Quantifying Uncertainty
Lucor, D., Su, C.-H., and Karniadakis, G.E.. "Generalized Polynomial Chaos and Random Oscillators" 60
(3).
2004.
pp. 571--596.
We present a new approach to obtain solutions for general random oscillators using a broad class of polynomial chaos expansions, which are more efficient than the classical Wiener Hermite expansions. The approach is general but here we present results for linear oscillators only with random forcing or random coefficients. In this context, we are able to obtain relatively sharp error estimates in the representation of the stochastic input as well as the solution. We have also performed computational comparisons with Monte Carlo simulations which show that the new approach can be orders of magnitude faster. especially for compact distributions. Copyright (C) 2004 John Wiley Sons, Ltd.
Le Maitre, O.P., Knio, O.M., Najm, H.N., and Ghanem, R.G.. "Uncertainty propagation using Wiener-Haar expansions" J. Comput. Phys.. 197
(1).
2004.
pp. 28--57.
An uncertainty quantification scheme is constructed based on generalized Polynomial Chaos (PC) representations. Two such representations are considered, based on the orthogonal projection of uncertain data and solution variables using either a Haar or a Legendre basis. Governing equations for the unknown coefficients in the resulting representations are derived using a Galerkin procedure and then integrated in order to determine the behavior of the stochastic process. The schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of Rayleigh-Bénard instability. For situations involving random parameters close to a critical point, the computational implementations show that the Wiener-Haar (WHa) representation provides more robust predictions that those based on a Wiener-Legendre (WLe) decomposition. However, when the solution depends smoothly on the random data, the WLe scheme exhibits superior convergence. Suggestions regarding future extensions are finally drawn based on these experiences.
Le Maitre, O.P., Najm, H.N., Ghanem, R.G., and Knio, O.M.. "Multi-resolution analysis of wiener-type uncertainty propagation schemes" Journal of Computational Physics. 197
(2).
2004.
pp. 502--531.
A multi-resolution analysis (MRA) is applied to an uncertainty propagation scheme based on a generalized polynomial chaos (PC) representation. The MRA relies on an orthogonal projection of uncertain data and solution variables onto a multi-wavelet basis, consisting of compact piecewise-smooth polynomial functions. The coefficients of the expansion are computed through a Galerkin procedure. The MRA scheme is applied to the simulation of the Lorenz system having a single random parameter. The convergence of the solution with respect to the resolution level and expansion order is investigated. In particular, results are compared to two Monte-Carlo sampling strategies, demonstrating the superiority of the MRA. For more complex problems, however, the MRA approach may require excessive CPU times. Adaptive methods are consequently developed in order to overcome this drawback. Two approaches are explored: the first is based on adaptive refinement of the multi-wavelet basis, while the second is based on adaptive block-partitioning of the space of random variables. Computational tests indicate that the latter approach is better suited for large problems, leading to a more efficient, flexible and parallelizable scheme.
Xiu, D. and Karniadakis, G.E.. "Supersensitivity due to uncertain boundary conditions" International Journal for Numerical Methods in Engineering.
vol. 61.
NOV 28 2004.
pp. 2114--2138.
We study the viscous Burgers' equation subject to perturbations on the boundary conditions. Two kinds of perturbations are considered: deterministic and random. For deterministic perturbations, we show that small perturbations can result in O(1) changes in the location of the transition layer. For random perturbations, we solve the stochastic Burgers' equation using different approaches. First, we employ the Jacobi-polynomial-chaos, which is a subset of the generalized polynomial chaos for stochastic modeling. Converged numerical results are reported (up to seven significant digits), and we observe similar `stochastic supersensitivity' for the mean location of the transition layer. Subsequently, we employ up to fourth-order perturbation expansions. We show that even with small random inputs, the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output. Two types of distributions are considered: uniform distribution and a `truncated' Gaussian distribution with no tails. Various solution statistics, including the spatial evolution of probability density function at steady state, are studied. Copyright (C) 2004 John Wiley Sons, Ltd.
Le Maitre, O.P., Reagan, M.T., Debusschere, B.J., Najm, H.N., Ghanem, R.G., and Knio, O.M.. "Natural Convection in a Closed Cavity under Stochastic Non-Boussinesq Conditions" SIAM Journal on Scientific Computing. 26
(2).
2004.
pp. 375--394.
A stochastic projection method (SPM) is developed for quantitative propagation of uncertainty in compressible zero-Mach-number flows. The formulation is based on a spectral representation of uncertainty using the polynomial chaos (PC) system, and on a Galerkin approach to determining the PC coefficients. Governing equations for the stochastic modes are solved using a mass-conservative projection method. The formulation incorporates a specially tailored stochastic inverse procedure for exactly satisfying the mass-conservation divergence constraints. A brief validation of the zero-Mach-number solver is first performed, based on simulations of natural convection in a closed cavity. The SPM is then applied to analyze the steady-state behavior of the heat transfer and of the velocity and temperature fields under stochastic non-Boussinesq conditions.
