Quantifying Uncertainty
Knio, O.M. and Le Maitre, O.P.. "Uncertainty propagation in CFD using Polynomial Chaos decomposition" Fluid Dyn. Res.. 38
(9).
2006.
pp. 616--640.
Uncertainty quantification in CFD computations is receiving increased interest, due in large part to the increasing complexity of physical models, and the inherent introduction of random model data. This paper focuses on recent application of PC methods for uncertainty representation and propagation in CFD computations. The fundamental concept on which polynomial chaos (PC) representations are based is to regard uncertainty as generating a new set of dimensions, and the solution as being dependent on these dimensions. A spectral decomposition in terms of orthogonal basis functions is used, the evolution of the basis coefficients providing quantitative estimates of the effect of random model data. A general overview of PC applications in CFD is provided, focusing exclusively on applications involving the unreduced Navier-Stokes equations. Included in the present review are an exposition of the mechanics of PC decompositions, an illustration of various means of implementing these representations, and a perspective on the applicability of the corresponding techniques to propagate and quantify uncertainty in Navier-Stokes computations.
Wan, X. and Karniadakis, G.E.. "Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures" SIAM J. Sci. Comput.. 28
(3).
2006.
pp. 901--928.
We develop a multi-element generalized polynomial chaos (ME-gPC) method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs. Given a stochastic input with an arbitrary probability measure, its random space is decomposed into smaller elements. Subsequently, in each element a new random variable with respect to a conditional probability density function (PDF) is defined, and a set of orthogonal polynomials in terms of this random variable is constructed numerically. Then, the generalized polynomial chaos (gPC) method is implemented element-by-element. Numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost. Efficiency and convergence of ME-gPC are studied numerically by considering some commonly used random variables. ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities.
Yu, Y., Zhao, M., Lee, T., Pestieau , N., Bo, W., Glimm, J., and Grove, J.W.. "Uncertainty quantification for chaotic computational fluid dynamics" Journal of Computational Physics. 217
(1).
SEP 2006.
pp. 200--216.
We seek error models for simulations that model chaotic flow. Stable statistics for the solution and for the error are obtained after suitable averaging procedures.
We seek error models for simulations that model chaotic flow. Stable statistics for the solution and for the error are obtained after suitable averaging procedures. (c) 2006 Elsevier Inc. All rights reserved.
Ghanem, R.G. and Doostan, A.. "On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data" Journal of Computational Phyics. 217
(1).
2006.
pp. 63--81.
This paper investigates the predictive accuracy of stochastic models. In particular, a formulation is presented for the impact of data limitations associated with the calibration of parameters for these models, on their overall predictive accuracy. In the course of this development, a new method for the characterization of stochastic processes from corresponding experimental observations is obtained. Specifically, polynomial chaos representations of these processes are estimated that are consistent, in some useful sense, with the data. The estimated polynomial chaos coefficients are themselves characterized as random variables with known probability density function, thus permitting the analysis of the dependence of their values on further experimental evidence. Moreover, the error in these coefficients, associated with limited data, is propagated through a physical system characterized by a stochastic partial differential equation (SPDE). This formalism permits the rational allocation of resources in view of studying the possibility of validating a particular predictive model. A Bayesian inference scheme is relied upon as the logic for parameter estimation, with its computational engine provided by a Metropolis-Hastings Markov chain Monte Carlo procedure.
Asokan, B.V. and Zabaras, N.. "A stochastic variational multiscale method for diffusion in heterogeneous random media" Journal of Computational Physics. 218
(2).
2006.
pp. 654--676.
A stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media. The exact solution of the governing equations is split into two components: a coarse-scale solution that can be captured on a coarse mesh and a subgrid solution. A localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse-scale solution. The coarse-scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model. The generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models. Finally, various numerical examples are considered for evaluating the method.
Xiu, D. and Tartakovsky, D.M.. "Numerical Methods for Differential Equations in Random Domains" SIAM Journal on Scientific Computing. 28
(3).
2006.
pp. 1167--1185.
