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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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We develop computable a posteriori error estimates for the pointwise evaluation of linear functionals of a solution to a parameterized linear system of equations. These error estimates are based on a variational analysis applied to polynomial spectral methods for forward and adjoint problems. We also use this error estimate to define an improved linear functional and we prove that this improved functional converges at a much faster rate than the original linear functional given a pointwise convergence assumption on the forward and adjoint solutions. The advantage of this method is that we are able to use low order spectral representations for the forward and adjoint systems to cheaply produce linear functionals with the accuracy of a higher order spectral representation. The method presented in this paper also applies to the case where only the convergence of the spectral approximation to the adjoint solution is guaranteed. We present numerical examples showing that the error in this improved functional is often orders of magnitude smaller. We also demonstrate that in higher dimensions, the computational cost required to achieve a given accuracy is much lower using the improved linear functional. problems using Bayesian methods require accurate and efficient estimates of distributions or probabilities. For such problems, the moments of the spectral representation are useful only if the output distribution happens to have a particularly simple form, such as Gaussian. In [31,30], the computational efficiency of the inference problem was dramatically improved by sampling the spectral representation rather than the full model. While this approach is very appealing in terms of the computational cost, the reliability of the predictions relies on the pointwise accuracy of the spectral representation. This accuracy may be lacking for the low order spectral methods which are commonly used for high dimensional parameterized systems.Meanwhile, computational modeling is becoming increasingly reliant on a posteriori error estimates to provide a measure of reliability on the numerical predictions. This methodology has been developed for a variety of methods and is widely accepted in the analysis of discretization error for partial differential equations [4,15,21]. The adjoint-based (dual-weighted residual) method is motivated by the observation that often the goal of a simulation is to compute a small number of linear functionals of the solution, such as the average value in a region or the drag on an object, rather than controlling the error in a global norm. This method has been successfully extended to estimate numerical errors due to operator splittings [16], operator decomposition for multiscale/multiphysics applications [9,19,20], adaptive sampling algorithms [17,18], and inverse sensitivity analysis [5,8]. It was also used in [32] to estimate the error in moments of linear functionals for the stochastic Galerkin approximation of a partial differential equation. In [7], the present authors used adjoint-based analysis to ...

a b s t r a c tThe aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments. The governing equations are based on Darcy's law with stochastic permeability. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen-Loève expansion. Mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product Hermite polynomials is used in the stochastic dimensions. Error analysis is performed and experimentally verified with numerical simulations. Computational results include incompressible and slightly compressible single and two-phase flow.