cost, developing time, and effort. With an abundance of COTS software packages to choose from, the problem now is how to systematically evaluate, rank, and select a COTS product that best meets the software project requirements and at the same time can leverage off the current corporate information technology architectural environment. This paper describes a systematic process for decision support in evaluating and ranking COTS software. Performed right after the requirements analysis, this process provides the evaluators with more concise, structural, and step-by-step activities for determining the best COTS software product with manageable risk. In addition, the process is presented in phases that are flexible to allow for customization or tailoring to meet various projects' requirements.
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Computer Methods in Applied Mechanics and Engineering
Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to project a continuous finite element space into a given discontinuous space, and then applies a discontinuous Galerkin formulation. The projection leads to parameterization of the discontinuous degrees-of-freedom by their continuous counterparts and has a variational multiscale interpretation. This significantly reduces the computational burden and, at the same time, little or no degradation of the solution occurs. In fact, the new method produces improved solutions compared with the traditional discontinuous Galerkin method in some situations. © 2005 Elsevier B.V. All rights reserved.
Proposed for publication in the SIAM Journal on Scientific Computing.
We report the implementation of an iterative scheme for calculating the Optimized Effective Potential (OEP). Given an energy functional that depends explicitly on the Kohn-Sham wave functions, and therefore, implicitly on the local effective potential appearing in the Kohn-Sham equations, a gradient-based minimization is used to find the potential that minimizes the energy. Previous work has shown how to find the gradient of such an energy with respect to the effective potential in the zero-temperature limit. We discuss a density-matrix-based derivation of the gradient that generalizes the previous results to the finite temperature regime, and we describe important optimizations used in our implementation. We have applied our OEP approach to the Hartree-Fock energy expression to perform Exact Exchange (EXX) calculations. We report our EXX results for common semiconductors and ordered phases of hydrogen at zero and finite electronic temperatures. We also discuss issues involved in the implementation of forces within the OEP/EXX approach.
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This document provides verification test results for normal, lognormal, and uniform distributions that are used in Sandia's Latin Hypercube Sampling (LHS) software. The purpose of this testing is to verify that the sample values being generated in LHS are distributed according to the desired distribution types. The testing of distribution correctness is done by examining summary statistics, graphical comparisons using quantile-quantile plots, and format statistical tests such as the Chisquare test, the Kolmogorov-Smirnov test, and the Anderson-Darling test. The overall results from the testing indicate that the generation of normal, lognormal, and uniform distributions in LHS is acceptable.
We propose two new multilinear operators for expressing the matrix compositions that are needed in the Tucker and PARAFAC (CANDECOMP) decompositions. The first operator, which we call the Tucker operator, is shorthand for performing an n-mode matrix multiplication for every mode of a given tensor and can be employed to concisely express the Tucker decomposition. The second operator, which we call the Kruskal operator, is shorthand for the sum of the outer-products of the columns of N matrices and allows a divorce from a matricized representation and a very concise expression of the PARAFAC decomposition. We explore the properties of the Tucker and Kruskal operators independently of the related decompositions. Additionally, we provide a review of the matrix and tensor operations that are frequently used in the context of tensor decompositions.
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Mathematical Programming
We present a series of related robust optimization models for placing sensors in municipal water networks to detect contaminants that are maliciously or accidentally injected. We formulate sensor placement problems as mixed-integer programs, for which the objective coefficients are not known with certainty. We consider a restricted absolute robustness criteria that is motivated by natural restrictions on the uncertain data, and we define three robust optimization models that differ in how the coefficients in the objective vary. Under one set of assumptions there exists a sensor placement that is optimal for all admissible realizations of the coefficients. Under other assumptions, we can apply sorting to solve each worst-case realization efficiently, or we can apply duality to integrate the worst-case outcome and have one integer program. The most difficult case is where the objective parameters are bilinear, and we prove its complexity is NP-hard even under simplifying assumptions. We consider a relaxation that provides an approximation, giving an overall guarantee of near-optimality when used with branch-and-bound search. We present preliminary computational experiments that illustrate the computational complexity of solving these robust formulations on sensor placement applications.
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