Wan, X., Xiu, D., and Karniadakis, G.E.. "Stochastic Solutions for the Two-Dimensional Advection-Diffusion Equation" SIAM Journal on Scientific Computing. 26
(2).
2004.
pp. 578--590.
In this paper, we solve the two-dimensional advection-diffusion equation with random transport velocity. The generalized polynomial chaos expansion is employed to discretize the equation in random space while the spectral hp element method is used for spatial discretization. Numerical results which demonstrate the convergence of generalized polynomial chaos are presented. Specifically, it appears that the fast convergence rate in the variance is the same as that of the mean solution in the Jacobi-chaos unlike the Hermite-chaos. To this end, a new model to represent compact Gaussian distributions is also proposed.
Zhang, D. and Lu, Z.. "An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Lo\`eve and polynomial expansions" Journal of Computational Physics. 194
(2).
2004.
pp. 773--794.
In this study, we attempt to obtain higher-order solutions of the means and (co)variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of Karhunen-Loève decomposition, polynomial expansion, and perturbation methods. We first decompose the log hydraulic conductivity Y = ln Ks as an infinite series on the basis of a set of orthogonal Gaussian standard random variables ξi. The coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity. We then write head as an infinite series whose terms h(n) represent head of nth order in terms of ÏY, the standard deviation of Y, and derive a set of recursive equations for h(n). We then decompose h(n) with polynomial expansions in terms of the products of n Gaussian random variables. The coefficients in these series are determined by substituting decompositions of Y and h(n) into those recursive equations. We solve the mean head up to fourth-order in ÏY and the head variances up to third-order in ÏY2. We conduct Monte Carlo (MC) simulation and compare MC results against approximations of different orders from the moment-equation approach based on Karhunen-Loève decomposition (KLME). We also explore the validity of the KLME approach for different degrees of medium variability and various correlation scales. It is evident that the KLME approach with higher-order corrections is superior to the conventional first-order approximations and is computationally more efficient than the Monte Carlo simulation.
Field, R.V. and Grigoriu, M.. "On the accuracy of the polynomial chaos approximation" Probabilisitc Engineering Mechanics. 19
(1-2).
JAN-APR 2004.
pp. 65--80.
Polynomial chaos representations for non-Gaussian random variables and stochastic processes are infinite series of Hermite polynomials of standard Gaussian random variables with deterministic coefficients. Finite truncations of these series are referred to as polynomial chaos (PC) approximations. This paper explores features and limitations of PC approximations. Metrics are developed to assess the accuracy of the PC approximation. A collection of simple, but relevant examples is examined in this paper. The number of terms in the PC approximations used in the examples exceeds the number of terms retained in most current applications. For the examples considered, it is demonstrated that (1) the accuracy of the PC approximation improves in some metrics as additional terms are retained, but does not exhibit this behavior in all metrics considered in the paper, (2) PC approximations for strictly stationary, non-Gaussian stochastic processes are initially nonstationary and gradually may approach weak stationarity as the number of terms retained increases, and (3) the development of PC approximations for certain processes may become computationally demanding, or even prohibitive, because of the large number of coefficients that need to be calculated. However, there have been many applications in which PC approximations have been successful. (C) 2003 Elsevier Ltd. All rights reserved.
Reagan, M.T., Najm, H.N., Debusschere, B.J., Le Maitre, O.P., Knio, O.M., and Ghanem, R.G.. "Spectral stochastic uncertainty quantification in chemical systems" Combustion Theory and Modelling. 8
(3).
SEP 2004.
pp. 607--632.
Uncertainty quantification (UQ) in the computational modelling of physical systems is important for scientific investigation, engineering design, and model validation. We have implemented an `intrusive' UQ technique in which (1) model parameters and field variables are modelled as stochastic quantities, and are represented using polynomial chaos (PC) expansions in terms of Hermite polynomial functions of Gaussian random variables, and (2) the deterministic model equations are reformulated using Galerkin projection into a set of equations for the time evolution of the field variable PC mode strengths. The mode strengths relate specific parametric uncertainties to their effects on model outputs. In this work, the intrusive reformulation is applied to homogeneous ignition using a detailed chemistry model through the development of a reformulated pseudospectral chemical source term. We present results analysing the growth of uncertainty during the ignition process. We also discuss numerical issues pertaining to the accurate representation of uncertainty with truncated PC expansions, and ensuing stability of the time integration of the chemical system.
Choi, S.K., Grandhi, R.V., Canfield, R.A., and Pettit, C.L.. "Polynomial chaos expansion with Latin hypercube sampling for estimating response variability" AIAA Journal. 42
(6).
JUN 2004.
pp. 1191--1198.
