Quantifying Uncertainty
Asokan, B.V. and Zabaras, N.. "Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations" Journal of Computational Physics.
vol. 202.
2005.
pp. 94--133.
Wan, X. and Karniadakis, G.E.. "An adaptive multi-element generalized Polynomial Chaos method for stochastic differential equations" J. Comput. Phys.. 209
(2).
2005.
pp. 617--642.
We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method for Legendre-chaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of ME-gPC is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized polynomial chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the Kraichnan-Orszag three-mode problem, as well as advection-diffusion problems. The new method is similar to spectral element method for deterministic problems but with h-p discretization of the random space
Lucor, D. and Karniadakis, G.E.. "Adaptive Generalized Polynomial Chaos for Nonlinear Random Oscillators" SIAM J. Sci. Comput.. 26
(2).
2005.
pp. 720--735.
The solution of nonlinear random oscillators subject to stochastic forcing is investigated numerically. In particular, solutions to the random Duffing oscillator with random Gaussian and non-Gaussian excitations are obtained by means of the generalized polynomial chaos (GPC). Adaptive procedures are proposed to lower the increased computational cost of the GPC approach in large-dimensional spaces. Adaptive schemes combined with the use of an enriched representation of the system improve the accuracy of the GPC approach by reordering the random modes according to their magnification by the system.
Debusschere, B.J., Najm, H.N., Pebay, P.P., Knio, O.M., Ghanem, R.G., and Le Maitre, O.P.. "Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes" SIAM J. Sci. Comput.. 26
(2).
2005.
pp. 698--719.
This paper gives an overview of the use of polynomial chaos (PC) expansions to represent stochastic processes in numerical simulations. Several methods are presented for performing arithmetic on, as well as for evaluating polynomial and nonpolynomial functions of variables represented by PC expansions. These methods include Taylor series, a newly developed integration method, as well as a sampling-based spectral projection method for nonpolynomial function evaluations. A detailed analysis of the accuracy of the PC representations, and of the different methods for nonpolynomial function evaluations, is performed. It is found that the integration method offers a robust and accurate approach for evaluating nonpolynomial functions, even when very high-order information is present in the PC expansions.
Soize, C. and Ghanem, R.G.. "Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure" SIAM Journal on Scientific Computing. 26
(2).
2005.
pp. 395--410.
The basic random variables on which random uncertainties can in a given model depend can be viewed as defining a measure space with respect to which the solution to the mathematical problem can be defined. This measure space is defined on a product measure associated with the collection of basic random variables. This paper clarifies the mathematical structure of this space and its relationship to the underlying spaces associated with each of the random variables. Cases of both dependent and independent basic random variables are addressed. Bases on the product space are developed that can be viewed as generalizations of the standard polynomial chaos approximation. Moreover, two numerical constructions of approximations in this space are presented along with the associated convergence analysis.
Xiu, D., Kevrekidis, I.G., and Ghanem, R.G.. "An Equation-Free, Multiscale Approach to Uncertainty Quantification" Computing in Science and Engineering. 7
(3).
2005.
pp. 16--23.
Recently, interest has grown in developing efficient computational methods (both sampling and nonsampling) for studying ordinary or partial differential equations with random inputs. Stochastic Galerkin (SG) methods based on generalized polynomial chaos (gPC) representations have several appealing features. However, when the model equations are complicated, the numerical implementation of such algorithms can become highly nontrivial, and care is needed to design robust and efficient solvers for the resulting systems of equations. The authors' equation- and Galerkin-free computational approach to uncertainty quantification (UQ) for dynamical systems lets them conduct UQ computations without explicitly deriving the SG equations for the gPC coefficients. They use short bursts of appropriately initialized ensembles of simulations with the basic model to estimate the quantities required in SG algorithms.
Chen, Q.-Y., Gottlieb, D., and Hesthaven, J.S.. "Uncertainty analysis for the steady-state flows in a dual throat nozzle" Journal of Computational Physics. 204
(1).
