gPC Expansion Research Articles

Stochastic Analysis of the Impact of Substrate Compression on the Performance of Textile Antennas

One of the many adverse effects modifying the performance of textile antennas in real operating conditions is substrate compression. Therefore, this communication presents a stochastic collocation method (SCM) that either relies on the generalized polynomial chaos (gPC) expansion or on a novel Hermite-Pade approximant. The method is introduced to rigorously quantify the effect of random variations in the height and the permittivity of the substrate on the figures of merit of a textile antenna. Next, the joint height and permittivity probability distribution of a compressible substrate are characterized by means of a new measurement setup based on a resonant-perturbation technique. Finally, the method is validated for a probe-fed GPS textile antenna. It is shown that Hermite-Pade approximants model the highly nonlinear relationship between these substrate random variables and the figures of merit of the antenna more efficiently than the gPC. Moreover, a Kolmogorov–Smirnoff test proves that the resulting distributions of the antenna’s figures of merit are as accurate as those obtained by means of a Monte-Carlo (MC) analysis, with demonstrated speedup factors up to 123.

Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations

We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called non-intrusive sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a functional of the parametric solution is then computed via compressive sensing methods from samples of functionals of the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sufficient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coefficients of the gpc expansion are contained in certain weighted $\ell_p$-spaces for $0<p\leq 1$. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the $L_2$, resp. $L_\infty$ convergence rates afforded by best $s$-term approximations of the parametric solution up to logarithmic factors.

Fragility Analysis of Base-Isolated Liquid Storage Tanks under Random Sinusoidal Base Excitation Using Generalized Polynomial Chaos Expansion–Based Simulation

AbstractGeneralized polynomial chaos (gPC) expansion–based simulation technique is used to investigate the influence of input parameter uncertainty, on peak response quantities and fragility curves of base-isolated liquid storage tanks. Unidirectional horizontal sinusoidal base excitation is considered to develop the fragility curves for the base-isolated liquid storage tanks. Extensively used laminated rubber bearing (LRB), with linear force-deformation behavior, is considered as the isolation system. The liquid storage tank is modeled using a widely accepted lumped mass model. The failure of the liquid storage tank is defined corresponding to the elastic buckling of the tank wall. The uncertainties are considered in the isolator parameters and in the base excitation. Considerable difference in the peak response estimation is observed when the input parameters are represented using different probability distributions, especially when the uncertainties are higher. It is also observed that when the uncerta...

Constant Jacobian Matrix-Based Stochastic Galerkin Method for Probabilistic Load Flow

An intrusive spectral method of probabilistic load flow (PLF) is proposed in the paper, which can handle the uncertainties arising from renewable energy integration. Generalized polynomial chaos (gPC) expansions of dependent random variables are utilized to build a spectral stochastic representation of PLF model. Instead of solving the coupled PLF model with a traditional, cumbersome method, a modified stochastic Galerkin (SG) method is proposed based on the P-Q decoupling properties of load flow in power system. By introducing two pre-calculated constant sparse Jacobian matrices, the computational burden of the SG method is significantly reduced. Two cases, IEEE 14-bus and IEEE 118-bus systems, are used to verify the computation speed and efficiency of the proposed method.

Integration of fault diagnosis and control based on a trade-off between fault detectability and closed loop performance

This paper presents a novel methodology for simultaneous optimal tuning of a fault detection and diagnosis (FDD) algorithm and a feedback controller for a chemical plant in the presence of stochastic parametric faults. The key idea is to propagate the effect of time invariant stochastic uncertainties onto the measured variables by using a Generalized Polynomial Chaos (gPC) expansion and the nonlinear first principles’ model of the process. A bi-level optimization is proposed for achieving a trade-off between the fault detectability and the closed loop process variability. The goal of the outer level optimization is to seek a trade-off between the efficiency of detecting a fault and the closed loop performance, while the inner level optimization is designed to optimally calibrate the FDD algorithm. The proposed method is illustrated by a continuous stirred tank reactor (CSTR) system with a fault consisting of stochastic and intermittent variations in the inlet concentration. Beyond achieving improved trade-offs between fault detectability and control, it is shown that the computational cost of the gPC model based method is lower than the Monte Carlo type sampling based approaches, thus demonstrating the potential of the gPC method for dealing with large problems and real-time applications.

