Quantifying Uncertainty
Weiner, N.. "The homogeneous chaos" Amer. J. Math.. 60
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1938.
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Xiu, D. and Karniadakis, G.E.. "The Weiner-Askey Polynomial Chaos for stochastic differential equations" SIAM J. Sci. Comput.. 24
(2).
2002.
pp. 619--644.
We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos. Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error. Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs.
Xiu, D. and Karniadakis, G.E.. "Modeling uncertainty in steady state diffusion problems via generalized chaos" Computer Methods in Applied Mechanics and Engineering. 191
(43).
2002.
pp. 4927--4948.
We present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs. In particular, we focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. The stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener [Amer. J. Math. 60 (1938) 897]. A Galerkin projection in random space is applied to derive the equations in the weak form. The resulting set of deterministic equations for each random mode is solved iteratively by a block Gauss-Seidel iteration technique. Both discrete and continuous random distributions are considered, and convergence is verified in model problems and against Monte Carlo simulations. (C) 2002 Elsevier Science B.V. All rights reserved.
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.
Jardak, M., Su, C.-H., and Karniadakis, G.E.. "Spectral Polynomial Chaos Solutions of the Stochastic Advection Equation" Journal of Scientific Computing. 17
(1-4).
2002.
pp. 319--338.
We present a new algorithm based on Wiener-Hermite functionals combined with Fourier collocation to solve the advection equation with stochastic transport velocity. We develop different stategies of representing the stochastic input, and demonstrate that this approach is orders of magnitude more efficient than Monte Carlo simulations for comparable accuracy.
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
Ghanem, R.G.. "Ingredients for a general purpose stochastic finite element formulation" Computer Methods in Applied Mechanics and Engineering.
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Ghanem, R.G. and Dham, S.. "Stochastic finite element analysis for multiphase flow in heterogeneous porous media" Transport in Porous Media.
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Xiu, D. and Karniadakis, G.E.. "Modeling uncertainty in flow simulations via generalized Polynomial Chaos" J. Comput. Phys.. 187
(1).
2003.
pp. 137--167.
We present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach can be considered as a generalization of the original polynomial chaos expansion, first introduced by Wiener [Am. J. Math. 60 (1938) 897]. The original method employs the Hermite polynomials (one of the 13 members of the Askey scheme) as the basis in random space. The algorithm is applied to micro-channel flows with random wall boundary conditions, and to external flows with random freestream. Efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by Monte Carlo simulations. It is shown that the generalized polynomial chaos method promises a substantial speed-up compared with the Monte Carlo method. The utilization of different type orthogonal polynomials from the Askey scheme also provides a more efficient way to represent general non-Gaussian processes compared with the original Wiener-Hermite expansions.
Ogura, H.. "Orthogonal functionals of the Poisson process" IEEE Transactions on Information Theory. 18
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1972.
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Siegel, A., Imamura, T., and Meecham, W.C.. "WienerâHermite expansion in model turbulence in the late decay stage," Journal of Mathematical Physics.
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Meecham, W.C. and Jeng, D.T.. "Use of the Wiener-Hermite expansion for nearly normal turbulence" Journal of Fluid Mechanics.
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Crow, S.C. and Canavan, G.H.. "Relationship between a Wiener-Hermite expansion and an energy cascade" Journal of Fluid Mechanics.
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Matthies, H.G., Brenner, C.E., Bucher, C.G., and Soares, C.G.. "Uncertainties in probabilistic numerical analysis of structures and solids - stochastic finite-elements" Structural Safety. 19
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1997.
pp. 283--336.
Le Maitre, O.P., Knio, O.M., Debusschere, B.J., Najm, H.N., and Ghanem, R.G.. "A multigrid solver for two-dimensional stochastic diffusion equations" Methods in Applied Mechanics and Engineering.
vol. 192.
2003.
pp. 4723--4744.
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.
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.
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.
Orszag, S.A. and Bissonnette, L.R.. "Dynamical properties of truncated Wiener-Hermite expansions" Physics Fluids.
vol. 10.
1967.
pp. 2603--2613.
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.
Nobile, F., Tempone, R., and Webster, C.G.. "A stochastic collocation method for elliptic partial differential equations with random input data" SIAM Journal on Numerical Analysis. 43
(3).
2007.
pp. 1005--1034.
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.
Liu, W.K., Belytschko, T., and Mani, A.. "Random field finite elements" International Journal for Numerical Methods in Engineering.
vol. 23.
1986.
pp. 1831--1845.
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.
Schwab, C. and Todor, R.A.. "Sparse finite elements for stochastic elliptic problems: higher order moments" Computing. 71
(1).
2003.
pp. 43--63.
We define the higher order moments associated to the stochastic solution of an elliptic BVP in D â Rd with stochastic input data. We prove that the k-th moment solves a deterministic problem in Dk â Rdk, for which we discuss well-posedness and regularity. We discretize the deterministic k-th moment problem using sparse grids and, exploiting a spline wavelet basis, we propose an efficient algorithm, of logarithmic-linear complexity, for solving the resulting 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.
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.