2005.
pp. 378--398.
It is well known that the steady state of an isentropic flow in a dual-throat nozzle with equal throat areas is not unique. In particular there is a possibility that the flow contains a shock wave, whose location is determined solely by the initial condition. In this paper, we consider cases with uncertainty in this initial condition and use generalized polynomial chaos methods to study the steady-state solutions for stochastic initial conditions. Special interest is given to the statistics of the shock location. The polynomial chaos (PC) expansion modes are shown to be smooth functions of the spatial variable x, although each solution realization is discontinuous in the spatial variable x. When the variance of the initial condition is small, the probability density function of the shock location is computed with high accuracy. Otherwise, many terms are needed in the PC expansion to produce reasonable results due to the slow convergence of the PC expansion, caused by non-smoothness in random space.
Asokan, B.V. and Zabaras, N.. "Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations" Journal of Computational Physics. 202
(1).
2005.
pp. 94--133.
An extension of the deterministic variational multiscale (VMS) approach with algebraic subgrid scale (SGS) modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic Navier-Stokes equations. The stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method (SSFEM). Generalized polynomial chaos and Karhunen-Loève expansion techniques are used for representation of uncertain quantities. The proposed stabilized method is then applied to various standard advection-diffusion and fluid-flow examples with uncertainty in essential boundary conditions. Comparisons are drawn between the numerical solutions and Monte-Carlo/analytical solutions wherever possible.
Asokan, B.V. and Zabaras, N.. "Using stochastic analysis to capture unstable equilibrium in natural convection" Journal of Computational Physics. 208
(1).
2005.
pp. 134--153.
A stabilized stochastic finite element implementation for the natural convection system of equations under Boussinesq assumptions with uncertainty in inputs is considered. The stabilized formulations are derived using the variational multiscale framework assuming a one-step trapezoidal time integration rule. The stabilization parameters are shown to be functions of the time-step size. Provision is made for explicit tracking of the subgrid-scale solution through time. A support-space/stochastic Galerkin approach and the generalized polynomial chaos expansion (GPCE) approach are considered for input-output uncertainty representation. Stochastic versions of standard Rayleigh-Benard convection problems are used to evaluate the approach. It is shown that for simulations around critical points, the GPCE approach fails to capture the highly non-linear input uncertainty propagation whereas the support-space approach gives fairly accurate results. A summary of the results and findings is provided.
Asokan, B.V. and Zabaras, N.. "Using stochastic analysis to capture unstable equilibrium in natural convection" Journal of Computational Physics. 208
(1).
2005.
pp. 134--153.
In recent years, there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems. Spectral stochastic methods and Monte-Carlo based sampling methods are two approaches that have been used to analyze these problems. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view. This is especially true in the context of realistic thermal flow problems, where uncertainties in the topology of the boundary domain, boundary flux conditions and heterogeneous physical properties usually require high-dimensional random descriptors. The sparse grid collocation method based on the Smolyak algorithm offers a viable alternate method for solving high-dimensional stochastic partial differential equations. An extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation. We show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions. Equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured. Comparison of the present method with the generalized polynomial chaos expansion and Monte-Carlo methods are mad
Matthies, H.G. and Keese, A.. "Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations" Computer Methods in Applied Mechanics and Engineering. 194
(12-16).
2005.
pp. 1295--1331.
Stationary systems modelled by elliptic partial differential equations-linear as well as nonlinear-with stochastic coefficients (random fields) are considered. The mathematical setting as a variational problem, existence theorems, and possible discretisations-in particular with respect to the stochastic part-are given and investigated with regard to stability. Different and increasingly sophisticated computational approaches involving both Wiener's polynomial chaos as well as the Karhunen-Loeve expansion are addressed in conjunction with stochastic Galerkin procedures, and stability within the Galerkin framework is established. New and effective algorithms to compute the mean and covariance of the solution are proposed. The similarities and differences with better known Monte Carlo methods are exhibited, as well as alternatives to integration in high-dimensional spaces. Hints are given regarding the numerical implementation and parallelisation. Numerical examples serve as illustration. (C) 2004 Elsevier B.V. All rights reserved.