A Pseudospectral Collocation Approach for Flood Inundation Modelling with Random Input Fields

In this study, an efficient framework of pseudospectral collocation approach combined with the generalized polynomial chaos (gPC) and Karhunen-Loevè expansion (gPC/KLE) was introduced to examine the flood flow fields within a two-dimensional flood modelling system. In the proposed framework, the heterogeneous random input field (logarithmic Manning’s roughness) was approximated by the normalized KLE and the output field of flood flow depth was represented by the gPC expansion, whose coefficients were obtained with a nodal set construction via Smolyak sparse grid quadrature. In total, 3 scenarios (with different levels of input spatial variability) were designed for gPC/KLE application and the results from Monte Carlo simulations were provided as the benchmark for comparison. This study demonstrated that the gPC/KLE approach could predict the statistics of flood flow depth (i.e., means and standard deviations) with significantly less computational requirement than MC; it also outperformed the probabilistic collocation method (PCM) with KLE (PCM/KLE) in terms of fitting accuracy. This study made the first attempt to apply gPC/KLE to flood inundation field and evaluated the effects of key parameters (like the number of eigenpairs and the order of gPC expansion) on model performances.

Stochastic collocation-based finite element of structural nonlinear dynamics with application in composite structures

Stochastic analysis of structures having nonlinearity by means of sampling methods leads to expensive cost in term of computational time. In contrast, non-sampling methods based on the spectral representation of uncertainty are very efficient with comparable accurate results. In this pa- per, the application of spectral methods to nonlinear dynamics of structures with random parameters is investigated. The impact of the parameter randomness on structural responses has been consid- ered. To this end, uncertain parameters and the structure responses are represented using the gPC expansions with unknown deterministic coefficients and random orthogonal polynomial basis. The deterministic finite element model of the structure is used as black-box and it is executed on a set of random collocation points. As the sample structure responses are estimated, a nonlinear optimization process is employed to calculate the unknown coefficients. The method has this main advantage that can be used for complicated nonlinear structural dynamic problems for which the deterministic FEM model has been already developed. Furthermore, it is very time efficient in comparison with sampling methods, as MC simulations. The application of the method is applied to the nonlinear transient analysis of composite beam structures including uncertain quadratic random damping. The results show that the proposed method can capture the large range of uncertainty in input parameters as well as in structural dynamic responses while it is too time-efficient.

An ME-PC Enhanced HDMR Method for Efficient Statistical Analysis of Multiconductor Transmission Line Networks

An efficient method for statistically characterizing multiconductor transmission line (MTL) networks subject to a large number of manufacturing uncertainties is presented. The proposed method achieves its efficiency by leveraging a high-dimensional model representation (HDMR) technique that approximates observables (quantities of interest in MTL networks, such as voltages/currents on mission-critical circuits) in terms of iteratively constructed component functions of only the most significant random variables (parameters that characterize the uncertainties in MTL networks, such as conductor locations and widths, and lumped element values). The efficiency of the proposed scheme is further increased using a multielement probabilistic collocation (ME-PC) method to compute the component functions of the HDMR. The ME-PC method makes use of generalized polynomial chaos (gPC) expansions to approximate the component functions, where the expansion coefficients are expressed in terms of integrals of the observable over the random domain. These integrals are numerically evaluated and the observable values at the quadrature/collocation points are computed using a fast deterministic simulator. The proposed method is capable of producing accurate statistical information pertinent to an observable that is rapidly varying across a high-dimensional random domain at a computational cost that is significantly lower than that of gPC or Monte Carlo methods. The applicability, efficiency, and accuracy of the method are demonstrated via statistical characterization of frequency-domain voltages in parallel wire, interconnect, and antenna corporate feed networks.

Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings

In this paper we develop a set of stochastic numerical schemes for hyperbolic and transport equations with diffusive scalings and subject to random inputs. The schemes are asymptotic preserving (AP), in the sense that they preserve the diffusive limits of the equations in discrete setting, without requiring excessive refinement of the discretization. Our stochastic AP schemes are extensions of the well-developed deterministic AP schemes. To handle the random inputs, we employ generalized polynomial chaos (gPC) expansion and combine it with stochastic Galerkin procedure. We apply the gPC Galerkin scheme to a set of representative hyperbolic and transport equations and establish the AP property in the stochastic setting. We then provide several numerical examples to illustrate the accuracy and effectiveness of the stochastic AP schemes.