Physical phenomena in domains with rough boundaries play an important role in a variety of applications. Often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both. This topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we propose a novel computational framework, which is based on using stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively, and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a stochastic Galerkin method and Monte Carlo simulations to solve the transformed stochastic problem. We demonstrate our approach by applying it to an elliptic problem in single- and double-connected random domains, and comment on the accuracy and convergence of the numerical methods.
Tartakovsky, D.M. and Xiu, D.. "Stochastic analysis of transport in tubes with rough walls" Journal of Computational Physics. 217
(1).
2006.
pp. 248--259.
Flow and transport in tubes with rough surfaces play an important role in a variety of applications. Often the topology of such surfaces cannot be accurately described in all of its relevant details due to either insufficient data or measurement errors or both. In such cases, this topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we use a computational framework, which is based on stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a generalized polynomial chaos and Monte Carlo simulations to solve the transformed stochastic problem. We use our approach to describe transport of a passive scalar in Stokes' flow and to quantify the corresponding predictive uncertainty.
Wan, X. and Karniadakis, G.E.. "Beyond Wiener---Askey Expansions: Handling Arbitrary PDFs" Journal of Scientific Computing. 27
(1-3).
2006.
pp. 455--464.
In this paper we present a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with stochastic inputs with arbitrary probability measures. Based on the decomposition of the random space of the stochastic inputs, we construct numerically a set of orthogonal polynomials with respect to a conditional probability density function (PDF) in each element and subsequently implement generalized Polynomial Chaos (gPC) locally. Numerical examples show that ME-gPC exhibits both p- and h-convergence for arbitrary probability measures.
Lin, G., Su, C.-H., and Karniadakis, G.E.. "Predicting shock dynamics in the presence of uncertainties" Journal of Computational Physics. 217
(1).
2006.
pp. 260--276.
We revisit the classical aerodynamics problem of supersonic flow past a wedge but subject to random inflow fluctuations or random wedge oscillations around its apex. We first obtain analytical solutions for the inviscid flow, and subsequently we perform stochastic simulations treating randomness both as a steady as well as a time-dependent process. We use a multi-element generalized polynomial chaos (ME-gPC) method to solve the two-dimensional stochastic Euler equations. A Galerkin projection is employed in the random space while WENO discretization is used in physical space. A key issue is the characteristic flux decomposition in the stochastic framework for which we propose different approaches. The results we present show that the variance of the location of perturbed shock grows quadratically with the distance from the wedge apex for steady randomness. However, for a time-dependent random process the dependence is quadratic only close to the apex and linear for larger distances. The multi-element version of polynomial chaos seems to be more effective and more efficient in stochastic simulations of supersonic flows compared to the global polynomial chaos method.
Lin, G., Grinberg, L., and Karniadakis, G.E.. "Numerical studies of the stochastic Korteweg-de Vries equation" Journal of Computational Physics. 213
(2).
2006.
pp. 676--703.
We present numerical solutions of the stochastic Korteweg-de Vries equation for three cases corresponding to additive time-dependent noise, multiplicative space-dependent noise and a combination of the two. We employ polynomial chaos for discretization in random space, and discontinuous Galerkin and finite difference for discretization in physical space. The accuracy of the stochastic solutions is investigated by comparing the first two moments against analytical and Monte Carlo simulation results. Of particular interest is the interplay of spatial discretization error with the stochastic approximation error, which is examined for different orders of spatial and stochastic approximation.
Schwab, C. and Todor, R.A.. "Karhunen-Lo\`eve approximation of random fields by generalized fast multipole methods" Journal of Computational Physics. 217
(1).
2006.
pp. 100--122.
Beran, P.S., Pettit, C.L., and Millman, D.R.. "Uncertainty quantification of limit-cycle oscillations" Jounral of Computational Physics. 217
(1).
2006.
pp. 217--247.
