Showing papers on "Polynomial chaos published in 2009"

Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics

Abstract: The quantification of uncertainty in computational fluid dynamics (CFD) predictions is both a significant challenge and an important goal. Probabilistic uncertainty quantification (UQ) methods have been used to propagate uncertainty from model inputs to outputs when input uncertainties are large and have been characterized probabilistically. Polynomial chaos (PC) methods have found increased use in probabilistic UQ over the past decade. This review describes the use of PC expansions for the representation of random variables/fields and discusses their utility for the propagation of uncertainty in computational models, focusing on CFD models. Many CFD applications are considered, including flow in porous media, incompressible and compressible flows, and thermofluid and reacting flows. The review examines each application area, focusing on the demonstrated use of PC UQ and the associated challenges. Cross-cutting challenges with time unsteadiness and long time horizons are also discussed.

Fast numerical methods for stochastic computations: A review

Abstract: This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces. AMS subject classifications: 41A10, 60H35, 65C30, 65C50

Recent Advances in Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Analysis and Design

Abstract: Non-intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability. PCE estimates coefficients for known orthogonal polynomial basis functions based on a set of response function evaluations, using sampling, linear regression, tensor-product quadrature, or Smolyak sparse grid approaches. SC, on the other hand, forms interpolation functions for known coefficients, and requires the use of structured collocation point sets derived from tensor product or sparse grids. When tailoring the basis functions or interpolation grids to match the forms of the input uncertainties, exponential convergence rates can be achieved with both techniques for a range of probabilistic analysis problems. In addition, analytic features of the expansions can be exploited for moment estimation and stochastic sensitivity analysis. In this paper, the latest ideas for tailoring these expansion methods to numerical integration approaches will be explored, in which expansion formulations are modified to best synchronize with tensor-product quadrature and Smolyak sparse grids using linear and nonlinear growth rules. The most promising stochastic expansion approaches are then carried forward for use in new approaches for mixed aleatory-epistemic UQ, employing second-order probability approaches, and design under uncertainty, employing bilevel, sequential, and multifidelity approaches.

Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification

Abstract: Non-intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability PCE estimates coefficients for known orthogonal polynomial basis functions based on a set of response function evaluations, using sampling, linear regression, tensor-product quadrature, or Smolyak sparse grid approaches SC, on the other hand, forms interpolation functions for known coefficients, and requires the use of structured collocation point sets derived from tensor-products or sparse grids When tailoring the basis functions or interpolation grids to match the forms of the input uncertainties, exponential convergence rates can be achieved with both techniques for general probabilistic analysis problems In this paper, we explore relative performance of these methods using a number of simple algebraic test problems, and analyze observed differences In these computational experiments, performance of PCE and SC is shown to be very similar, although when differences are evident, SC is the consistent winner over traditional PCE formulations This stems from the practical difficulty of optimally synchronizing the formof the PCE with the integration approach being employed, resulting in slight over- or under-integration of prescribed expansion form With additional nontraditional tailoring of PCE form, it is shown that this performance gap can be reduced, and in some cases, eliminated

A comparative study of uncertainty propagation methods for black-box-type problems

Abstract: A wide variety of uncertainty propagation methods exist in literature; however, there is a lack of good understanding of their relative merits. In this paper, a comparative study on the performances of several representative uncertainty propagation methods, including a few newly developed methods that have received growing attention, is performed. The full factorial numerical integration, the univariate dimension reduction method, and the polynomial chaos expansion method are implemented and applied to several test problems. They are tested under different settings of the performance nonlinearity, distribution types of input random variables, and the magnitude of input uncertainty. The performances of those methods are compared in moment estimation, tail probability calculation, and the probability density function construction, corresponding to a wide variety of scenarios of design under uncertainty, such as robust design, and reliability-based design optimization. The insights gained are expected to direct designers for choosing the most applicable uncertainty propagation technique in design under uncertainty.

