Polynomial Chaos Decomposition Research Articles

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modeled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection–extraction problem.

Stochastic multi-scale modeling of carbon fiber reinforced composites with polynomial chaos

A stochastic multi-scale modeling framework for uncertainty quantification of carbon fiber reinforced composites with a non-intrusive method called Polynomial Chaos Decomposition with Differentiation (PCDD) is presented. The performance behavior and reliability of the composites are dependent on its constituents properties (fiber and matrix properties) in addition to ply-orientation, ply-thickness, and loading conditions; hence, the uncertainties are considered using two stages: i) micro-scale modeling, and ii) macro-scale modeling. In the first stage, stochastic micro-scale modeling with PCDD is carried out to obtain the effective material properties of a lamina that are influenced by the uncertainties in its constituents. Then, these stochastic effective material properties are considered along with the uncertainties in geometrical properties of the laminate such as ply thicknesses and ply orientation to determine the stochastic performance of the laminate. The framework was applied for multi-scale buckling analysis and reliability estimation. Another approach for polynomial chaos called Stochastic Point Collocation (COLL) was also studied, and the results obtained with PCDD and COLL were compared with a large number of Latin Hypercube Sampling simulations. The results demonstrated the computational superiority of the proposed framework to achieve high accuracy stochastic response models as well as invaluable information about composites using a stochastic approach.

A new non-intrusive polynomial chaos using higher order sensitivities

This paper proposes a new non-intrusive method for uncertainty quantification called Polynomial Chaos Decomposition with Differentiation (PCDD) that uses higher-order sensitivities of the response. In PCDD, the polynomial chaos expansion (PCE) of the response is differentiated with respect to the basis random variables using multi-indices. This differentiation results in a system of linear equations which can then be solved to determine the expansion coefficients. Here, the higher accuracy, Modified Forward Finite Difference (ModFFD) that involves representation of the response using Taylor expansion of order equal to the chaos-order is used in combination with PCE. Therefore, the total number of samples required with this method is equal to the number of terms in the PCE. To verify the validity of this new technique, two analytical problems and two stochastic composite laminate problems were studied. The results of the analytical problems showed that the accuracy of PCDD using ModFFD is similar to that of PCDD using analytical sensitivities, which in addition is comparable to the exact results. For composite laminate problems, the PCDD displayed very high accuracy comparable to 50,000 Latin Hypercube Samples, which underlines the computational efficiency of this proposed method.

Statistical moments of Gaussian kernel correlation sum and weighted least square estimator of correlation dimension and noise level

In this paper, using non-central chi-squared distribution and polynomial chaos decomposition of a log-normal random variable, we derive the exact expressions for the covariance and variance of the Gaussian kernel correlation sum (Gkcs). The obtained results are combined with U-statistics theory and non-linear models theory to construct the exact confidence intervals for the correlation dimension and for the noise level in the case where deterministic time series are corrupted by additive Gaussian noise. The theoretical results are tested on two continuous chaotic dynamics corrupted by Gaussian noise for different values of signal to noise ratio (SNR). An application to a real data time series has been also conducted.

Sensitivity analysis of parametric uncertainties and modeling errors in computational-mechanics models by using a generalized probabilistic modeling approach

Engineering analyses of structures may be confronted with many sources of uncertainty, which may be of different types, such as parametric uncertainties versus modeling errors, and which may pertain to different structural components when complex structures are analyzed. Soize, Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions, Int. J. Numer. Meth. Eng., 81:939–970, 2010 has recently introduced a generalized probabilistic modeling approach, which can individually represent parametric uncertainties and modeling errors and which can individually represent sources of uncertainty pertaining to different structural components of complex structures. In this paper, we propose to augment this generalized probabilistic modeling approach with a stochastic sensitivity analysis in order to quantify and gain insight into separate impacts of distinct sources of uncertainty on quantities of interest. We demonstrate the proposed methodology by applying it to two computational-mechanics models involving uncertainty.