A computationally efficient procedure for quantifying uncertainty and finding significant parameters of uncertainty models is presented. To deal with the random nature of input parameters of structural models, several efficient probabilistic methods are investigated. Specifically, the polynomial chaos expansion with Latin hypercube sampling is used to represent the response of an uncertain system. Latin hypercube sampling is employed for evaluating the generalized Fourier coefficients of the polynomial chaos expansion. Because the key challenge in uncertainty analysis is to find the most significant components that drive response variability, analysis of variance is employed to find the significant parameters of the approximation model. Several analytical examples and a large finite element model of a joined-wing are used to verify the effectiveness of this procedure.
Red-Horse, J.R. and Benjamin, A.S.. "A probabilistic approach to uncertainty quantification with limited information" Reliability Engineering & System Safety. 85
(1-3).
JUL-SEP 2004.
pp. 183--190.
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 and reducible uncertainty. (C) 2004 Published by Elsevier Ltd.
Jardak, M. and Ghanem, R.G.. "Spectral stochastic homogenization of divergence-type PDEs" Computer Methods in Applied Mechanics and Engineering. 193
(6--8).
2004.
pp. 429-447.
This paper presents a formulation and numerical analysis of the homogenization of stochastic PDEs. The framework of homogenization is adopted to describe an effective medium that is equivalent in some sense to a heterogeneous medium of interest. The parameters of the resulting homogeneous medium are described as stochastic processes characterized by their polynomial chaos decomposition. The formulation yields a chaos decomposition for the predicted behavior of the homogeneous medium that captures, in addition to the effect of heterogeneity, the effect of variability. Once this description has been computed, various statistics of the solution can be efficiently evaluated. (C) 2003 Elsevier B.V. All rights reserved.
Lucor, D. and Karniadakis, G.E.. "Predictability and uncertainty in flow-structure interactions" European Journal of Mechanics B-Fluids. 23
(1).
JAN-FEB 2004.
pp. 41--49.
Direct numerical simulation advances in the field of flow-structure interactions are reviewed both from a deterministic and stochastic point of view. First, results of complex wake flows resulting from vibrating cylindrical bluff bodies in linear and exponential sheared flows are presented. On the structural side, non-linear modeling of cable structures with variable tension is derived and applied to the problem of a catenary riser of complex shape. Finally, a direct approach using Polynomial Chaos to modeling uncertainty associated with flow-structure interaction is also described. The method is applied to the two-dimensional flow-structure interaction case of an elastically mounted cylinder with random structural parameters subject to vortex-induced vibrations. (C) 2003 Elsevier SAS. All rights reserved.
Babuska, I.M., Tempone, R., and Zouraris, G.E.. "Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations" SIAM J. Numer. Anal.. 42
(2).
2004.
pp. 800--825.
We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the com- putations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method gener- ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The sec- ond method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h- or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.
Keywords: stochastic elliptic equation ; perturbation estimates ; Karhunen--Loeve expansion ; finite elements ; Monte Carlo method ; $k\times h$-version ; $p\times h$-version ; expected value ; error estimates
Hossain, F., Anagnostou, E.N., and Lee, K.H.. "A non-linear and stochastic response surface method for Bayesian estimation of uncertainty in soil moisture simulation from a land surface model" Nonlinear processes in Geophysics. 11
(4).
2004.
pp. 427--440.
This study presents a simple and efficient scheme for Bayesian estimation of uncertainty in soil moisture simulation by a Land Surface Model (LSM). The scheme is assessed within a Monte Carlo (MC) simulation framework based on the Generalized Likelihood Uncertainty Estimation (GLUE) methodology. A primary limitation of using the GLUE method is the prohibitive computational burden imposed by uniform random sampling of the model's parameter distributions. Sampling is improved in the proposed scheme by stochastic modeling of the parameters' response surface that recognizes the non-linear deterministic behavior between soil moisture and land surface parameters. Uncertainty in soil moisture simulation (model output) is approximated through a Hermite polynomial chaos expansion of normal random variables that represent the model's parameter (model input) uncertainty. The unknown coefficients of the polynomial are calculated using limited number of model simulation runs. The calibrated polynomial is then used as a fast-running proxy to the slower-running LSM to predict the degree of representativeness of a randomly sampled model parameter set. An evaluation of the scheme's efficiency in sampling is made through comparison with the fully random MC sampling (the norm for GLUE) and the nearest-neighborhood sampling technique. The scheme was able to reduce computational burden of random MC sampling for GLUE in the ranges of 10\%-70\%. The scheme was also found to be about 10\% more efficient than the nearest-neighborhood sampling method in predicting a sampled parameter set's degree of representativeness. The GLUE based on the proposed sampling scheme did not alter the essential features of the uncertainty structure in soil moisture simulation. The scheme can potentially make GLUE uncertainty estimation for any LSM more efficient as it does not impose any additional structural or distributional assumptions.