Chaos: From Simple Models To Complex Systems

Abstract: First Encounter with Chaos The Language of Dynamical Systems Examples of Chaotic Behaviors Probabilistic Approach to Chaos Characterization of Chaotic Dynamical Systems From Order to Chaos in Dissipative Systems Chaos in Hamiltonian Systems Chaos and Information Theory Coarse-Grained Information and Large Scale Predictability Chaos, Numerical Computations and Experiments Chaos in Few-Degrees of Freedom Systems Spatiotemporal Chaos Turbulence as a Dynamical Systems Problem Chaos and Statistical Mechanics: Fermi-Pasta-Ulam a Case Study.

Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations

Abstract: Uncertainty quantification appears today as a crucial point in numerous branches of science and engineering. In the last two decades, a growing interest has been devoted to a new family of methods, called spectral stochastic methods, for the propagation of uncertainties through physical models governed by stochastic partial differential equations. These approaches rely on a fruitful marriage of probability theory and approximation theory in functional analysis. This paper provides a review of some recent developments in computational stochastic methods, with a particular emphasis on spectral stochastic approaches. After a review of different choices for the functional representation of random variables, we provide an overview of various numerical methods for the computation of these functional representations: projection, collocation, Galerkin approaches…. A detailed presentation of Galerkin-type spectral stochastic approaches and related computational issues is provided. Recent developments on model reduction techniques in the context of spectral stochastic methods are also discussed. The aim of these techniques is to circumvent several drawbacks of spectral stochastic approaches (computing time, memory requirements, intrusive character) and to allow their use for large scale applications. We particularly focus on model reduction techniques based on spectral decomposition techniques and their generalizations.

Uncertainty quantification in chemical systems

Abstract: SUMMARY We demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non-isothermal ignition of a methane–air system. We employ Bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process. We analyze the time evolution of moments and probability density functions of the solution. We also examine the role and significance of dependence among the uncertain parameters. We finish with a discussion of the role of non-linearity and the performance of the algorithm. Copyright q 2009 John Wiley & Sons, Ltd.

Robust Aerodynamic Design Optimization Using Polynomial Chaos

Abstract: This paper investigates the potential of polynomial chaos methods, when used in conjunction with computational fluid dynamics, to quantify the effects of uncertainty in the computational aerodynamic design process. The technique is shown to be an efficient and accurate means of simulating the inherent uncertainty and variability in manufacturing and flow conditions and thus can provide the basis for computationally feasible robust optimization with computational fluid dynamics. This paper presents polynomial chaos theory and the nonintrusive spectral projection implementation, using this to demonstrate polynomial chaos as a basis for robust optimization, focusing on the problem of maximizing the lift-to-drag ratio of a two-dimensional airfoil while minimizing its sensitivity to uncertainty in the leading-edge thickness. The results demonstrate that the robustly optimized designs are significantly less sensitive to input variation, compared with nonrobustly optimized airfoils. The results also indicate that the inherent geometric uncertainty could degrade the on-design as well as the offdesign performance of the nonrobust airfoil. This leads to the further conclusion that the global optimum for some design problems is unreachable without accounting for uncertainty.

Characterization of reservoir simulation models using a polynomial chaos‐based ensemble Kalman filter

Abstract: [1] Model-based predictions of flow in porous media are critically dependent on assumptions and hypotheses that are not always based on first principles and that cannot necessarily be justified on the basis of known prevalent physics. Constitutive models, for instance, fall under this category. While these predictive tools are usually calibrated using observational data, the scatter in the resulting parameters has typically been ignored. In this paper, this scatter is used to construct stochastic process models of the parameters which are then used as the cornerstone in a novel model validation methodology useful for ascertaining the confidence in model-based predictions. The uncertainties are first quantified by representing the unknown model parameters via their polynomial chaos decompositions. These are descriptions of stochastic processes in terms of their coordinates with respect to an orthogonal basis. This is followed by a filtering step to update these representations with measurements as they become available. In order to account for the non-Gaussian nature of model parameters and model errors, an adaptation of the ensemble Kalman filter is developed. Instead of propagating an ensemble of model states forward in time as is suggested within the framework of the ensemble Kalman filter, the proposed approach allows the propagation of a stochastic representation of unknown variables using their respective polynomial chaos decompositions. The model is propagated forward in time by solving the system of partial differential equations using a stochastic projection method. Whenever measurements are available, the proposed data assimilation technique is used to update the stochastic parameters of the numerical model. The proposed method is applied to a black oil reservoir simulation model where measurements are used to stochastically characterize the flow medium and to verify the model validity with specified confidence bounds. The updated model can then be employed to forecast future flow behavior.