Exposure assessment of one-year-old child to 3G tablet in uplink mode and to 3G femtocell in downlink mode using polynomial chaos decomposition

So far, the assessment of the exposure of children, in the ages 0–2 years old, to relatively new radio-frequency (RF) technologies, such as tablets and femtocells, remains an open issue. This study aims to analyse the exposure of a one year-old child to these two sources, tablets and femtocells, operating in uplink (tablet) and downlink (femtocell) modes, respectively. In detail, a realistic model of an infant has been used to model separately the exposures due to (i) a 3G tablet emitting at the frequency of 1940 MHz (uplink mode) placed close to the body and (ii) a 3G femtocell emitting at 2100 MHz (downlink mode) placed at a distance of at least 1 m from the infant body. For both RF sources, the input power was set to 250 mW. The variability of the exposure due to the variation of the position of the RF sources with respect to the infant body has been studied by stochastic dosimetry, based on polynomial chaos to build surrogate models of both whole-body and tissue specific absorption rate (SAR), which makes it easy and quick to investigate the exposure in a full range of possible positions of the sources. The major outcomes of the study are: (1) the maximum values of the whole-body SAR (WB SAR) have been found to be 9.5 mW kg−1 in uplink mode and 65 μW kg−1 in downlink mode, i.e. within the limits of the ICNIRP Guidelines; (2) in both uplink and downlink mode the highest SAR values were approximately found in the same tissues, i.e. in the skin, eye and penis for the whole-tissue SAR and in the bone, skin and muscle for the peak SAR; (3) the change in the position of both the 3G tablet and the 3G femtocell significantly influences the infant exposure.

Polynomial Chaos expansion for subsurface flows with uncertain soil parameters

The effects of uncertainty in hydrological laws are studied on subsurface flows modeled by Richards’ equation. The empirical parameters of the water content and the hydraulic conductivity are considered as uncertain inputs of the model. One-dimensional infiltration problems are treated and the influence of the variability of the input parameters on the position and the spreading of the wetting front is evaluated. A Polynomial Chaos (PC) expansion is used to represent the output quantities and permits to significantly reduce the number of simulations in comparison with a classical Monte-Carlo method. A non-intrusive spectral projection supplies the coefficients of the PC decomposition. Three test cases with different hydrological laws are presented and demonstrate that second order PC expansions are sufficient to represent our quantities of interest owing to smooth dependences for the considered problems. Our results show a correlation between the position and the spreading of the wetting front and an amplification of the input uncertainty for all models. For each test case, five configurations with variable initial saturation state are investigated. The global sensitivity analysis indicates that the relative influence of an input parameter changes according to the output quantity considered and the initial saturation of the soil. The impact of the assumed distributions for the parameters is also briefly illustrated.

Generalized polynomial chaos decomposition and spectral methods for the stochastic Stokes equations

In this paper we propose and analyze a high order numerical method to solve the Stokes equations with random coefficients. A stochastic Galerkin approach, based on a finite dimensional Karhunen–Loève decomposition technique for the stochastic inputs, is used to reduce the original stochastic Stokes equations into a set of deterministic equations for the expansion coefficients. Then a PN×PN-2 spectral method, together with a block Jacobi iteration is applied to solve the resulting problem. We establish the well-posedness of the weak formulation and its discrete counterpart. Moreover, we provide a rigorous convergence analysis and demonstrate exponential convergence with respect to the degrees of the polynomials chaos expansion used for the approximation in the random direction. Finally, a series of numerical tests is presented to support the theoretical results.

Sensitivity study of dynamic systems using polynomial chaos

Global sensitivity has mainly been analyzed in static models, though most physical systems can be described by differential equations. Very few approaches have been proposed for the sensitivity of dynamic models and the only ones are local. Nevertheless, it would be of great interest to consider the entire uncertainty range of parameters since they can vary within large intervals depending on their meaning. Other advantage of global analysis is that the sensitivity indices of a given parameter are evaluated while all the other parameters can be varied. In this way, the relative variability of each parameter is taken into account, revealing any possible interactions. This paper presents the global sensitivity analysis for dynamic models with an original approach based on the polynomial chaos (PC) expansion of the output. The evaluation of the PC expansion of the output is less expensive compared to direct simulations. Moreover, at each time instant, the coefficients of the PC decomposition convey the parameter sensitivity and then a sensitivity function can be obtained. The PC coefficients are determined using non-intrusive methods. The proposed approach is illustrated with some well-known dynamic systems.

Stochastic Uncertainty Quantification of Eddy Currents in the Human Body by Polynomial Chaos Decomposition

The finite element method can be used to compute the electromagnetic fields induced in the human body by environmental extremely low frequency (ELF) fields. However, the electric properties of tissues are not precisely known and may vary depending on the individual, his/her age and other physiological parameters. In this paper, we account for the uncertainties on the conductivities of the brain tissues and spread them out to the induced fields by means of a nonintrusive approach based on Hermite polynomial chaos, with the finite element method as a black box. After showing the convergence of the method, we compute the probability to be over the thresholds defined by the international guidelines for limiting exposure to electromagnetic fields published by ICNIRP.