Marzouk, Y.M., Najm, H.N., and Rahn, L.A.. "Stochastic spectral methods for efficient Bayesian solution of inverse problems" Journal of Computational Physics. 224
(2).
2007.
pp. 560--586.
We present a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos (PC) expansions to represent random variables. Evaluation of integrals over the unknown parameter space is recast, more efficiently, as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion. The accuracy of posterior estimates is examined with respect to the order of the PC representation, the choice of PC basis, and the decomposition of the support of the prior. The computational cost of the new scheme shows significant gains over direct sampling.
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.
Le Maitre, O.P., Knio, O.M., Najm, H.N., and Ghanem, R.G.. "A stochastic projection method for fluid flow. I: basic formulation" Journal of Computational Physics. 173
(2).
2001.
pp. 481--511.
Le Maitre, O.P., Reagan, M.T., Najm, H.N., Ghanem, R.G., and Knio, O.M.. "A stochastic projection method for fluid flow II.: random process" Journal of Computational Physics. 181
(1).
2002.
pp. 9--44.
An uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes. The scheme relies on spectral representation of uncertainty using the polynomial chaos (PC) system. The solver combines a Galerkin procedure for the determination of PC coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations. Implementation of the numerical scheme is illustrated through simulations of natural convection in a 2D square cavity with stochastic temperature distribution at the cold wall. The properties of the uncertainty representation scheme are analyzed, and the predictions are contrasted with results obtained using a Monte Carlo approach.
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.
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.
Ghanem, R.G. and Red-Horse, J.R. "Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach" Physica D. 133
(1-4).
1999.
pp. 137--144.
This paper presents an efficient procedure for characterizing the solution of evolution equations with stochastic coefficients. These typically model the behavior of physical systems whose properties are modeled as spatially or temporally varying stochastic processes described within the framework of probability theory. The concepts of projection, orthogonality and weak convergence are exploited in a manner which directly mimics deterministic finite element solutions except that, in the stochastic case, inner products refer to expectation operations. Specifically, the KarhunenâLoève expansion is used to discretize these processes into a denumerable set of random variables, thus providing a denumerable function space in which the problem is cast. The polynomial chaos expansion is then used to represent the solution in this space, and the coefficients in the expansion are evaluated as generalized Fourier coefficients via a Galerkin procedure in the Hilbert space of random variables.
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., 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.
Ganapathysubramanian, B. and Zabaras, N.. "Sparse grid collocation schemes for stochastic natural convection problems" J. Comput. Phys.. 225
(1).
2007.
pp. 652--685.
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 made.
Zabaras, N. and Ganapathysubramanian, B.. "A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach" Journal of Computational Physics. 227
(9).
2008.
pp. 4697--4735.
Experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions. Recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties. These advances raise the possibility of solving the corresponding stochastic inverse problem: the problem of designing/estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information. A scalable, parallel methodology for stochastic inverse/design problems is developed in this article. The representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology. A novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems. The stochastic inverse/design problem is transformed to a deterministic optimization problem in a larger-dimensional space that is subsequently solved using deterministic optimization algorithms. The design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems, thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes. Various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework.
Ganapathysubramanian, B. and Zabaras, N.. "A seamless approach towards stochastic modeling: Sparse grid collocation and data driven input models" Finite Elements in Analysis and Design. 44
(5).
2008.
pp. 298--320.
Many physical systems of fundamental and industrial importance are significantly affected by the underlying fluctuations/variations in boundary, initial conditions as well as variabilities in operating and surrounding conditions. There has been increasing interest in analyzing and quantifying the effects of uncertain inputs in the solution of partial differential equations that describe these physical phenomena. Such analysis naturally leads to a rigorous methodology to design/control physical processes in the presence of multiple sources of uncertainty. A general application of these ideas to many significant problems in engineering is mainly limited by two issues. The first is the significant effort required to convert complex deterministic software/legacy codes into their stochastic counterparts. The second bottleneck to the utility of stochastic modeling is the construction of realistic, viable models of the input variability. This work attempts to demystify stochastic modeling by providing easy-to-implement strategies to address these two issues. In the first part of the paper, strategies to construct realistic input models that encode the variabilities in initial and boundary conditions as well as other parameters are provided. In the second part of the paper, we review recent advances in stochastic modeling and provide a road map to trivially convert any deterministic code into its stochastic counterpart. Several illustrative examples showcasing the ease of converting deterministic codes to stochastic codes are provided.
Acharjee, S. and Zabaras, N.. "A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes" Computers and Structures. 85
(5-6).
2007.
pp. 244--254.
Large deformation processes are inherently complex considering the non-linear phenomena that need to be accounted for. Stochastic analysis of these processes is a formidable task due to the numerous sources of uncertainty and the various random input parameters. As a result, uncertainty propagation using intrusive techniques requires tortuous analysis and overhaul of the internal structure of existing deterministic analysis codes. In this paper, we present an approach called non-intrusive stochastic Galerkin (NISG) method, which can be directly applied to presently available deterministic legacy software for modeling deformation processes with minimal effort for computing the complete probability distribution of the underlying stochastic processes. The method involves finite element discretization of the random support space and piecewise continuous interpolation of the probability distribution function over the support space with deterministic function evaluations at the element integration points. For the hyperelastic-viscoplastic large deformation problems considered here with varying levels of randomness in the input and boundary conditions, the NISG method provides highly accurate estimates of the statistical quantities of interest within a fraction of the time required using existing Monte Carlo methods.