Stochastic analysis of unsaturated flow with probabilistic collocation method

Abstract: [1] In this study, we present an efficient approach, called the probabilistic collocation method (PCM), for uncertainty analysis of flow in unsaturated zones, in which the constitutive relationship between the pressure head and the unsaturated conductivity is assumed to follow the van Genuchten-Mualem model. Spatial variability of soil parameters leads to uncertainty in predicting flow behaviors. The aim is to quantify the uncertainty associated with flow quantities such as the pressure head and the effective saturation. In the proposed approach, input random fields, i.e., the soil parameters, are represented via the Karhunen-Loeve expansion, and the flow quantities are expressed by polynomial chaos expansions (PCEs). The coefficients in the PCEs are determined by solving the equations for a set of carefully selected collocation points in the probability space. To illustrate this approach, we use two-dimensional examples with different input variances and correlation scales and under steady state and transient conditions. We also demonstrate how to deal with multiple-input random parameters. To validate the PCM, we compare the resulting mean and variance of the flow quantities with those from Monte Carlo (MC) simulations. The comparison reveals that the PCM can accurately estimate the flow statistics with a much smaller computational effort than the MC.

Domain decomposition of stochastic PDEs: Theoretical formulations

Abstract: We present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations. The methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space. The probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes. The spatial decomposition is achieved through a Schur-complement-based domain decomposition. The methodology aims to exploit the full potential of high-performance computing platforms by reducing discretization errors with high-resolution numerical model in conjunction to giving due regards to uncertainty in the system. The mathematical formulation is numerically validated with an example of waves in random media. Copyright © 2008 John Wiley & Sons, Ltd.

Design of Composite Wings Including Uncertainties : A Probabilistic Approach

Abstract: A probabilistic method is developed to optimize the design of an idealized composite wing through consideration of the uncertainties in the material properties, fiber-direction angle, and ply thickness. The polynomial chaos expansion method is used to predict the mean, variance, and probability density function of the flutter speed, making use of an efficient Latin hypercube sampling technique. One-dimensional, two-dimensional, and three-dimensional polynomial chaos expansions are introduced into the probabilistic flutter model for different combinations of material, fiber-direction-angle, and ply-thickness uncertainties. The results are compared with Monte Carlo simulation and it is found that the probability density functions obtained using second- and third-order polynomial chaos expansion models compare well but require much less computation. A reliability criterion is defined, indicating the probability of failure due to flutter, and is used to determine successfully the optimal robust design of the composite wing.

Stable chaos

Abstract: Stable chaos is a generalization of the chaotic behaviour exhibited by cellular automata to continuous-variable systems and it owes its name to an underlying irregular and yet linearly stable dynamics. In this review we discuss analogies and differences with the usual deterministic chaos and introduce several tools for its characterization. Some examples of transitions from ordered behavior to stable chaos are also analyzed to further clarify the underlying dynamical properties. Finally, two models are specifically discussed: the diatomic hard-point gas chain and a network of globally coupled neurons.

Uncertainty analysis of complex structural systems

Abstract: In this paper methods for uncertainty propagation for complex structural systems are discussed. Only a limited class of methods is available for this purpose. It is shown that only numerical procedures allow the uncertainty assessment and reliability estimation of such systems. In particular, the perturbation procedure, the Karhunen–Loeve reduction scheme, the polynomial chaos expansion, direct and advanced Monte Carlo simulation (MCS), as well as random matrix theory meet these requirements. It is shown that advanced MCS procedures proved to be the most versatile approach. Copyright © 2009 John Wiley & Sons, Ltd.