Stochastic data assimilation of the random shallow water model loads with uncertain experimental measurements

This paper is concerned with the estimation of a parametric probabilistic model of the random displacement source field at the origin of seaquakes in a given region. The observation of the physical effects induced by statistically independent realizations of the seaquake random process is inherent with uncertainty in the measurements and a stochastic inverse method is proposed to identify each realization of the source field. A statistical reduction is performed to drastically lower the dimension of the space in which the random field is sought and one is left with a random vector to identify. An approximation of the vector components is determined using a polynomial chaos decomposition, solution of an optimality system to identify an optimal representation. A second order gradient-based optimization technique is used to efficiently estimate this statistical representation of the unknown source while accounting for the non-linear constraints in the model parameters. This methodology allows the uncertainty associated with the estimates to be quantified and avoids the need for repeatedly solving the forward model.

Some aspects of probabilistic modeling, identification and propagation of uncertainties in computational mechanics

In this paper, we present some aspects relative to the types of uncertainties, the variability of real systems, the types of probabilistic approaches and of the representations for the probabilistic models of uncertainties, the construction of the probabilistic models using the maximum entropy principle. We then present the nonparametric probabilistic approach of uncertainties for elliptic problems, for 3D continuous dynamical systems with geometrical nonlinearities induced by large displacements and for low- and mediumfrequency vibroacoustics of a complex system with experimental validations. Finally, a generalized probabilistic approach of uncertainties in computational dynamics using the random matrix theory and polynomial chaos decompositions is presented.

Stochastic Uncertainty Quantification of the Conductivity in EEG Source Analysis by Using Polynomial Chaos Decomposition

The electroencephalogram (EEG) is one of the techniques used for the non-invasive diagnosis of patients suffering from epilepsy. EEG source localization identifies the neural activity, starting from measured EEG. This numerical localization procedure has a resolution, which is difficult to determine due to uncertainties in the EEG forward models. More specifically, the conductivities of the brain and the skull in the head models are not precisely known. In this paper, we propose the use of a non-intrusive stochastic method based on a polynomial chaos decomposition for quantifying the possible errors introduced by the uncertain conductivities of the head tissues. The accuracy and computational advantages of this non-intrusive method for EEG source analysis is illustrated. Further, the method is validated by means of Monte Carlo simulations.

Polynomial chaos expansions for optimal control of nonlinear random oscillators

The polynomial chaos decomposition of stochastic variables and processes is implemented in conjunction with optimal polynomial control of nonlinear dynamical systems. The procedure is demonstrated on a base-excited system whereby ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its Karhunen–Loeve expansion. An adaptive scheme for stochastic approximation with polynomial chaos bases is proposed which is based on a displacement–velocity norm and is applied to the identification of phase orbits of nonlinear oscillators. This approximation is then integrated in the design of an optimal polynomial controller, allowing for the efficient estimation of statistics and probability density functions of quantities of interest. Numerical investigations are carried out employing the polynomial chaos expansions and the Lyapunov asymptotic stability condition based control policy. The results reveal that the performance, as gaged by probabilistic quantities of interest, of the controlled oscillators is greatly improved. A comparative study is also presented against the classical stochastic optimal control, whereby statistical linearization based LQG is employed to design the optimal controller. It is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators.

Some aspects of probabilistic modeling, identification and propagation of uncertainties in computational mechanics

In this paper, we present some aspects relative to the types of uncertainties, the variability of real systems, the types of probabilistic approaches and of the representations for the probabilistic models of uncertainties, the construction of the probabilistic models using the maximum entropy principle. We then present the nonparametric probabilistic approach of uncertainties for elliptic problems, for 3D continuous dynamical systems with geometrical nonlinearities induced by large displacements and for low- and medium-frequency vibroacoustics of a complex system with experimental validations. Finally, a generalized probabilistic approach of uncertainties in computational dynamics using the random matrix theory and polynomial chaos decompositions is presented.

Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions

AbstractA new generalized probabilistic approach of uncertainties is proposed for computational model in structural linear dynamics and can be extended without difficulty to computational linear vibroacoustics and to computational non‐linear structural dynamics. This method allows the prior probability model of each type of uncertainties (model‐parameter uncertainties and modeling errors) to be separately constructed and identified. The modeling errors are not taken into account with the usual output‐prediction‐error method, but with the nonparametric probabilistic approach of modeling errors recently introduced and based on the use of the random matrix theory. The theory, an identification procedure and a numerical validation are presented. Then a chaos decomposition with random coefficients is proposed to represent the prior probabilistic model of random responses. The random germ is related to the prior probability model of model‐parameter uncertainties. The random coefficients are related to the prior probability model of modeling errors and then depends on the random matrices introduced by the nonparametric probabilistic approach of modeling errors. A validation is presented. Finally, a future perspective is introduced when experimental data are available. The prior probability model of the random coefficients can be improved in constructing a posterior probability model using the Bayesian approach. Copyright © 2009 John Wiley & Sons, Ltd.

Characterization of reservoir simulation models using a polynomial chaos‐based ensemble Kalman filter

Model‐based predictions of flow in porous media are critically dependent on assumptions and hypotheses that are not always based on first principles and that cannot necessarily be justified on the basis of known prevalent physics. Constitutive models, for instance, fall under this category. While these predictive tools are usually calibrated using observational data, the scatter in the resulting parameters has typically been ignored. In this paper, this scatter is used to construct stochastic process models of the parameters which are then used as the cornerstone in a novel model validation methodology useful for ascertaining the confidence in model‐based predictions. The uncertainties are first quantified by representing the unknown model parameters via their polynomial chaos decompositions. These are descriptions of stochastic processes in terms of their coordinates with respect to an orthogonal basis. This is followed by a filtering step to update these representations with measurements as they become available. In order to account for the non‐Gaussian nature of model parameters and model errors, an adaptation of the ensemble Kalman filter is developed. Instead of propagating an ensemble of model states forward in time as is suggested within the framework of the ensemble Kalman filter, the proposed approach allows the propagation of a stochastic representation of unknown variables using their respective polynomial chaos decompositions. The model is propagated forward in time by solving the system of partial differential equations using a stochastic projection method. Whenever measurements are available, the proposed data assimilation technique is used to update the stochastic parameters of the numerical model. The proposed method is applied to a black oil reservoir simulation model where measurements are used to stochastically characterize the flow medium and to verify the model validity with specified confidence bounds. The updated model can then be employed to forecast future flow behavior.

Reduced Chaos decomposition with random coefficients of vector-valued random variables and random fields

We develop a stochastic functional representation that is adapted to problems involving various forms of epistemic uncertainties including modeling error and data paucity. The new representation builds on the polynomial Chaos decomposition and eventually yields a Karhunen–Loeve expansion with random multiplicative coefficients. In this expansion, one set of uncertainty is captured in the usual manner, as uncorrelated scalar random variables. Another component of the uncertainty, statistically independent from the first, is captured by constructing the, usually deterministic, functions in the KL expansion as random functions. We think of the first set of uncertainties as associated with a coarse scale model, and of the second set as associated with subscale fluctuations not captured in the coarse scale description.

Identification of random shapes from images through polynomial chaos expansion of random level set functions

AbstractIn this paper, an efficient method is proposed for the identification of random shapes in a form suitable for numerical simulation within the extended stochastic finite element method (X‐SFEM). The method starts from a collection of images representing different outcomes of the random shape to identify. The key point of the method is to represent the random geometry in an implicit manner using the level set technique. In this context, the problem of random geometry identification is equivalent to the identification of a random level set function, which is a random field. This random field is represented on a polynomial chaos (PC) basis and various efficient numerical strategies are proposed in order to identify the coefficients of its PC decomposition. The performance of these strategies is evaluated through some ‘manufactured’ problems and useful conclusions are provided. The propagation of geometrical uncertainties in structural analysis using the X‐SFEM is finally examined. Copyright © 2009 John Wiley & Sons, Ltd.

Risk Analysis of Structures in Presence of Stochastic Fields of Deterioration: Flowchart for Coupling Inspection Results and Structural Reliability

Inspection by non-destructive testing techniques of existing structures is not perfect and it has become a common practice to model their reliability in terms of receiver operating characteristic (ROC) curves. This paper suggests a method of building ROC curves in case of random fields of defects on a structure by using polynomial chaos decomposition. Knowledge of spatial distribution of defects allows a reliability analysis to be performed. When selecting stochastic finite element analysis to solve this problem, the format is the same as the one chosen for modelling inspections results. The paper shows how to link these quantities (ie. reliability and inspection results) in a risk analysis by using polynomial chaos decomposition as a common language.