The Dakota 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 user's manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies.
This article is concerned with the approximation of high-dimensional functions by kernel-based methods. Motivated by uncertainty quantification, which often necessitates the construction of approximations that are accurate with respect to a probability density function of random variables, we aim at minimizing the approximation error with respect to a weighted $L^p$-norm. We present a greedy procedure for designing computer experiments based upon a weighted modification of the pivoted Cholesky factorization. The method successively generates nested samples with the goal of minimizing error in regions of high probability. Numerical experiments validate that this new importance sampling strategy is superior to other sampling approaches, especially when used with non-product probability density functions. We also show how to use the proposed algorithm to efficiently generate surrogates for inferring unknown model parameters from data.
Jakeman, John D.; Eldred, Michael S.; Geraci, G.; Gorodetsky, A.
In this paper, we present an adaptive algorithm to construct response surface approximations of high-fidelity models using a hierarchy of lower fidelity models. Our algorithm is based on multiindex stochastic collocation and automatically balances physical discretization error and response surface error to construct an approximation of model outputs. This surrogate can be used for uncertainty quantification (UQ) and sensitivity analysis (SA) at a fraction of the cost of a purely high-fidelity approach. We demonstrate the effectiveness of our algorithm on a canonical test problem from the UQ literature and a complex multi-physics model that simulates the performance of an integrated nozzle for an unmanned aerospace vehicle. We find that when the input-output response is sufficiently smooth our algorithm produces approximations that can be up to orders of magnitude more accurate than single fidelity approximations for a fixed computational budget.
Polynomial chaos expansions (PCE) are well-suited to quantifying uncertainty in models parameterized by independent random variables. The assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate PCE coefficients. In contrast, the application of PCE to models of dependent variables is much more challenging. Three approaches can be used to construct PCE of models of dependent variables. The first approach uses mapping methods where measure transformations, such as the Nataf and Rosenblatt transformation, can be used to map dependent random variables to independent ones; however we show that this can significantly degrade performance since the Jacobian of the map must be approximated. A second strategy is the class of dominating support methods. In these approaches a PCE is built using independent random variables whose distributional support dominates the support of the true dependent joint density; we provide evidence that this approach appears to produce approximations with suboptimal accuracy. A third approach, the novel method proposed here, uses Gram–Schmidt orthogonalization (GSO) to numerically compute orthonormal polynomials for the dependent random variables. This approach has been used successfully when solving differential equations using the intrusive stochastic Galerkin method, and in this paper we use GSO to build PCE using a non-intrusive stochastic collocation method. The stochastic collocation method treats the model as a black box and builds approximations of the input–output map from a set of samples. Building PCE from samples can introduce ill-conditioning which does not plague stochastic Galerkin methods. To mitigate this ill-conditioning we generate weighted Leja sequences, which are nested sample sets, to build accurate polynomial interpolants. We show that our proposed approach, GSO with weighted Leja sequences, produces PCE which are orders of magnitude more accurate than PCE constructed using mapping or dominating support methods.
In the context of the DARPA funded project SEQUOIA we are interested in the design under uncertainty of a jet engine nozzle subject to the performance requirements of a reconnaissance mission for a small unmanned military aircraft. This design task involves complex and expensive aero-thermo-structural computational analyses where it is of a paramount importance to also include the effect of the uncertain variables to obtain reliable predictions of the device’s performance. In this work we focus on the forward propagation analysis which is a key part of the design under uncertainty workflow. This task cannot be tackled directly by means of single fidelity approaches due to the prohibitive computational cost associated to each realization. We report here a summary of our latest advancement regarding several multilevel and multifidelity strategies designed to alleviate these challenges. The overall goal of these techniques is to reduce the computational cost of analyzing a high-fidelity model by resorting to less accurate, but less computationally demanding, lower fidelity models. The features of these multifidelity UQ approaches are initially illustrated and demonstrated on several model problems and afterward for the aero-thermo-structural analysis of the jet engine nozzle.
Predictive analysis of complex computational models, such as uncertainty quantification (UQ), must often rely on using an existing database of simulation runs. In this paper we consider the task of performing low-multilinear-rank regression on such a database. Specifically we develop and analyze an efficient gradient computation that enables gradient-based optimization procedures, including stochastic gradient descent and quasi-Newton methods, for learning the parameters of a functional tensor-train (FT). We compare our algorithms with 22 other nonparametric and parametric regression methods on 10 real-world data sets and show that for many physical systems, exploiting low-rank structure facilitates efficient construction of surrogate models. We use a number of synthetic functions to build insight into behavior of our algorithms, including the rank adaptation and group-sparsity regularization procedures that we developed to reduce overfitting. Finally we conclude the paper by building a surrogate of a physical model of a propulsion plant on a naval vessel.