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., Wan, X., Su, C.-H., and Karniadakis, G.E.. "Stochastic Computational Fluid Mechanics" Computing in Science and Engineering. 9
(2).
2007.
pp. 21--29.
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.
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.
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.
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.
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
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.
Witteveen, J.A.S., Sarkar, S., and Bijl, H.. "Modeling physical uncertainties in dynamic stall induced fluid-structure interaction of turbine blades using arbitrary polynomial chaos" Computers and Structures. 85
(11-14).
2007.
pp. 866--878.
A nonlinear dynamic problem of stall induced flutter oscillation subject to physical uncertainties is analyzed using arbitrary polynomial chaos. A single-degree-of-freedom stall flutter model with torsional oscillation is considered subject to nonlinear aerodynamic loads in the dynamic stall regime and nonlinear structural stiffness. The analysis of the deterministic aeroelastic response demonstrated that the problem is sensitive to variations in structural natural frequency and structural nonlinearity. The effect of uncertainties in these parameters is studied. Arbitrary polynomial chaos is employed in which appropriate expansion polynomials are constructed based on the statistical moments of the uncertain input. The arbitrary polynomial chaos results are compared with Monte Carlo simulations.
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.
Ghanmi, S., Bouazizi, M.-L., and Bouhaddi, N.. "Robustness of mechanical systems against uncertainties" Finite Elements in Analysis and Design. 43
(9).
2007.
pp. 715--731.
In this paper, one can propose a method which takes into account the propagation of uncertainties in the finite element models in a multi-objective optimization procedure. This method is based on the coupling of stochastic response surface method (SRSM) and a genetic algorithm provided with a new robustness criterion. The SRSM is based on the use of stochastic finite element method (SFEM) via the use of the polynomial chaos expansion (PC). Thus, one can avoid the use of Monte Carlo simulation (MCS) whose costs become prohibitive in the optimization problems, especially when the finite element models are large and have a considerable number of design parameters. The objective of this study is on one hand to quantify efficiently the effects of these uncertainties on the responses variability or the cost functions which one wishes to optimize and on the other hand, to calculate solutions which are both optimal and robust with respect to the uncertainties of design parameters. In order to study the propagation of input uncertainties on the mechanical structure responses and the robust multi-objective optimization with respect to these uncertainty, two numerical examples were simulated. The results which relate to the quantification of the uncertainty effects on the responses variability were compared with those obtained by the reference method (REF) using MCS and with those of the deterministic response surfaces methodology (RSM). In the same way, the robust multi-objective optimization results resulting from the SRSM method were compared with those obtained by the direct optimization considered as reference (REF) and with RSM methodology. The SRSM method application to the response variability study and the robust multi-objective optimization gave convincing results.
Hover, F.S.. "Brief paper: Gradient dynamic optimization with Legendre chaos" Automatica. 44
(1).
2008.
pp. 135--140.
The polynomial chaos approach for stochastic simulation is applied to trajectory optimization, by conceptually replacing random variables with free variables. Using the gradient method, we generate with low computational cost an accurate parametrization of optimal trajectories.
Le Maitre, O.P., Najm, H.N., P\'ebay, P.P., Ghanem, R.G., and Knio, O.M.. "Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems" SIAM Journal on Scientific Computing. 29
(2).
2007.
pp. 864--889.
This paper presents a multi-resolution approach for the propagation of parametric uncertainty in chemical systems. It is motivated by previous studies where Galerkin formulations of Wiener-Hermite expansions were found to fail in the presence of steep dependences of the species concentrations with regard to the reaction rates. The multi-resolution scheme is based on representation of the uncertain concentration in terms of compact polynomial multi-wavelets, allowing for the control of the convergence in terms of polynomial order and resolution level. The resulting representation is shown to greatly improve the robustness of the Galerkin procedure in presence of steep dependences. However, this improvement comes with a higher computational cost which drastically increases with the number of uncertain reaction rates. To overcome this drawback an adaptive strategy is proposed to control locally (in the parameter space) and in time the resolution level. The efficiency of the method is demonstrated for an uncertain chemical system having eight random parameters.
Canuto, C. and Kozubek, T.. "A fictitious domain approach to the numerical solution of PDEs in stochastic domains" Numer. Math.. 107
(2).
2007.
pp. 257--293.
We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is âspectralâ in the polynomial chaos order, in any subdomain which does not contain the random boundaries.
Ghanem, R.G., Doostan, A., and Red-Horse, J.R. "A probabilistic construction of model validation" Computer Methods in Applied Mechanics and Engineering. 197
(29-32).
2008.
pp. 2585--2595.
