This report discusses the application of a Bayesian calibration analysis to data from the QASPR project at Sandia. The goal is to use experimental measurements of a response value to obtain information about the “best” values for some of the inputs which go into the corresponding simulator. The simulator is treated as an expensive black-box model, so that only a finite number of runs are available. Towards this end, a fast Gaussian process response surface approximation is used as an emulator for the simulator. It is shown how the Bayesian framework will allow us to explicitly account for uncertainty present in the experiments, the response surface approximation, and the results. In addition to simple point estimates, we can obtain information on marginal and joint confidence intervals/regions as well as correlations among the various parameters. A method for handling multiple, correlated, response measures (such as occur over time) is also developed.

Surfpack is a general-purpose software library of multidimensional function approximation methods for applications such as data visualization, data mining, sensitivity analysis, uncertainty quantification, and numerical optimization. Surfpack is primarily intended for use on sparse, irregularly-spaced, n-dimensional data sets where classical function approximation methods are not applicable. Surfpack is under development at Sandia National Laboratories, with a public release of Surfpack version 1.0 in August 2006. This paper provides an overview of Surfpack’s function approximation methods along with some of its software design attributes. In addition, this paper provides some simple examples to illustrate the utility of Surfpack for data trend analysis, data visualization, and optimization.

Response surface approximations (RSA) are often used as inexpensive replacements for computationally expensive computer simulations. Once a RSA has been computed, it is cheap to evaluate this “meta-model” or surrogate, and thus the RSA is often used in a variety of contexts, including optimization and uncertainty quantification. Usually, some method of sampling points over the input domain is used to generate samples of the input variables. These samples are run through the computer simulation. A response surface approximation is then generated based on the sample points. This report presents a study investigating the dependency of the response surface method on the sampling type. The purpose of this study was to address the question: Does a particular RSA type perform better (in terms of a better fit) if a particular sampling method is used? The RSA types examined were kriging, polynomial regression, and multivariate adaptive regression splines (MARS). The sampling types examined were Latin Hypercube, Halton, Hammersley, Centroidal Voronoi Tesselation (CVT), and standard Monte Carlo sampling. The example problems were a 5-dimensional version of Rosenbrock’s function and the Paviani function. RSA of the three response surface types were developed based on the five sampling methods. Performance was compared using ANOVA techniques.

Response surface functions are often used as simple and inexpensive replacements for computationally expensive computer models that simulate the behavior of a complex system over some parameter space. Progressive response surfaces are built up incrementally as global information is added from new sample points added to the previous points in the parameter space. As the response surfaces are globally upgraded, indications of the convergence of the response surface approximation to the exact (fitted) function can be inferred. Sampling points can be incrementally added in a structured or unstructured fashion. Whatever the approach, it is usually desirable to sample the entire parameter space uniformly (at least in early stages of sampling). At later stages of sampling, depending on the nature of the quantity being resolved it may be desirable to continue sampling uniformly (progressive response surfaces), or to switch to a focusing/economizing strategy of preferentially sampling certain regions of the parameter space based on information gained in previous stages of sampling (adaptive response surfaces). Here we consider progressive response surfaces where a balanced representation of global response over the parameter space is desired. We use Kriging and Moving-Least-Squares methods to fit Halton quasi-Monte-Carlo data samples and interpolate over the parameter space. On 2-D test problems we use the response surfaces to compute various response measures and assess the accuracy/applicability of heuristic error estimates based on convergence behavior of the computed response quantities. Where applicable we apply Richardson Extrapolation for estimates of error and asymptotic convergence, and assess the accuracy of these estimates. We seek to develop a robust methodology for constructing progressive response surface approximations with reliable error estimates.

This paper provides an overview of several approaches to formulating and solving o ptimiz ation u nder uncer tainty (OUU) en gineering design problems. In addition, the topic of high -performance computing and OUU is addressed, with a discussion of the coarse - and fine -grained parallel computing opportunities in the various OUU problem formulations. The OUU approaches covered her e are: sampling -based OUU, surr o- gate model -based OUU, analytic reliabil it y-based OUU (also known as reliability -based design optimiz a- tion) , polynomial chaos -based OUU, and st ochastic perturbation -based OUU.

