Quantifying Uncertainty
Xiu, D. and Hesthaven, J.S.. "High-order collocation methods for differential equations with random inputs" SIAM J. Sci. Comput.. 27
(3).
2005.
pp. 1118--1139.
Mathelin, L., Hussaini, M., and Zang, T.. "Stochastic Approaches to Uncertainty Quantification in CFD Simulations" Num. Alg.. 38
(1-3).
2005.
pp. 209--236.
This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.
Babuska, I.M., Tempone, R., and Zouraris, G.E.. "Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation" Computer Methods in Applied Mechanics and Engineering.
vol. 194.
2005.
pp. 1251--1294.
Babuska, I.M., Nobile, F., and Tempone, R.. "A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data" SIAM J. Numer. Anal.. 45
(3).
2007.
pp. 1005--1034.
In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms ( input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in I. Babuska, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825 and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the ``probability error'' with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.
Xiu, D. and Sherwin, S.J.. "Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network" Journal of Computational Physics. 226
(2).
2007.
pp. 1385--1407.
Reduced models of human arterial networks are an efficient approach to analyze quantitative macroscopic features of human arterial flows. The justification for such models typically arise due to the significantly long wavelength associated with the system in comparison to the lengths of arteries in the networks. Although these types of models have been employed extensively and many issues associated with their implementations have been widely researched, the issue of data uncertainty has received comparatively little attention. Similar to many biological systems, a large amount of uncertainty exists in the value of the parameters associated with the models. Clearly reliable assessment of the system behaviour cannot be made unless the effect of such data uncertainty is quantified. In this paper we present a study of parametric data uncertainty in reduced modelling of human arterial networks which is governed by a hyperbolic system. The uncertain parameters are modelled as random variables and the governing equations for the arterial network therefore become stochastic. This type stochastic hyperbolic systems have not been previously systematically studied due to the difficulties introduced by the uncertainty such as a potential change in the mathematical character of the system and imposing boundary conditions. We demonstrate how the application of a high-order stochastic collocation method based on the generalized polynomial chaos expansion, combined with a discontinuous Galerkin spectral/hp element discretization in physical space, can successfully simulate this type of hyperbolic system subject to uncertain inputs with bounds. Building upon a numerical study of propagation of uncertainty and sensitivity in a simplified model with a single bifurcation, a systematical parameter sensitivity analysis is conducted on the wave dynamics in a multiple bifurcating human arterial network. Using the physical understanding of the dynamics of pulse waves in these types of networks we are able to provide an insight into the results of the stochastic simulations, thereby demonstrating the effects of uncertainty in physiologically accurate human arterial networks.
Brutman, L.. "Lebesgue functions for polynomial interpolationâa survey" Annals of Numerical Mathematicss.
vol. 4.
1997.
pp. 111--127.
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.
Huang, S. and Kou, X.. "An extended stochastic response surface method for random field problems" Acta Mechanica Sinica. 23
(4).
AUG 2007.
pp. 445--450.
An efficient and accurate uncertainty propagation methodology for mechanics problems with random fields is developed in this paper. This methodology is based on the stochastic response surface method (SRSM) which has been previously proposed for problems dealing with random variables only. This paper extends SRSM to problems involving random fields or random processes fields. The favorable property of SRSM lies in that the deterministic computational model can be treated as a black box, as in the case of commercial finite element codes. Numerical examples are used to highlight the features of this technique and to demonstrate the accuracy and efficiency of the proposed method. A comparison with Monte Carlo simulation shows that the proposed method can achieve numerical results close to those from Monte Carlo simulation while dramatically reducing the number of deterministic finite element runs.
Hossain, F., Anagnostou, E.N., and Lee, K.H.. "A non-linear and stochastic response surface method for Bayesian estimation of uncertainty in soil moisture simulation from a land surface model" Nonlinear processes in Geophysics. 11
(4).
2004.
pp. 427--440.