We describe a procedure to assess the predictive accuracy of process models subject to approximation error and uncertainty. The proposed approach is a functional analysis-based probabilistic approach for which we represent random quantities using polynomial chaos expansions (PCEs). The approach permits the formulation of the uncertainty assessment in validation, a significant component of the process, as a problem of approximation theory. It has two essential parts. First, a statistical procedure is implemented to calibrate uncertain parameters of the candidate model from experimental or model-based measurements. Such a calibration technique employs PCEs to represent the inherent uncertainty of the model parameters. Based on the asymptotic behavior of the statistical parameter estimator, the associated PCE coefficients are then characterized as independent random quantities to represent epistemic uncertainty due to lack of information. Second, a simple hypothesis test is implemented to explore the validation of the computational model assumed for the physics of the problem. The above validation path is implemented for the case of dynamical system validation challenge exercise. (C) 2007 Elsevier B.V. All rights reserved.
Ghanem, R.G.. "Probabilistic characterization of transport in heterogeneous media" Computer Methods in Applied Mechanics and Engineering. 158
(3-4).
JUN 8 1998.
pp. 199--220.
The mechanics of transport and flow in a random porous medium are addressed in this paper. The hydraulic properties of the porous medium are modeled as spatial random processes. The random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented. Thus, the hydraulic processes are discretized using the spectral Karhunen - Loeve expansion. This expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes. These amplitudes form a basis in the manifold associated with the random processes. The concentrations over the whole domain are also random processes, with unknown probabilistic structure. These processes are represented using the Polynomial Chaos basis. This is a basis in the functional space described by all second order random variables. The deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator. Once the spatio-temporal variation of the concentrations has been specified in terms of the Polynomial Chaos expansion, individual realizations can be readily computed. (C) 1998 Elsevier Science S.A.
Ghanem, R.G. and Spanos, P.D.. "POLYNOMIAL CHAOS IN STOCHASTIC FINITE-ELEMENTS" Jounral of Applied Mechanics-transactions of the ASME. 57
(1).
MAR 1990.
pp. 197--202.
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.
Sakamoto, S and Ghanem, R.G.. "Polynomial chaos decomposition for the simulation of non-Gaussian nonstationary stochastic processes" Journal of Engineering Mechanics-ASCE. 128
(2).
FEB 2002.
pp. 190--201.
A method is developed for representing and synthesizing random processes that have been specified by their two-point correlation function and their nonstationary marginal probability density functions. The target process is represented as a polynomial transformation of an appropriate Gaussian process. The target correlation structure is decomposed according to the Karhunen-Loeve expansion of the underlying Gaussian process. A sequence of polynomial transformations in this process is then used to match the one-point marginal probability density functions. The method results in a representation of a stochastic process that is particularly well suited for implementation with the spectral stochastic finite element method as well as for general purpose simulation of realizations of these processes.
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.
Li, R. and Ghanem, R.G.. "Adaptive Polynomial Chaos expansions applied to statistics of extremes in nonlinear random vibration" Prob. Engrg. Mech.. 13
(2).
1998.
pp. 125--136.
This paper presents a new module towards the development of efficient computational stochastic mechanics. Specifically, the possibility of an adaptive polynomial chaos expansion is investigated. Adaptivity in this context refers to retaining, through an iterative procedure, only those terms in a representation of the solution process that are significant to the numerical evaluation of the solution. The technique can be applied to the calculation of statistics of extremes for nongaussian processes. The only assumption involved is that these processes be the response of a nonlinear oscillator excited by a general stochastic process. The proposed technique is an extension of a technique developed by the second author for the solution of general nonlinear random vibration problems. Accordingly, the response process is represented using its Karhunen-Loeve expansion. This expansion allows for the optimal encapsulation of the information contained in the stochastic process into a set of discrete random variables. The response process is then expanded using the polynomial chaos basis, which is a complete orthogonal set in the space of second-order random variables. The time dependent coefficients in this expansion are then computed by using a Galerkin projection scheme which minimizes the approximation error involved in using a finite-dimensional subspace. These coefficients completely characterize the solution process, and the accuracy of the approximation can be assessed by comparing the contribution of successive coefficients. A significant contribution of this paper is the development and implimentation of adaptive schemes for the polynomial chaos expansion. These schemes permit the inclusion of only those terms in the expansion that have a significant contribution. (C) 1997 Elsevier Science Ltd.
Ghanem, R.G. and Sarkar, A.. "Reduced models for the medium-frequency dynamics of stochastic systems" Journal of the Acoustical Society of America. 113
(2).
FEB 2003.
pp. 834--846.
In this paper, a frequency domain vibration analysis procedure of a randomly parametered structural system is described for the medium-frequency range. In this frequency range, both traditional modal analysis and statistical energy analysis (SEA) procedures well-suited for low- and high-frequency vibration analysis respectively, lead to computational and conceptual difficulties. The uncertainty in the structural system can be attributed to various reasons such as the coupling of the primary structure with a variety of secondary systems for which conventional modeling is not practical. The methodology presented in the paper consists of coupling probabilistic reduction methods with dynamical reduction methods. In particular, the Karhunen-Loeve and Polynomial Chaos decompositions of stochastic processes are coupled with an operator decomposition scheme based on the spectrum of an energy operator adapted to the frequency band of interest. (C) 2003 Acoustical Society of America.
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.
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.
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.
