Publications

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The Dakota (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a exible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quanti cation with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a exible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a user's manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies.

The Dakota (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a exible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quanti cation with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a exible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the Dakota software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of Dakota-related research publications in the areas of surrogate-based optimization, uncertainty quanti cation, and optimization under uncertainty that provide the foundation for many of Dakota's iterative analysis capabilities.

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%E2%80%9CNaive%E2%80%9D or straight-forward Kriging implementations can often perform poorly in practice. The relevant features of the robustly accurate and efficient Kriging and Gradient Enhanced Kriging (GEK) implementations in the DAKOTA software package are detailed herein. The principal contribution is a novel, effective, and efficient approach to handle ill-conditioning of GEK's %E2%80%9Ccorrelation%E2%80%9D matrix, RN%CC%83, based on a pivoted Cholesky factorization of Kriging's (not GEK's) correlation matrix, R, which is a small sub-matrix within GEK's RN%CC%83 matrix. The approach discards sample points/equations that contribute the least %E2%80%9Cnew%E2%80%9D information to RN%CC%83. Since these points contain the least new information, they are the ones which when discarded are both the easiest to predict and provide maximum improvement of RN%CC%83's conditioning. Prior to this work, handling ill-conditioned correlation matrices was a major, perhaps the principal, unsolved challenge necessary for robust and efficient GEK emulators. Numerical results demonstrate that GEK predictions can be significantly more accurate when GEK is allowed to discard points by the presented method. Numerical results also indicate that GEK can be used to break the curse of dimensionality by exploiting inexpensive derivatives (such as those provided by automatic differentiation or adjoint techniques), smoothness in the response being modeled, and adaptive sampling. Development of a suitable adaptive sampling algorithm was beyond the scope of this work; instead adaptive sampling was approximated by omitting the cost of samples discarded by the presented pivoted Cholesky approach.

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The DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. DAKOTA contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the DAKOTA toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a theoretical manual for selected algorithms implemented within the DAKOTA software. It is not intended as a comprehensive theoretical treatment, since a number of existing texts cover general optimization theory, statistical analysis, and other introductory topics. Rather, this manual is intended to summarize a set of DAKOTA-related research publications in the areas of surrogate-based optimization, uncertainty quantification, and optimization under uncertainty that provide the foundation for many of DAKOTA's iterative analysis capabilities.

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Climate models have a large number of inputs and outputs. In addition, diverse parameters sets can match observations similarly well. These factors make calibrating the models difficult. But as the Earth enters a new climate regime, parameters sets may cease to match observations. History matching is necessary but not sufficient for good predictions. We seek a 'Pareto optimal' ensemble of calibrated parameter sets for the CCSM climate model, in which no individual criteria can be improved without worsening another. One Multi Objective Genetic Algorithm (MOGA) optimization typically requires thousands of simulations but produces an ensemble of Pareto optimal solutions. Our simulation budget of 500-1000 runs allows us to perform the MOGA optimization once, but with far fewer evaluations than normal. We devised an analytic test problem to aid in the selection MOGA settings. The test problem's Pareto set is the surface of a 6 dimensional hypersphere with radius 1 centered at the origin, or rather the portion of it in the [0,1] octant. We also explore starting MOGA from a space-filling Latin Hypercube sample design, specifically Binning Optimal Symmetric Latin Hypercube Sampling (BOSLHS), instead of Monte Carlo (MC). We compare the Pareto sets based on: their number of points, N, larger is better; their RMS distance, d, to the ensemble's center, 0.5553 is optimal; their average radius, {mu}(r), 1 is optimal; their radius standard deviation, {sigma}(r), 0 is optimal. The estimated distributions for these metrics when starting from MC and BOSLHS are shown in Figs. 1 and 2.

Latin Hypercube Sampling (LHS) and Jittered Sampling (JS) both achieve better convergence than standard Monte Carlo Sampling (MCS) by using stratification to obtain a more uniform selection of samples, although LHS and JS use different stratification strategies. The "Koksma-Hlawka-like inequality" bounds the error in a computed mean in terms of the sample design's discrepancy, which is a common metric of uniformity. However, even the "fast" formulas available for certain useful L2 norm discrepancies require O (N2M) operations, where M is the number of dimensions and N is the number of points in the design. It is intuitive that "space-filling" designs will have a high degree of uniformity. In this paper we propose a new metric of the space-filling property, called "Binning Optimality," which can be evaluated in O(N log(N)) operations. A design is "Binning Optimal" in base 6. if when you recursively divide the hypercube into bM congruent disjoint sub-cubes, each sub-cube of a particular generation has the same number of points until the sub-cubes are small enough that they all contain either 0 or 1 points. The O(Nlog(N)) cost of determining if a design is binning optimal comes from quick-sorting the points into Morton order, i.e. sorting the points according to their position on a space-filling Z-curve. We also present a O(N log(N)) fast algorithm to generate Binning Optimal Symmetric Latin Hypercube Sample (BOSLHS) designs. These BOSLHS designs combine the best features of. and are superior in several metrics to. standard LHS and JS designs. Our algorithm takes significantly less than 1 second to generate M = 8 dimensional space-filling LHS designs with N = 216 = 65536 points compared to previous work which requires "minutes" to generate designs with N = 100 points.© 2010 by the American Institute of Aeronautics and Astronautics, Inc.

Surfpack is a library of multidimensional function approximation methods useful for efficient surrogate-based sensitivity/uncertainty analysis or calibration/optimization. I will survey current Surfpack meta-modeling capabilities for continuous variables and describe recent progress generalizing to both continuous and categorical factors, including relevant test problems and analysis comparisons.

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The DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. DAKOTA contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic finite element methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the DAKOTA toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a user's manual for the DAKOTA software and provides capability overviews and procedures for software execution, as well as a variety of example studies.

The DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. DAKOTA contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic finite element methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the DAKOTA toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a reference manual for the commands specification for the DAKOTA software, providing input overviews, option descriptions, and example specifications.