Quantifying Uncertainty
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.