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.
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.
Ghanem, R.G. and Pellissetti, M.. "Adaptive data refinement in the spectral stochastic finite element method" Communications in Numerical Methods in Engineering. 18
(2).
FEB 2002.
pp. 141--151.
One version of the stochastic finite element method involves representing the solution with respect to a basis in the space of random variables and evaluating the co-ordinates of the solution with respect to this basis by relying on Hilbert space projections. The approach results in an explicit dependence of the solution on certain statistics of the data. The error in evaluating these statistics, which is directly related to the amount of available data, can be propagated into errors in computing probabilistic measures of the solution. This provides the possibility of controlling the approximation error, due to limitations in the data, in probabilistic statements regarding the performance of the system under consideration. In addition to this error associated with data resolution, is added the more traditional error, associated with mesh resolution. This latter also contributes to polluting the estimated probabilities associated with the problem. The present paper will develop the above concepts and indicate how they can be coupled in order to yield a more meaningful and useful measure of approximation error in a given problem.
Doostan, A., Ghanem, R.G., and Red-Horse, J.R. "Stochastic model reduction for chaos representations" Computer Methods in Applied Mechanics and Engineering. 196
(37-40).
2007.
pp. 3951--3966.
This paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic Galerkin schemes for the solution of partial differential equations with stochastic coefficients. In particular, an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation (SPDE) in the Hilbert space spanned by Wiener chaos. A description of the stochastic aspect of the problem on two well-separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale. With such algorithms at hand, the solution of SPDE's becomes both computationally manageable and efficient. Moreover, a solid foundation is thus provided for the adaptive error control in stochastic Galerkin procedures. Different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics.
Ghanem, R.G, Saad, G., and Doostan, A.. "Efficient solution of stochastic systems: Application to the embankment dam problem" Structural Safety. 29
(3).
2007.
pp. 238--251.
The embankment dam problem of the benchmark study is treated using the newly developed Stochastic Model Reduction for Polynomial Chaos Representations method. The elastic and shear moduli of the material, in the present problem, are modeled as two stochastic processes that are explicit functions of the same process possessing a relatively low correlation length. The state of the system can thus be viewed as a function defined on a high dimensional space, associated with the fluctuations of the underlying process. In such a setting, the spectral stochastic finite element method (SSFEM) for the specified spatial discretization is computationally prohibitive. The approach adopted in this paper enables the stochastic characterization of a fine mesh problem based on the high dimensional polynomial chaos solution of a coarse mesh analysis. After relatively reducing the dimensionality of the problem through a Karhunen-Loeve representation of the stochastic variables, the SSFEM solution consisting of a high dimensional polynomial in Gaussian independent variables is obtained for the coarse mesh problem. Then the attained solution is used to define a new basis for solving the fine mesh problem. The paper presents some new algorithms for the estimation of chaos coefficients in the presence of complex non-Gaussian dependencies. A numerical convergence study is presented together with a discussion of the results.
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.
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., Xiu, D., Su, C.-H., and Karniadakis, G.E.. "Predictability and uncertainty in CFD" International Journal for Numerical Methods in Fluids. 43
(5).
OCT 20 2003.
pp. 483--505.
CFD has reached some degree of maturity today, but the new question is how to construct simulation error bars that reflect uncertainties of the physical problem, in addition to the usual numerical inaccuracies. We present a fast Polynomial Chaos algorithm to model the input uncertainty and its propagation in incompressible flow simulations. The stochastic input is represented spectrally by Wiener-Hermite functionals, and the governing equations are formulated by employing Galerkin projections. The resulted system is deterministic, and therefore existing solvers can be used in this new context of stochastic simulations. The algorithm is applied to a second-order oscillator and to a flow-structure interaction problems. Open issues and extensions to general random distributions are presented. Copyright (C) 2003 John Wiley Sons, Ltd.
Ganapathysubramanian, B. and Zabaras, N.. "Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method" Journal of Computational Physics. 226
(1).
SEP 10 2007.
pp. 326--353.
In recent years, there has been intense interest in understanding various physical phenomena in random heterogeneous media. Any accurate description/simulation of a process in such media has to satisfactorily account for the twin issues of randomness as well as the multilength scale variations in the material properties. An accurate model of the material property variation in the system is an important prerequisite towards complete characterization of the system response. We propose a general methodology to construct a data-driven, reduced-order model to describe property variations in realistic heterogeneous media. This reduced-order model then serves as the input to the stochastic partial differential equation describing thermal diffusion through random heterogeneous media. A decoupled scheme is used to tackle the problems of stochasticity and multilength scale variations in properties. A sparse-grid collocation strategy is utilized to reduce the solution of the stochastic partial differential equation to a set of deterministic problems. A variational multiscale method with explicit subgrid modeling is used to solve these deterministic problems. An illustrative example using experimental data is provided to showcase the effectiveness of the proposed methodology. (C) 2007 Elsevier Inc. All rights reserved.
Xiu, D. and Shen, J.. "An efficient spectral method for acoustic scattering from rough surfaces" Communicationns in Computational Physcis. 2
(1).
FEB 2007.
pp. 54--72.
