John Red-Horse

Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach

Abstract This paper presents an efficient procedure for characterizing the solution of evolution equations with stochastic coefficients. These typically model the behavior of physical systems whose properties are modeled as spatially or temporally varying stochastic processes described within the framework of probability theory. The concepts of projection, orthogonality and weak convergence are exploited in a manner which directly mimics deterministic finite element solutions except that, in the stochastic case, inner products refer to expectation operations. Specifically, the Karhunen–Loeve expansion is used to discretize these processes into a denumerable set of random variables, thus providing a denumerable function space in which the problem is cast. The polynomial chaos expansion is then used to represent the solution in this space, and the coefficients in the expansion are evaluated as generalized Fourier coefficients via a Galerkin procedure in the Hilbert space of random variables.

Analysis of Eigenvalues and Modal Interaction of Stochastic Systems

An eigenvalue spectral analysis of stochastic engineering systems is presented. A comparative numerical study between approximations based on Monte Carlo sampling, a Taylor series-based perturbation approach, and the polynomial chaos representation is conducted. It is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method, especially for the higher modes. The differences of accuracy in the two methods are more pronounced as the system variability increases. Moreover, the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors. In addition, a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes.

A probabilistic approach to uncertainty quantification with limited information.

Abstract Many safety assessments depend upon models that rely on probabilistic characterizations about which there is incomplete knowledge. For example, a system model may depend upon the time to failure of a piece of equipment for which no failures have actually been observed. The analysts in this case are faced with the task of developing a failure model for the equipment in question, having very limited knowledge about either the correct form of the failure distribution or the statistical parameters that characterize the distribution. They may assume that the process conforms to a Weibull or log-normal distribution or that it can be characterized by a particular mean or variance, but those assumptions impart more knowledge to the analysis than is actually available. To address this challenge, we propose a method where random variables comprising equivalence classes constrained by the available information are approximated using polynomial chaos expansions (PCEs). The PCE approximations are based on rigorous mathematical concepts developed from functional analysis and measure theory. The method has been codified in a computational tool, AVOCET, and has been applied successfully to example problems. Results indicate that it should be applicable to a broad range of engineering problems that are characterized by both irreducible andreducible uncertainty.

Probabilistic Approach to NASA Langley Research Center Multidisciplinary Uncertainty Quantification Challenge Problem

A multifaceted exploration of the NASA Langley Research Center Multidisciplinary Uncertainty Quantification Challenge Problem aimed at examining the suitability of a probabilistic characterization of the epistemic uncertainties included with the problem statement is pursued. In the process, subproblems A through D are treated, and the opportunities and challenges associated with a probabilistic description are delineated as they pertain to each uncertainty characterization, uncertainty propagation, and uncertainty management, as well as to sensitivity analysis. All epistemic variables are replaced with random variables, and an equivalent effective uncertainty model with no epistemic component is identified. A Bayesian approach for parameter inference and an update that is applied directly on the probability densities of the various uncertainty variables are pursued, and sampling techniques for uncertainty propagation are used. A conditional expectation approach and a method based on the reduction of epist...

A Probabilistic Approach to the NASA Langley Multidisciplinary Uncertainty Quantification Challenge Problem

We pursue a multi-faceted exploration of the Challenge problem aimed at examining the suitability of a probabilistic characterization of the epistemic uncertainties included with the problem statement. In the process, we treat subproblems A through D and delineate the opportunities and challenges associated with a probabilistic description as it pertains to each of uncertainty characterization, uncertainty propagation, and uncertainty management, as well as to sensitivity analysis and to design. We replace all epistemic variables with random variables whose initial (prior) distribution is uniform and we rely chiefly on two mathematical constructs to pursue our analysis. According to the first construction, we pursue a Bayesian approach for parameter inference and update that is applied directly on the probability densities of the various uncertainty variables, and use sampling techniques for uncertainty propagation. The second construction is based on an adapted polynomial chaos expansions (PCE), that reflects acquired knowledge about the input-output map. Throughout the work, issues of efficiency are tackled with a combination of sparse quadrature methods and MCMC while proximity between probability densities is gaged using the Kullback-Leibler distance and new measures that are adapted to the structure and requirements of the Challenge problem.

Bayesian methods for estimating the reliability in complex hierarchical networks (interim report).

Current work on the Integrated Stockpile Evaluation (ISE) project is evidence of Sandias commitment to maintaining the integrity of the nuclear weapons stockpile. In this report, we undertake a key element in that process: development of an analytical framework for determining the reliability of the stockpile in a realistic environment of time-variance, inherent uncertainty, and sparse available information. This framework is probabilistic in nature and is founded on a novel combination of classical and computational Bayesian analysis, Bayesian networks, and polynomial chaos expansions. We note that, while the focus of the effort is stockpile-related, it is applicable to any reasonably-structured hierarchical system, including systems with feedback.

Orthogonal representations of stochastic processes and their propagation in mechanics

The paper reviews the representation of stochastic processes used to model coefficients in partial differential equations. In particular, methods based on the Karhunen-Loeve and Polynomial Chaos expansions are presented. It is shown that these representations permit both the efficient characterization and management of the uncertainty in the solution of the PDE. Management in this context is construed to mean error estimation and control. Furthermore, generalizations of the Karhunen-Loeve expansion to representing processes in Sobolev spaces is also presented. These processes are essential in applications to mechanics since, in this case, many functions of interest are constrained with regards to their smoothness and differentiability.