Quantifying Uncertainty
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.
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.
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.
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.
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.
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
Nobile, F., Tempone, R., and Webster, C.G.. "A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data" SIAM J. Numer. Anal.. 46
(5).
2008.
pp. 2309--2345.
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. BabuËka, R. Tempone, and G. E. Zouraris, J. Numer. Anal., 42 (2004), pp. 800â825] 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 âprobability errorâ 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.
Keywords: collocation techniques ; stochastic PDEs ; finite elements ; uncertainty quantification ; sparse grids ; Smolyak approximation ; multivariate polynomial approximation
Ding, Y., Li, T., Zhang, D., and Zhang, P.. "Adaptive Stroud Stochastic Collocation Method for Flow in Random Porous Media via Karhunen-Loeve Expansion" Communications in Comp. Phys.. 4
(1).
2008.
pp. 102--123.
In this paper we develop a Stochastic Collocation Method (SCM) for flow in randomly heterogeneous porous media. At first, the Karhunen-Lo\`eve expansion is taken to decompose the log transformed hydraulic conductivity field, which leads to a stochastic PDE that only depends on a finite number of i.i.d. Gaussian random variables. Based on the eigenvalue decay property and a rough error estimate of Stroud cubature in SCM, we propose to subdivide the leading dimensions in the integration space for random variables to increase the accuracy. We refer to this approach as \it adaptive Stroud SCM. One- and two-dimensional steady-state single phase flow examples are simulated with the new method, and comparisons are made with other stochastic methods, namely, the Monte Carlo method, the tensor product SCM, and the quasi-Monte Carlo SCM. The results indicate that the adaptive Stroud SCM is more efficient and the statistical moments of the hydraulic head can be more accurately estimated.