Roger Ghanem

Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure

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.

Ingredients for a general purpose stochastic finite elements implementation

Abstract The Stochastic Finite Element Method has seen a number of incarnations over the past few years, ranging from perturbation-based methods, and Neumann-expansion based methods, to methods for evaluating bounds on the response variability, and more recent functional expansions combined with error minimization techniques. Of these, only the last class of methods provides an explicit expression to the solution of the problem, as opposed to evaluating a global norm of this solution, and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method. The main obstacle in developing a substantial user-community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest. This paper presents the main ingredients for developing a general purpose version of the Spectral Stochastic Finite Element Method.

Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes

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.

Polynomial Chaos in Stochastic Finite Elements

A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of a convergent orthogonal expansion of the process. The solution process is viewed as an element in the Hilbert space of random functions, in which a sequence of projection operators is identified as the polynomial chaos of consecutive orders. Thus, the solution process is represented by its projections onto the spaces spanned by these polynomials

Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection

Abstract A spectral formalism has been developed for the “non-intrusive” analysis of parametric uncertainty in reacting-flow systems. In comparison to conventional Monte Carlo analysis, this method quantifies the extent, dependence, and propagation of uncertainty through the model system and allows the correlation of uncertainties in specific parameters to the resulting uncertainty in detailed flame structure. For the homogeneous ignition chemistry of a hydrogen oxidation mechanism in supercritical water, spectral projection enhances existing Monte Carlo methods, adding detailed sensitivity information to uncertainty analysis and relating uncertainty propagation to reaction chemistry. For 1-D premixed flame calculations, the method quantifies the effect of each uncertain parameter on total uncertainty and flame structure, and localizes the effects of specific parameters within the flame itself. In both 0-D and 1-D examples, it is clear that known empirical uncertainties in model parameters may result in large uncertainties in the final output. This has important consequences for the development and evaluation of combustion models. This spectral formalism may be extended to multidimensional systems and can be used to develop more efficient “intrusive” reformulations of the governing equations to build uncertainty analysis directly into reacting flow simulations.

Numerical solution of spectral stochastic finite element systems

Abstract This paper addresses the issues involved in solving systems of linear equations which arise in the context of the spectral stochastic finite element (SSFEM) formulation. Two efficient solution procedures are presented that dramatically reduce the amount of computations involved in numerically solving these problems. A brief review is first provided of the underlying spectral approach which highlights the peculiar structure of the matrices generated and how their properties are related to both the level of approximation involved as well as to the convergence behavior of the proposed solution procedure. The differences between these matrices from their deterministic finite element counterparts are illustrated. An iterative solution scheme is proposed, which utilizes their specific properties for efficient memory management and enhanced convergence behavior. Results from numerical tests are presented. Comparisons with standard algorithms illustrate the efficiency of the proposed algorithm. The second solution procedure presented in this paper is based on hierarchical basis concepts. Results from numerical tests are again provided, and the limitations of this approach are assessed. The performance of both proposed algorithms indicates that the linear algebraic systems from the underlying SSFEM formulation can be solved with considerably less effort in memory and computation time than their size suggests. Furthermore, the data structures and the hierarchical concept introduced in this study are found to have great potential for the future development of adaptive procedures in stochastic FEM.

On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data

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.

Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach

Abstract 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 Karhunen–Loeve 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.

Probabilistic characterization of transport in heterogeneous media

Abstract 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.

Stochastic finite element analysis for multiphase flow in heterogeneous porous media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.