Surrogate-based optimization methods have become established as effective techniques for engineering design problems through their ability to tame nonsmoothness and reduce computational expense. In recent years, supporting mathematical theory has been developed to provide the foundation of provable convergence for these methods. One of the requirements of this provable convergence theory involves consistency between the surrogate model and the underlying truth model that it approximates. This consistency can be enforced through a variety of correction approaches, and is particularly essential in the case of surrogate-based optimization with model hierarchies. First-order additive and multiplicative corrections currently exist which satisfy consistency in values and gradients between the truth and surrogate models at a single point. This paper demonstrates that first-order consistency can be insufficient to achieve acceptable convergence rates in practice and presents new second-order additive, multiplicative, and combined corrections which can significantly accelerate convergence. These second-order corrections may enforce consistency with either the actual truth model Hessian or its finite difference, quasi-Newton, or Gauss-Newton approximation.

This paper provides an overview of several approaches to formulating and solving optimization under uncertainty (OUU) engineering design problems. In addition, the topic of high-performance computing and OUU is addressed, with a discussion of the coarse- and fine-grained parallel computing opportunities in the various OUU problem formulations. The OUU approaches covered here are: sampling-based OUU, surrogate model-based OUU, analytic reliability-based OUU (also known as reliability-based design optimization), polynomial chaos-based OUU, and stochastic perturbation-based OUU.

The intent of this paper is to provide an overview of modern design of experiments (DOE) techniques that can be applied in computational engineering design studies. The term modern refers to DOE techniques specifically designed for use with deterministic computer simulations. In addition, this term is used to contrast classical DOE techniques that were developed for laboratory and field experiments that possess random error sources. Several types of modern DOE methods are described including pseudo-Monte Carlo sampling, quasi-Monte Carlo sampling, Latin hypercube sampling, orthogonal array sampling, and Hammersley sequence sampling.

In this paper, several formulations for optimization under uncertainty are presented. In addition to the direct nesting of uncertainty quantification within optimization, formulations are presented for surrogate-based optimization under uncertainty in which the surrogate model appears at the optimization level, at the uncertainty quantification level, or at both levels. These surrogate models encompass both data fit and hierarchical surrogates. The DAKOTA software framework is used to provide the foundation for prototyping and initial benchmarking of these formulations. A critical component is the extension of algorithmic techniques for deterministic surrogate-based optimization to these surrogate-based optimization under uncertainty formulations. This involves the use of sequential trust regionbased approaches to manage the extent of the approximations and verify the approximate optima. Two analytic test problems and one engineering problem are solved using the different methodologies in order to compare their relative merits. Results show that surrogate-based optimization under uncertainty formulations show promise both in reducing the number of function evaluations required and in mitigating the effects of nonsmooth response variations.

Optimization under uncertainty (OUU) was performed on an inertial confinement fusion (ICF) capsule model using the ALEGRA shock physics simulation code coupled with the DAKOTA software toolkit. The OUU results demonstrate how the inclusion of an uncertain variable in the ICF capsule design problem yields a different optimal design than would have been found with conventional optimization methods. In this particular ICF capsule design, maximum implosion performance of the ICF capsule was sacrificed to obtain robust implosion performance. These results provide a proof-of-concept demonstration of the utility of OUU methods in ICF capsule design and motivate future OUU algorithm development and ICF capsule design 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, analytic reliability, and stochastic finite element methods; parameter estimation with nonlinear least squares methods; and sensitivity 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.

In this paper, several formulations for optimization under uncertainty are presented. In addition to the direct nesting of uncertainty quantification within optimization, formulations are presented for surrogate-based optimization under uncertainty in which the surrogate model appears at the optimization level, at the uncertainty quantification level, or at both levels. These surrogate models encompass both data fit and hierarchical surrogates. The DAKOTA software framework is used to provide the foundation for prototyping and initial benchmarking of these formulations. A critical component is the extension of algorithmic techniques for deterministic surrogate-based optimization to these surrogate-based optimization under uncertainty formulations. This involves the use of sequential trust regionbased approaches to manage the extent of the approximations and verify the approximate optima. Two analytic test problems and one engineering problem are solved using the different methodologies in order to compare their relative merits. Results show that surrogate-based optimization under uncertainty formulations show promise both in reducing the number of function evaluations required and in mitigating the effects of nonsmooth response variations.