An efficient and accurate spectral method is presented for scattering problems with rough surfaces. A probabilistic framework is adopted by modeling the surface roughness as random process. An improved boundary perturbation technique is employed to transform the original Helmholtz equation in a random domain into a stochastic Helmholtz equation in a fixed domain. The generalized polynomial chaos (gPC) is then used to discretize the random space; and a Fodrier-Legendre method to discretize the physical space. These result in a highly efficient and accurate spectral algorithm for acoustic scattering from rough surfaces. Numerical examples are presented to illustrate the accuracy and efficiency of the present algorithm.
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.
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.
Sudret, B.. "Global sensitivity analysis using polynomial chaos expansions" Reliability Engineering & System Safety. 93
(7).
JUL 2008.
pp. 964--979.
Global sensitivity analysis (SA) aims at quantifying the respective effects of input random variables (or combinations thereof) onto the variance of the response of a physical or mathematical model. Among the abundant literature on sensitivity measures, the Sobol' indices have received much attention since they provide accurate information for most models. The paper introduces generalized polynomial chaos expansions (PCE) to build surrogate models that allow one to compute the Sobol' indices analytically as a post-processing of the PCE coefficients. Thus the computational cost of the sensitivity indices practically reduces to that of estimating the PCE coefficients. An original non intrusive regression- based approach is proposed, together with an experimental design of minimal size. Various application examples illustrate the approach, both from the field of global SA (i.e. well-known benchmark problems) and from the field of stochastic mechanics. The proposed method gives accurate results for various examples that involve up to eight input random variables, at a computational cost which is 2-3 orders of magnitude smaller than the traditional Monte Carlo-based evaluation of the Sobol' indices. (C) 2007 Elsevier Ltd. All rights reserved.
Witteveen, J.A.S., Loeven, A., Sarkar, S., and Bijl, H.. "Probabilistic collocation for period-1 limit cycle oscillations" Journal of Sound and Vibration. 311
(1-2).
MAR 18 2008.
pp. 421--439.
In this paper probabilistic collocation for limit cycle oscillations (PCLCO) is proposed. Probabilistic collocation (PC) is a non-intrusive approach to compute the polynomial chaos description of uncertainty numerically. Polynomial chaos can require impractical high orders to approximate long-term time integration problems, due to the fast increase of required polynomial chaos order with time. PCLCO is a PC formulation for modeling the long-term stochastic behavior of dynamical systems exhibiting a periodic response, i.e. a limit cycle oscillation (LCO). In the PC method deterministic time series are computed at collocation points in probability space. In PCLCO, PC is applied to a time-independent parametrization of the periodic response of the deterministic solves instead of to the time-dependent functions themselves. Due to the time-independent parametrization the accuracy of PCLCO is independent of time. The approach is applied to period-1 oscillations with one main frequency subject to a random parameter. Numerical results are presented for the harmonic oscillator, a two-dof airfoil flutter model and the fluid-structure interaction of an elastically mounted cylinder. (C) 2007 Elsevier Ltd. All rights reserved.
Paffrath, M. and Wever, U.. "Adapted polynomial chaos expansion for failure detection" Journal of Computational Physics. 226
(1).
SEP 10 2007.
pp. 263--281.
In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator-prey model and a chemical reaction problem. (C) 2007 Elsevier Inc. All rights reserved.
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.
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.
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.
Xiu, D.. "Fast numerical methods for robust optimal design" Engineering Optimization. 40
(6).
2008.
pp. 489--504.
A fast numerical approach for robust design optimization is presented. The core of the method is based on the state-of-the-art fast numerical methods for stochastic computations with parametric uncertainty. These methods employ generalized polynomial chaos (gPC) as a high-order representation for random quantities and a stochastic Galerkin (SG) or stochastic collocation (SC) approach to transform the original stochastic governing equations to a set of deterministic equations. The gPC-based SG and SC algorithms are able to produce highly accurate stochastic solutions with (much) reduced computational cost. It is demonstrated that they can serve as efficient forward problem solvers in robust design problems. Possible alternative definitions for robustness are also discussed. Traditional robust optimization seeks to minimize the variance (or standard deviation) of the response function while optimizing its mean. It can be shown that although variance can be used as a measure of uncertainty, it is a weak measure and may not fully reflect the output variability. Subsequently a strong measure in terms of the sensitivity derivatives of the response function is proposed as an alternative robust optimization definition. Numerical examples are provided to demonstrate the efficiency of the gPC-based algorithms, in both the traditional weak measure and the newly proposed strong measure.
Creamer, D.B.. "On closure schemes for polynomial chaos expansions of stochastic differential equations" Waves in Random and Complex Media. 18
(2).
MAY 2008.
pp. 197--218.