Fault detection and diagnosis with parametric uncertainty using generalized polynomial chaos

This paper presents a new methodology to identify and diagnose intermittent stochastic faults occurring in a process. A generalized polynomial chaos (gPC) expansion representing the stochastic inputs is employed in combination with the nonlinear mechanistic model of the process to calculate the resulting statistical distribution of measured variables that are used for fault detection and classification. A Galerkin projection based stochastic finite difference analysis is utilized to transform the stochastic mechanistic equation into a coupled deterministic system of equations which is solved numerically to obtain the gPC expansion coefficients. To detect and recognize faults, the probability density functions (PDFs) and joint confidence regions (JCRs) of the measured variables to be used for fault detection are obtained by substituting samples from a random space into the gPC expansions. The method is applied to a two dimensional heat transfer problem with faults consisting of stochastic changes combined with step change variations in the thermal diffusivity and in a boundary condition. The proposed methodology is compared with a Monte Carlo (MC) simulations based approach to illustrate its advantages in terms of computational efficiency as well as accuracy.

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

We propose a new fully Bayesian method to efficiently obtain the spectral representation of a spatial random field, which can conduct spatial–stochastic basis selection and evaluation of generalized Polynomial Chaos (gPC) expansions when the number of the available basis functions is significantly larger than the size of the training data-set. We develop a fully Bayesian stochastic procedure, called mixed shrinkage prior (MSP), which performs both basis selection and coefficient evaluation simultaneously. MSP involves assigning a prior probability to the gPC structure and assigning conjugate priors to the expansion coefficients that can be thought of as mixtures of Ridge–LASSO shrinkage priors, in augmented form. The method offers a number of advantages over existing compressive sensing methods in gPC literature, such that it recovers possible sparse structures in both stochastic and spatial domains while the resulting expansion can be re-used directly to economically obtain results at any spatial input values. Yet, it inherits all the advantages of Bayesian model uncertainty methods, e.g. accounts for uncertainty about basis significance and provides interval estimation through posterior distributions. A unique highlight of the MSP procedure is that it can address heterogeneous sparsity in the spatial domain for different random dimensions. Furthermore, it yields a compromise between Ridge and LASSO regressions, and hence combines a weak (l2-norm) and strong (l1-norm) shrinkage, in an adaptive, data-driven manner. We demonstrate the good performance of the proposed method, and compare it against other existing compressive sensing ones on elliptic stochastic partial differential equations.

CONSTRUCTING SURROGATE MODELS OF COMPLEX SYSTEMS WITH ENHANCED SPARSITY: QUANTIFYING THE INFLUENCE OF CONFORMATIONAL UNCERTAINTY IN BIOMOLECULAR SOLVATION.

Biomolecules exhibit conformational fluctuations near equilibrium states, inducing uncertainty in various biological properties in a dynamic way. We have developed a general method to quantify the uncertainty of target properties induced by conformational fluctuations. Using a generalized polynomial chaos (gPC) expansion, we construct a surrogate model of the target property with respect to varying conformational states. To alleviate the high-dimensionality of the corresponding stochastic space, we propose a method to increase the sparsity of the gPC expansion by defining a set of conformational "active space" random variables. With the increased sparsity, we employ the compressive sensing method to accurately construct the surrogate model. We demonstrate the performance of the surrogate model by evaluating fluctuation-induced uncertainty in solvent-accessible surface area for the bovine trypsin inhibitor protein system and show that the new approach offers more accurate statistical information than standard Monte Carlo approaches. Furthermore, the constructed surrogate model also enables us to directly evaluate the target property under various conformational states, yielding a more accurate response surface than standard sparse grid collocation methods. In particular, the new method provides higher accuracy in high-dimensional systems, such as biomolecules, where sparse grid performance is limited by the accuracy of the computed quantity of interest. Our new framework is generalizable and can be used to investigate the uncertainty of a wide variety of target properties in biomolecular systems.

Collocation-based Stochastic Modeling of Uncertain Geometric Mistuning in Bladed Rotor

In this paper, a collocation–based stochastic model is proposed to analysis mistuned bladed–rotors with geometrical parameter uncertainties. The uncertain parameters are approximated by the Karhunen-Loe‘ve expansion with predefined correlation functions. The generalized polynomial chaos (gPC) expansion possessing random orthogonal basis is served as uncertain dynamic responses. A set of collocation points are generated from the roots of higher order random polynomial basis and the whole FEM model of the rotor is executed on each point to estimate the dynamic responses. These estimations then are used to calculate the coefficients of the gPC expansions. The most attractive feature of the method is that only multiple solutions of the original deterministic FEM model of the rotor are required. A numerical case study is presented in which the geometrical parameters of a blisk are considered as uncertain parameters with spatial variation. The probability distributions of the random natural frequencies are evaluated for rotational and non-rotational blisk. The results show high accuracy compared to the Monte Carlo simulations with 500 realizations. The impact of the parameter uncertainty on the Campbell diagram during the start up and start down are presented.