Reagan, M.T., Najm, H.N., Pebay, P.P., Knio, O.M., and Ghanem, R.G.. "Quantifying uncertainty in chemical systems modeling" International Journal of Chemical Kinetics. 37
(6).
JUN 2005.
pp. 368--382.
This study compares two techniques for uncertainty quantification in chemistry computations, one based on sensitivity analysis and error propagation, and the other on stochastic analysis using polynomial chaos techniques. The two constructions are studied in the context of H-2-O-2 ignition under supercritical-water conditions. They are compared in terms of their prediction of uncertainty in species concentrations and the sensitivity of selected species concentrations to given parameters. The formulation is extended to one-dimensional reacting-flow simulations. The computations are used to study sensitivities to both reaction rate pre-exponentials and enthalpies, and to examine how this information must be evaluated in light of known, inherent parametric uncertainties in simulation parameters. The results indicate that polynomial chaos methods provide similar first-order information to conventional sensitivity analysis, while preserving higher-order information that is needed for accurate uncertainty quantification and for assigning confidence intervals on sensitivity coefficients. These higher-order effects can be significant, as the analysis reveals substantial uncertainties in the sensitivity coefficients themselves. © 2005 Wiley Periodicals, Inc.
Millman, D.R., King, P.I., and Beran, P.S.. "Airfoil pitch-and-plunge bifurcation behavior with Fourier chaos expansions" Journal of Aircraft. 42
(2).
MAR-APR 2005.
pp. 376--384.
A stochastic projection method is employed to obtain the probability distribution of pitch angle of an airfoil in pitch and plunge subject to probabilistic uncertainty in both the initial pitch angle and the cubic spring coefficient of the restoring pitch force. Historically, the selected basis for the stochastic projection method has been orthogonal polynomials, referred to as the polynomial chaos. Such polynomials, however, result in unacceptable computational expense for applications involving oscillatory motion, and a new basis, the Fourier chaos, is introduced for computing limit-cycle oscillations. Unlike the polynomial chaos expansions, which cannot predict limit-cycle oscillations, the Fourier chaos expansions predict both subcritical and supercritical responses even with low-order expansions and high-order nonlinearities. Bifurcation diagrams generated with this new approximate method compare well to Monte Carlo simulations.
Ghanem, R.G., Masri, S., Pellissetti, M., and Wolfe, R.. "Identification and prediction of stochastic dynamical systems in a polynomial chaos basis" Computer Methods in Applied Mechanics and Engineering. 194
(12-16).
2005.
pp. 1641--1654.
Non-parametric system identification techniques have been proposed for constructing predictive models of dynamical systems without detailed knowledge of the mechanisms of energy transfer and dissipation. In a class of such models, multi-dimensional Chebychev polynomials in the state variables are fitted to the observed dynamical state of the system. Due to the approximative nature of this non-parametric model as well as to various other sources of uncertainty such as measurement errors and non-anticipative excitations, the parameters of the model exhibit a scatter that is treated here in a probabilistic context. The statistics of these coefficients are related to the physical properties of the model being analyzed, and are used to endow the model predictions with a probabilistic structure. They are also used to obtain a parsimonious characterization of the predictive model while maintaining a desirable level of accuracy. The proposed methodology is quite simple and robust. (C) 2004 Elsevier B.V. All rights reserved.
Su, Q. and Strunz, K.. "Stochastic circuit modelling with Hermite polynomial chaos" Electronic Letters. 41
(21).
OCT 13 2005.
pp. 1163--1165.