Different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit-cycle oscillation, subject to parametric variability. The aeroelastic system is that of a rigid airfoil, supported by pitch and plunge structural coupling, with nonlinearities in the component in pitch. The nonlinearities are adjusted to permit the formation of a either a subcritical or supercritical branch of limit-cycle oscillations. Uncertainties are specified in the cubic coefficient of the torsional spring and in the initial pitch angle of the airfoil. Stochastic projections of the time-domain and cyclic equations governing system response are carried out, leading to both intrusive and non-intrusive computational formulations. Non-intrusive formulations are examined using stochastic projections derived from Wiener expansions involving Haar wavelet and B-spline bases, while Wiener-Hermite expansions of the cyclic equations are employed intrusively and non-intrusively. Application of the B-spline stochastic projection is extended to the treatment of aerodynamic nonlinearities, as modeled through the discrete Euler equations. The methodologies are compared in terms of computational cost, convergence properties, ease of implementation, and potential for application to complex aeroelastic systems.
Hou, T.Y., Luo , W., Rozovskii, B., and Zhou, H.-M.. "Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics" Journal of Computational Physics. 216
(2).
2006.
pp. 687--706.
In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier-Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner. The separation principle effectively reduces a stochastic equation to its associated propagator, a system of deterministic equations for the coefficients of the Wiener Chaos expansion. Simple formulas for statistical moments of the stochastic solution are presented. These formulas only involve the solutions of the propagator. We demonstrate that for short time solutions the numerical methods based on the Wiener Chaos expansion are more efficient and accurate than those based on the Monte Carlo simulations.
Pettit, C.L. and Beran, P.S.. "Spectral and multiresolution Wiener expansions of oscillatory stochastic processes" Journal of Sound and Vibration. 294
(4-5).
JUL 25 2006.
pp. 752--779.
Wiener chaos expansions are being evaluated for the representation of stochastic variability in the response of nonlinear aeroelastic systems, which often exhibit limit cycles. Preliminary studies with a simple nonlinear aeroelastic computational model have shown that the standard non-intrusive Wiener Hermite expansion fails to maintain time accuracy as the simulation evolves. Wiener-Hermite expansions faithfully reproduce the short-term characteristics of the process but consistently lose energy after several mean periods of oscillation. This energy loss remains even for very high-order expansions. To uncover the cause of this energy loss and to explore potential remedies, the more elementary problem of a sinusoid with random frequency is used herein to simulate the periodic response of an uncertain system. As time progresses, coefficients of the higher order terms in both the Wiener-Hermite and Wiener-Legendre expansions successively gain and lose dominance over the lower-order coefficients in a manner that causes any fixed-order expansion in terms of global basis functions to fail over a simulation time of sufficient duration. This characteristic behavior is attributed to the continually increasing frequency of the process in the random dimension. The recently developed Wiener-Haar expansion is found to almost entirely eliminate the loss of energy at large times, both for the sinusoidal process and for the response of a two degree-of-freedom nonlinear system, which is examined as a prelude to the stochastic simulation of aeroelastic limit cycles. It is also found that Mallat's pyramid algorithm is more efficient and accurate for evaluating Wiener-Haar expansion coefficients than Monte Carlo simulation or numerical quadrature.
Sachdeva, S.K., Nair, P.B., and Keane, A.J.. "Comparative study of projection schemes for stochastic finite element analysis" Computer Methods in Applied Mechanics and Engineering. 195
(19-22).
2006.
pp. 2371--2392.
We present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency. More specifically, we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic Krylov subspace. Numerical studies are presented for two problems: (1) static analysis of a plate with random Young's modulus and (2) settlement of a foundation supported on a randomly heterogeneous soil. Monte Carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared. We show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach, particularly for large-scale problems with many random variables. For the class of problems considered, we find that stochastic reduced basis methods can be up to orders of magnitude faster, while providing results of comparable or better accuracy.
Wan, X. and Karniadakis, G.E.. "Stochastic heat transfer enhancement in a grooved channel" Journal of Fluid Mechanics.
vol. 565.
OCT 2006.
pp. 255--278.