The propagation of waves in a medium having random inhomogeneities is studied using polynomial chaos (PC) expansions, wherein environmental variability is described by a spectral representation of a stochastic process and the wave field is represented by an expansion ill orthogonal random polynomials of the spectral components. A different derivation of this expansion is given using functional methods, resulting in a smaller set of equations determining the expansion coefficients, also derived by others. The connection with the PC expansion is new and provides insight into different approximation schemes for the expansion, which is in the correlation function, rather than the random variables. This separates the approximation to the wave function and the closure of the coupled equations (for approximating the chaos coefficients), allowing for approximation schemes other than the Usual PC truncation, e.g. by an extended Markov approximation. For small correlation lengths of the medium, low-order PC approximations provide accurate coefficients of ally order. This is different from the usual PC approximation, where, for example, the mean field might be well approximated while the wave function (which includes other coefficients) would not be. These ideas are illustrated in a geometrical optics problem for a medium with a simple correlation function.
Blatman, G. and Sudret, B.. "Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach" Comptes RendusS Mecanique. 336
(6).
JUN 2008.
pp. 518--523.
A method is proposed to build a sparse polynomial chaos (PC) expansion of a mechanical model whose input parameters are random. In this respect, an adaptive algorithm is described for automatically detecting the significant coefficients of the PC expansion. The latter can thus be computed by means of a relatively small number of possibly costly model evaluations, using a non-intrusive regression scheme (also known as stochastic collocation). The method is illustrated by a simple polynomial model, as well as the example of the deflection of a truss structure.
Augustin, F., Gilg, A., Paffrath, M., Rentrop, P., and Wever, U.. "A survey in mathematics for industry polynomial chaos for the approximation of uncertainties: Chances and limits" European Journal of Applied Mathematics. 19
(Part 2).
APR 2008.
pp. 149--190.
In technical applications, uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of Wiener (Amer. J. Math. 60(4),897-936, 1938) was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from academics and from industry. For each of the applications, chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.
Witteveen, J.A.S. and Bijl, H.. "Efficient quantification of the effect of uncertainties in advection-diffusion problems using polynomial chaos" Numerical Heat Transfer Part B-Fundamentals. 53
(5).
2008.
pp. 437--465.
Uncertainties in advection-diffusion heat transfer problems are modeled using polynomial chaos to increase the basic understanding of the effect of physical variability. The polynomial chaos method approximates the effect of uncertain parameters using a polynomial expansion in probability space. Since the computational work of an uncertainty analysis increases rapidly with the number of uncertain parameters to the equivalence of many deterministic simulations, strategies for efficient quantification of the effect of multiple uncertain parameters are needed. Three strategies are studied in this article. Results are presented for advection-diffusion problems of heat transfer in one-dimensional and two-dimensional pipe flows.
Ghosh, D. and Ghanem, R.G.. "Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions" International Journal for Numerical Methods is Engineering. 73
(2).
JAN 2008.
pp. 162--184.
Given their mathematical structure, methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory. This is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition. This enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non-continuous dependence of the solution on the stochastic parameters. In particular, given the polynomial structure of these approximations, the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated. An enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non-smooth functions by integrating a priori knowledge about their behaviour. The proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues. It is observed that through judicious selection of enrichment functions, the spectrum of such a random system can be more efficiently characterized, even for systems with closely spaced eigenvalues. Copyright (c) 2007 John Wiley & Sons, Ltd.
Gottlieb, D. and Xiu, D.. "Galerkin method for wave equations with uncertain coefficients" Communications in Computational Physics. 3
(2).
FEB 2008.
pp. 505--518.
Polynomial chaos methods (and generalized polynomial chaos methods) have been extensively applied to analyze PDEs that contain uncertainties. However this approach is rarely applied to hyperbolic systems. In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection. We consider a simple model of a scalar wave equation with random wave speed. We show that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. A consistent method of imposing the boundary conditions is proposed and its convergence is established. Numerical examples are presented to support the analysis.
Emery, A.F. and Bardot, D.. "Stochastic heat transfer in fins and transient cooling using polynomial chaos and wick products" Journal of Heat Transfer-Transactions of the AMSE. 129
(9).
SEP 2007.
pp. 1127--1133.
Stochastic heat transfer problems are often solved using a perturbation approach that yields estimates of mean values and standard deviations for properties and boundary conditions that are random variables. Methods based on polynomial chaos and Wick products can be used when the randomness is a random field or white noise to describe specific realizations and to determine the statistics of the response. Polynomial chaos is best suited for problems in which the properties are strongly correlated, while the Wick product approach is most effective for variables containing white noise components. A transient lumped capacitance cooling problem and a one-dimensional fin are analyzed by both methods to demonstrate their usefulness.
Ghanem, R.G. and Ghosh, D.. "Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition" International Journal for Numerical Methods is Engineering. 72
(4).
OCT 22 2007.
pp. 486--504.
A new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented. The eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis. An efficient method for estimating the coefficients with respect to this basis is proposed. The method uses a Galerkin-based approach by orthogonalizing the residual in the eigenvalue-eigenvector equation to the subspace spanned by the basis functions used for approximation. In this way, the stochastic problem is framed as a system of deterministic non-linear algebraic equations. This system of equations is solved using a Newton-Raphson algorithm. Although the proposed approach is not based on statistical sampling, the efficiency of the proposed method can be significantly enhanced by initializing the non-linear iterative process with a small statistical sample synthesized through a Monte Carlo sampling scheme. The proposed method offers a number of advantages over existing methods based on statistical sampling. First, it provides an approximation to the complete probabilistic description of the eigensolution. Second, it reduces the computational overhead associated with solving the statistical eigenvalue problem. Finally, it circumvents the dependence of the statistical solution on the quality of the underlying random number generator. Copyright (C) 2007 John Wiley & Sons, Ltd.