Hermite polynomial chaos is used to create models of electric circuit branches for the study of random changes of parameters. The proposed method allows for seamless integration with nodal analysis. An analogy of Fourier series and Hermite polynomial chaos expansion is introduced to explain the methodology Compared with root-sum-square and Monte Carlo methods, the proposed method is shown to be fast and accurate.
Xiu, D. and Kevrekidis, I.G.. "Equation-free, multiscale computation for unsteady random diffusion" Multiscale Modeling & Simulation. 4
(3).
2005.
pp. 915--935.
We present an ``equation-free'' multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A detailed fine-scale computation of such a problem requires discretization and solution of a large system of equations and can be prohibitively time consuming. To circumvent this difficulty, we propose an equation-free approach, where the fine-scale computation is conducted only for a (small) fraction of the overall time. The evolution of a set of appropriately defined coarse-grained variables (observables) is evaluated during the fine-scale computation, and ``projective integration'' is used to accelerate the integration. The choice of these coarse variables is an important part of the approach: they are the coefficients of pointwise polynomial expansions of the random solutions. Such a choice of coarse variables allows us to reconstruct representative ensembles of fine-scale solutions with ``correct'' correlation structures, which is a key to algorithm efficiency. Numerical examples demonstrating accuracy and efficiency of the approach are presented.
Hossain, F. and Anagnostou, E.N.. "Assessment of a stochastic interpolation based parameter sampling scheme for efficient uncertainty analyses of hydrologic models" Computers & Geosciences. 31
(4).
MAY 2005.
pp. 497--512.
This study assesses a stochastic interpolation based parameter sampling scheme for efficient uncertainty analyses of stream flow prediction by hydrologic models. The sampling scheme is evaluated within the generalised likelihood uncertainty estimation (GLUE; Beven and Binley, 1992) methodology. A primary limitation in using the GLUE method as an uncertainty tool 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 inherent non-linear parameter interactions. Uncertainty in discharge prediction (model output) is approximated through a Hermite polynomial chaos approximation of normal random variables that represent the model's parameter (model input) uncertainty. The unknown coefficients of the approximated polynomial are calculated using limited number of model simulation runs. The calibrated Hermite polynomial is then used as a fast-running proxy to the slower-running hydrologic model to predict the degree of representativeness of a randomly sampled model parameter set. An evaluation of the scheme's improvement in sampling is made over a medium-sized watershed in Italy using the TOPMODEL (Beven and Kirkby, 1979). Even for a very high (8) dimensional parameter uncertainty domain the scheme was consistently able to reduce computational burden of uniform sampling for GLUE by at least 15-25\%. It was also found to have significantly higher degree of consistency in sampling accuracy than the nearest neighborhood sampling method. The GLUE based on the proposed sampling scheme preserved the essential features of the uncertainty structure in discharge simulation. The scheme demonstrates the potential for increasing efficiency of GLUE uncertainty estimation for rainfall-runoff models as it does not impose any additional structural or distributional assumptions. (c) 2004 Elsevier Ltd. All rights reserved.
Frauenfelder, P., Schwab, C., and Todor, R.A.. "Finite elements for elliptic problems with stochastic coefficients" 194
(2-5, Sp. Iss. SI).
2005.
pp. 205--228.
We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen-Loeve (KL) expansion. An O(NlogN) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M, 1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r = r(M)) in an (M, r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r, up to 10 in moderate time. (C) 2004 Elsevier B.V. All rights reserved.
Xiu, D. and Hesthaven, J.S.. "High-order collocation methods for differential equations with random inputs" SIAM J. Sci. Comput.. 27
(3).
2005.
pp. 1118--1139.
Mathelin, L., Hussaini, M., and Zang, T.. "Stochastic Approaches to Uncertainty Quantification in CFD Simulations" Num. Alg.. 38
(1-3).
2005.
pp. 209--236.
This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.
Babuska, I.M., Tempone, R., and Zouraris, G.E.. "Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation" Computer Methods in Applied Mechanics and Engineering.
vol. 194.
2005.
pp. 1251--1294.