We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien-Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter E for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of E on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of E, which are skewed with their peaks increasing towards larger values of E as the Reynolds number increases. We then study the effect A multiple frequencies described by a periodically correlated random process. We find that the mean value of E is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing `smart' PDFs of the stochastic input that can result in improved heat transfer enhancement rates.
Williams, M.M.R.. "Polynomial chaos functions and stochastic differential equations" Annals of Nuclear Energy. 33
(9).
JUN 2006.
pp. 774--785.
The Karhunen-Loeve procedure and the associated polynomial chaos expansion have been employed to solve a simple first order stochastic differential equation which is typical of transport problems. Because the equation has an analytical solution, it provides a useful test of the efficacy of polynomial chaos. We find that the convergence is very rapid in some cases but that the increased complexity associated with many random variables can lead to very long computational times. The work is illustrated by exact and approximate solutions for the mean, variance and the probability distribution itself. The usefulness of a white noise approximation is also assessed. Extensive numerical results are given which highlight the weaknesses and strengths of polynomial chaos. The general conclusion is that the method is promising but requires further detailed study by application to a practical problem in transport theory. (c) 2006 Elsevier Ltd. All rights reserved.
Wan, X. and Karniadakis, G.E.. "Long-term behavior of polynomial chaos in stochastic flow simulations" Computer Methods in Applied Mechanics and Engineering. 195
(41-43).
2006.
pp. 5582--5596.
In this paper we focus on the long-term behavior of generalized polynomial chaos (gPC) and multi-element generalized polynomial chaos (ME-gPC) for partial differential equations with stochastic coefficients. First, we consider the one-dimensional advection equation with a uniform random transport velocity and derive error estimates for gPC and ME-gPC discretizations. Subsequently, we extend these results to other random distributions and high-dimensional random inputs with numerical verification using the algebraic convergence rate of ME-gPC. Finally, we apply our results to noisy flow past a stationary circular cylinder. Simulation results demonstrate that ME-gPC is effective in improving the accuracy of gPC for a long-term integration whereas high-order gPC cannot capture the correct asymptotic behavior. (c) 2005 Elsevier B.V. All rights reserved.
Sachdeva, S.K., Nair, P.B., and Keane, A.J.. "Hybridization of stochastic reduced basis methods with polynomial chaos expansions" Probabilitic Engineering Mechanics. 21
(2).
APR 2006.
pp. 182--192.
We propose a hybrid formulation combining stochastic reduced basis methods with polynomial chaos expansions for solving linear random algebraic equations arising from discretization of stochastic partial differential equations. Our objective is to generalize stochastic reduced basis projection schemes to non-Gaussian uncertainty models and simplify the implementation of higher-order approximations. We employ basis vectors spanning the preconditioned stochastic Krylov subspace to represent the solution process. In the present formulation, the polynomial chaos decomposition technique is used to represent the stochastic basis vectors in terms of multidimensional Hermite polynomials. The Galerkin projection scheme is then employed to compute the undetermined coefficients in the reduced basis approximation. We present numerical studies on a linear structural problem where the Youngs modulus is represented using Gaussian as well as lognormal models to illustrate the performance of the hybrid stochastic reduced basis projection scheme. Comparison studies with the spectral stochastic finite element method suggest that the proposed hybrid formulation gives results of comparable accuracy at a lower computational cost. (C) 2005 Elsevier Ltd. All rights reserved.
Creamer, D.B.. "On using polynomial chaos for modeling uncertainty in acoustic propagation" Journal of Acoustical Society of America. 119
(4).
APR 2006.
pp. 1979--1994.
The use of polynomial chaos for incorporating environmental variability into propagation models is investigated in the context of a simplified one-dimensional model, which is relevant for acoustic propagation when the random sound speed is independent of depth. Environmental variability is described by a spectral representation of a stochastic process and the chaotic representation of the wave field then consists of an expansion in terms of orthogonal random polynomials. Issues concerning implementation of the relevant equations, the accuracy of the approximation, uniformity of the expansion over the propagation range, and the computational burden necessary to evaluate different field statistics are addressed. When the correlation length of the environmental fluctuations is small, low-order expansions work well, while for large correlation lengths the convergence of the expansion is highly range dependent and requires high-order approximants. These conclusions also apply in higher-dimensional propagation problems.