Rupert, C.P. and Miller, C.T.. "An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems" Journal of Computational Physics. 226
(2).
OCT 2007.
pp. 2175--2205.
We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method. (c) 2007 Elsevier Inc. All rights reserved.
Li, H. and Zhang, D.. "Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods" Water Resources Research. 43
(9).
SEP 2007.
An efficient method for uncertainty analysis of flow in random porous media is explored in this study, on the basis of combination of Karhunen-Loeve expansion and probabilistic collocation method (PCM). The random log transformed hydraulic conductivity field is represented by the Karhunen-Loeve expansion and the hydraulic head is expressed by the polynomial chaos expansion. Probabilistic collocation method is used to determine the coefficients of the polynomial chaos expansion by solving for the hydraulic head fields for different sets of collocation points. The procedure is straightforward and analogous to the Monte Carlo method, but the number of simulations required in PCM is significantly reduced. Steady state flows in saturated random porous media are simulated with the probabilistic collocation method, and comparisons are made with other stochastic methods: Monte Carlo method, the traditional polynomial chaos expansion (PCE) approach based on Galerkin scheme, and the moment-equation approach based on Karhunen-Loeve expansion (KLME). This study reveals that PCM and KLME are more efficient than the Galerkin PCE approach. While the computational efforts are greatly reduced compared to the direct sampling Monte Carlo method, the PCM and KLME approaches are able to accurately estimate the statistical moments and probability density function of the hydraulic head.
Lucor, D., Meyers, J., and Sagaut, P.. "Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos" Journal of Fluid Mechanics.
vol. 585.
AUG 25 2007.
pp. 255--279.
We address the sensitivity of large-eddy simulations (LES) to parametric uncertainty in the subgrid-scale model. More specifically, we investigate the sensitivity of the LES statistical moments of decaying homogeneous isotropic turbulence to the uncertainty in the Smagorinsky model free parameter C-s (i.e. the Smagorinsky constant). Our sensitivity methodology relies on the non-intrusive approach of the generalized Polynomial Chaos (gPC) method; the gPC is a spectral non-statistical numerical method well-suited to representing random processes not restricted to Gaussian fields. The analysis is carried out at Re-lambda=100 and for different grid resolutions and C-s distributions. Numerical predictions are also compared to direct numerical simulation evidence. We have shown that the different turbulent scales of the LES solution respond differently to the variability in C-s. In particular, the study of the relative turbulent kinetic energy distributions for different C-s distributions indicates that small scales are mainly affected by changes in the subgrid-model parametric uncertainty.
Huang, S., Mahadevan, S., and Rebba, R.. "Collocation-based stochastic finite element analysis for random field problems" Probabilstic Engineering Mechanics. 22
(2).
APR 2007.
pp. 194--205.
A stochastic response surface method (SRSM) which has been previously proposed for problems dealing only with random variables is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen-Loeve (K-L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box. SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM. (C) 2006 Elsevier Ltd. All rights reserved.
Todor, R.A. and Schwab, C.. "Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients" IMA J. of Numer. Anal.. 27
(2).
APR 2007.
pp. 232--261.
A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a(x, omega) in a bounded domain D subset of < Ropf >(d) is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x is an element of D) and stochastic (omega is an element of Omega) variables in a(x, omega) via Karhunen-Loeve or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, omega) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneously.
Kim, D., Debusschere, B.J., and Najm, H.N.. "Spectral methods for parametric sensitivity in stochastic dynamical systems" Biophyscial Journal. 92
(2).
JAN 2007.
pp. 379--393.
Stochastic dynamical systems governed by the chemical master equation find use in the modeling of biological phenomena in cells, where they provide more accurate representations than their deterministic counterparts, particularly when the levels of molecular population are small. The analysis of parametric sensitivity in such systems requires appropriate methods to capture the sensitivity of the system dynamics with respect to variations of the parameters amid the noise from inherent internal stochastic effects. We use spectral polynomial chaos expansions to represent statistics of the system dynamics as polynomial functions of the model parameters. These expansions capture the nonlinear behavior of the system statistics as a result of finite-sized parametric perturbations. We obtain the normalized sensitivity coefficients by taking the derivative of this functional representation with respect to the parameters. We apply this method in two stochastic dynamical systems exhibiting bimodal behavior, including a biologically relevant viral infection model.
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.
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
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.
Deb, M.K., Babuska, I.M., and Oden, J.T. "Solution of stochastic partial differential equations using Galerkin finite element techniques" Comput. Methods Appl. Mech. Engrg..
vol. 190.
2001.
pp. 6359--6372.
This paper presents a framework for the construction of Galerkin approximations of elliptic boundary-value problems with stochastic input data. A variational formulation is developed which allows, among others, numerical treatment by the finite element method; a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized. The paper develops a foundation for treating stochastic partial differential equations (PDEs) which can be further developed in many directions