This study presents a simple and efficient scheme for Bayesian estimation of uncertainty in soil moisture simulation by a Land Surface Model (LSM). The scheme is assessed within a Monte Carlo (MC) simulation framework based on the Generalized Likelihood Uncertainty Estimation (GLUE) methodology. A primary limitation of using the GLUE method 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 non-linear deterministic behavior between soil moisture and land surface parameters. Uncertainty in soil moisture simulation (model output) is approximated through a Hermite polynomial chaos expansion of normal random variables that represent the model's parameter (model input) uncertainty. The unknown coefficients of the polynomial are calculated using limited number of model simulation runs. The calibrated polynomial is then used as a fast-running proxy to the slower-running LSM to predict the degree of representativeness of a randomly sampled model parameter set. An evaluation of the scheme's efficiency in sampling is made through comparison with the fully random MC sampling (the norm for GLUE) and the nearest-neighborhood sampling technique. The scheme was able to reduce computational burden of random MC sampling for GLUE in the ranges of 10\%-70\%. The scheme was also found to be about 10\% more efficient than the nearest-neighborhood sampling method in predicting a sampled parameter set's degree of representativeness. The GLUE based on the proposed sampling scheme did not alter the essential features of the uncertainty structure in soil moisture simulation. The scheme can potentially make GLUE uncertainty estimation for any LSM more efficient as it does not impose any additional structural or distributional assumptions.
Anile, A.M., Spinella, S., and Rinaudo, S.. "Stochastic response surface method and tolerance analysis in microelectronics" COMPEL-The International Journal for Computation and Mathematics in Electrical and Electronic Engineering. 22
(2).
2003.
pp. 314--327.
Tolerance analysis is a very important tool for chip design in the microelectronics industry. The usual method for tolerance analysis is Monte Carlo simulation, which, however, is extremely CPU intensive, because in order to yield statistically significant results, it needs to generate a large sample of function values. Here we report on another method, recently introduced in several fields, caged stochastic response surface method, which might be a viable alternative to Monte Carlo simulation for some classes of problems. The application considered here is on the tolerance analysis of the current of a submicrometer n(+)-n-n(+) diode as a function of the channel length and the channel doping. The numerical simulator for calculating the current is based on the energy transport hydrodynamical model introduced by Stratton, which is one of the most widely used in this field.
Isukapalli, S.S., Roy, A., and Georgopoulos, P.G.. "Efficient sensitivity/uncertainty analysis using the combined stochastic response surface method and automated differentiation: Application to environmental and biological systems" Risk Analysis. 20
(5).
OCT 2000.
pp. 591--602.
Estimation of uncertainties associated with model predictions is an important component of the application of environmental and biological models. ``Traditional'' methods for propagating uncertainty, such as standard Monte Carlo and Latin Hypercube Sampling, however, often require performing a prohibitive number of model simulations, especially for complex, computationally intensive models. Here, a computationally efficient method for uncertainty propagation, the Stochastic Response Surface Method (SRSM) is coupled with another method, the Automatic Differentiation of FORTRAN (ADIFOR). The SRSM is based on series expansions of model inputs and outputs in terms of a set of ``well-behaved'' standard random variables. The ADIFOR method is used to transform the model code into one that calculates the derivatives of the model outputs with respect to inputs or transformed inputs. The calculated model outputs and the derivatives at a set of sample points are used to approximate the unknown coefficients in the series expansions of outputs. A framework for the coupling of the SRSM and ADIFOR is developed and presented here. Two case studies are presented, involving (1) a physiologically based pharmacokinetic model for perchloroethylene for humans, and (2) an atmospheric photochemical model, the Reactive Plume Model. The results obtained agree closely with those of traditional Monte Carlo and Latin hypercube sampling methods, while reducing the required number of model simulations by about two orders of magnitude.
Nobile, F., Tempone, R., and Webster, C.G.. "An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data" SIAM J. Numer. Anal.. 46
(5).
2008.
pp. 2411--2442.
This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms (input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either ClenshawâCurtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are KarhunenâLo`ve truncations of âsmoothâ random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. It also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large-dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.
Keywords: collocation techniques ; PDEs with random data ; differential equations ; finite elements ; uncertainty quantification ; anisotropic sparse grids ; Smolyak sparse approximation ; multivariate polynomial approximation