Wu, B., Zhu, J., and Najm, F.N.. "Dynamic-range estimation" IEEE Transactions on Computer-Adied Design of Integrated Circuits and Systems. 25
(9).
SEP 2006.
pp. 1618--1636.
It has been widely recognized that the dynamic-range information of an application can be exploited to reduce the datapath bitwidth of either processors or application-specific integrated circuits and, therefore, the overall circuit area, delay, and power consumption. While recent proposals of analytical dynamic-range-estimation methods have shown significant advantages over the traditional profiling-based method in terms of runtime, it is argued here that the rather simplistic treatment of input correlation and system nonlinearity may lead to significant error. In this paper, three mathematical tools, namely Karhunen-Loe've expansion, polynomial chaos expansion, and independent component analysis are introduced, which enable not only the orthogonal decomposition of input random processes, but also the propagation of random processes through both linear and nonlinear systems with difficult constructs such as multiplications, divisions, and conditionals. It is shown that when applied to interesting nonlinear applications such as adaptive filters, polynomial filters, and rational filters, this method can produce complete accurate statistics of each internal variable, thereby allowing the synthesis of bitwidth with the desired tradeoff between circuit performance and signal-to-noise ratio.
Hover, F.S. and Triantafyllou, M.S.. "Application of polynomial chaos in stability and control" Automatica. 42
(5).
MAY 2006.
pp. 789--795.
The polynomial chaos of Wiener provides a framework for the statistical analysis of dynamical systems, with computational cost far superior to Monte Carlo simulations. It is a useful tool for control systems analysis because it allows probabilistic description of the effects of uncertainty, especially in systems having nonlinearities and where other techniques, such as Lyapunov's method, may fail. We show that stability of a system can be inferred from the evolution of modal amplitudes, covering nearly the full support of the uncertain parameters with a finite series. By casting uncertain parameters as unknown gains, we show that the separation of stochastic from deterministic elements in the response points to fast iterative design methods for nonlinear control. (c) 2006 Elsevier Ltd. All rights reserved.
Lovett, T.E., Ponci, F., and Monti, A.. "A polynomial chaos approach to measurement uncertainty" IEEE Transactions on Instrumentation and Measurement. 55
(3).
JUN 2006.
pp. 729--736.
Measurement uncertainty is traditionally represented in the form of expanded uncertainty as defined through the Guide to the Expression of Uncertainty in Measurement (GUM). The International Organization for Standardization GUM represents uncertainty through confidence intervals based on the variances and means derived from probability density functions. A new approach to the evaluation of measurement uncertainty based on the polynomial chaos theory is presented and compared with the traditional GUM method.
Millman, D.R., King, P.I., Maple, R.C., Beran, P.S., and Chilton, L.K.. "Estimating the probability of failure of a nonlinear aeroelastic system" Journal of Aircraft. 43
(2).
MAR-APR 2006.
pp. 504--516.
A limit-cycle oscillation (LCO) can be characterized by a subcritical or supercritical bifurcation, and bifurcations are shown to be discontinuities in the stochastic domain. The traditional polynomial-chaos-expansion method, which is a stochastic projection method, is too inefficient for estimating the LCO response surface because of the discontinuities associated with bifurcations. The objective of this research is to extend the stochastic projection method to include the construction of B-spline surfaces in the stochastic domain. The multivariate B-spline problem is solved to estimate the LCO response surface. A Monte Carlo simulation (MCS) is performed on this response surface to estimate the probability density function (PDF) of the LCO response. The stochastic projection method via B-splines is applied to the problem of estimating the PDF of a subcritical LCO response of a nonlinear airfoil in inviscid transonic flow. A probability of failure based upon certain failure criteria can then be computed from the estimated PDF. The stochastic algorithm provides a conservative estimate of the probability of failure of this aeroelastic system two orders of magnitude more efficiently than performing an MCS on the governing equations.