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.
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.
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.
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.
Balakrishnan, S., Georgopoulos, P., Banerjee, I., and Ierapetritou, M.. "Uncertainty considerations for describing complex reaction systems" AICHE Jorunal. 48
(12).
DEC 2002.
pp. 2875--2889.
Models that accurately describe chemical processes are often intricate involving numerous reacting species and reaction steps. For complex reaction mechanisms, output-species concentration profiles can change dramatically based on the set of values chosen for inputs if they are nondeterministic. A systematic uncertainty analysis can provide insight into the level of confidence of model estimates and aid mechanism reduction. Response surface methods and variants, thereof, require much fewer simulations for the adequate estimation of system uncertainty characteristics. This article focuses on reaction rate constant uncertainty using the stochastic response surface method (SRSM), whereby, uncertain outputs are expressed in terms of a polynomial chaos expansion of Hermite polynomials and engenders such useful properties as the mean and valiance and computation of sensitivity information. SRSM determines the uncertainty propagation characteristics very accurately, while using an order-of-magnitude fewer model simulations than traditional Monte Carlo techniques. Since uncertainty in kinetic rate parameters largely affects the reduction of kinetic models, a framework of analysis is also developed for mechanism reduction considering uncertainty using sensitivity information from SRSM to create good initial sets of reactions for the efficient solution of a multi-period optimization problem. Two case studies-an isothermal supercritical wet oxidation process and a nonisothermal H-2/CO/air combustion process-elucidate the application of this framework of analysis to complex kinetic mechanisms and illustrate the possible ease of computational burden associated with mechanism reduction under uncertainty.