Geotechnical probabilistic analysis by collocation-based stochastic response surface method: An Excel add-in implementation

Abstract: A general probabilistic method called collocation-based stochastic response surface method (CSRSM) was previously developed. It involves the propagation of input uncertainties through a computation model to arrive at a random output vector. It is assumed that the unknown random output can be expanded using a polynomial chaos basis with corresponding unknown coefficients. The unknown coefficients are evaluated using a collocation method because it has the important practical advantage of allowing existing deterministic numerical codes to be used as ‘black boxes’. The roots of the Hermite polynomial provide efficient collocation points to evaluate the coefficients in the stochastic response surface. An Excel add-in is developed to produce the basis functions (multi-dimensional Hermite polynomials) without resorting to symbolic algebra practitioners. This is a major practical advantage that would bring realistic probabilistic analyses within reach of the practitioners. Full Excel implementation details are i...

Efficient and Accurate Quantification of Uncertainty for Multiphase Flow With the Probabilistic Collocation Method

Abstract: Summary In this study, we explore an efficient and accurate method for uncertainty analysis of petroleum reservoir simulations. The essence of the approach is the combination of Karhunen-Loeve (KL) expansion and probabilistic collocation method. Monte Carlo (MC) simulation is the most common and straightforward approach for uncertainty quantification. It generates a large number of realizations of the underlying reservoir. Solving the multiple realizations leads to a large computational effort, especially for large-scale problems. We present an accurate and efficient alternative. In this approach, the underlying random fields, such as permeability and porosity are represented by the KL expansion and the resulting random fields (e.g., fluid saturations and pressures) or variables (e.g., hydrocarbon production) are expressed by the polynomial chaos expansions. The probabilistic collocation method (PCM) is used to determine the coefficients of the polynomial chaos expansions by solving for the fluid saturation and pressure fields via the original partial differential equations for selected sets of collocation points. This approach is nonintrusive because it results in independent deterministic differential equations, which, similar to the MC method, can be implemented with existing codes or simulators. However, the required number of simulations in the PCM is much less than that in the MC method. The approach is demonstrated with black-oil problems in heterogeneous reservoirs with the commercial Eclipse simulator. The accuracy, efficiency, and compatibility of this approach are compared against MC simulations. This study reveals that, while its computational efforts are greatly reduced compared to the MC method, the PCM is able to estimate accurately the statistical moments and probability density functions of the fluid saturations (and pressures) and the hydrocarbon production.

Dynamic Performance of a SCARA Robot Manipulator With Uncertainty Using Polynomial Chaos Theory

Abstract: This short paper outlines how polynomial chaos theory (PCT) can be utilized for manipulator dynamic analysis and controller design in a 4-DOF selective compliance assembly robot-arm-type manipulator with variation in both the link masses and payload. It includes a simple linear control algorithm into the formulation to show the capability of the PCT framework.

Epistemic Uncertainty in the Calculation of Margins

Abstract: Epistemic uncertainty, characterizing lack-of-knowledge, is often prevalent in engineering applications. However, the methods we have for analyzing and propagating epistemic uncertainty are not as nearly widely used or well-understood as methods to propagate aleatory uncertainty (e.g. inherent variability characterized by probability distributions). In this paper, we examine three methods used in propagating epistemic uncertainties: interval analysis, Dempster-Shafer evidence theory, and second-order probability. We demonstrate examples of their use on a problem in structural dynamics, specifically in the assessment of margins. In terms of new approaches, we examine the use of surrogate methods in epistemic analysis, both surrogate-based optimization in interval analysis and use of polynomial chaos expansions to provide upper and lower bounding approximations. Although there are pitfalls associated with surrogates, they can be powerful and efficient in the quantification of epistemic uncertainty.