A Stochastic Galerkin Method for Hamilton--Jacobi Equations with Uncertainty

We develop a class of stochastic numerical schemes for Hamilton--Jacobi equations with random inputs in initial data and/or the Hamiltonians. Since the gradient of the Hamilton--Jacobi equations gives a symmetric hyperbolic system, we utilize the generalized polynomial chaos (gPC) expansion with stochastic Galerkin procedure in random space and the Jin--Xin relaxation approximation in physical space for shock capturing. We provide an error estimate for the gPC stochastic Galerkin approximation to smooth solutions, and show that our numerical formulation preserves the symmetry and hyperbolicity of the underlying system, which allows one to efficiently quantify the uncertainty of the Hamilton--Jacobi equations due to random inputs, as demonstrated by the numerical examples.

Stochastic Dynamic Analysis of Structures with Spatially Uncertain Material Parameters

This paper investigates the uncertainty quantification in structural dynamic problems with spatially random variation in material and damping parameters. Uncertain and locally varying material parameters are represented as stochastic field by means of the Karhunen–Loève (KL) expansion. The stiffness and damping properties of the structure are considered uncertain. Stochastic finite element of structural modal analysis is performed in which modal responses are represented using the generalized polynomial chaos (gPC) expansion. Knowing the KL expansions of the random parameters, the nonintrusive technique is employed on a set of random collocation points where the structure deterministic finite element model is executed to estimate the unknown coefficients of the polynomial chaos expansions. A numerical case study is presented for a cantilever beam with random Young's modulus involving spatial variation. The proportional damping constants are estimated from the experimental modal analysis. The expected value, standard deviation, and probability distribution of the random eigenfrequencies and the damping ratios are evaluated. The results show high accuracy compared to the Monte-Carlo (MC) simulations with 3000 realizations. It is also demonstrated that the eigenfrequencies and the damping ratios are equally affected from material uncertainties.

Non-sampling inverse stochastic numerical–experimental identification of random elastic material parameters in composite plates

A non-sampling probability identification method based on the generalized polynomial chaos (gPC) expansion is adopted for estimating random parameters of composite plates form experimental eigenfrequencies. For that, the parameters and the eigenfrequencies are approximated using gPC expansion. Distribution functions of the eigenfrequencies are identified from experimental data employing the Bayesian inference. This identification is then used to construct a vector of random variables and an orthogonal basis for eigenfrequency expansions. The parameters are characterized by the gPC having unknown deterministic coefficients and the same random basis as the eigenfrequencies. The stochastic finite element simulation of the plates bears as the model from which the parameter coefficients are estimated via an inverse problem. The major advantage of the method is using deterministic identification procedure. An application is presented for which samples of orthotropic laminated plates are tested to identify E-moduli, shear modulus and the major Poisson's ratio from measured modal frequencies.

Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations

We analyze the convergence of compressive sensing based sampling techniques for the ecient evaluation of functionals of solutions for a class of high-dimensional, ane-parametric, linear operator equations which depend on possibly innitely many parameters. The proposed algorithms are based on so-called \non-intrusive sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, the parametric solution is then computed via compressive sensing methods from samples of (a functional of) the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sucient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coecients of gpc expansion are contained in certain weighted 'p-spaces for 0 < p 1. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the L2, resp. L1 convergence rates aorded by best s-term approximations of the parametric solution up to logarithmic factors.

Integration of fault diagnosis and control by finding a trade-off between the detectability of stochastic fault and economics

This paper presents a first principle model based methodology for simultaneous optimal tuning of a fault detection algorithm and a feedback controller. The key idea is to calculate the effect of stochastic input disturbances on the variability of the output variables by using a generalized polynomial chaos (gPC) expansion and a mechanic model of the process. A two-level optimization is proposed for simultaneously tuning the fault detection and controller algorithms. The goal of the outer level optimization is to find a trade-off between the efficiency for detecting faults and the closed loop performance, while the inner optimization is designed to optimally calibrate the fault detection algorithm. The proposed method is illustrated for a continuous stirred tank reactor (CSTR). The results show that the computational cost of the gPC-based method is significantly lower than a Monte Carlo (MC) simulation-based approach, thus demonstrating the potential of the gPC method for dealing with large problems.

BEST N-TERM GPC APPROXIMATIONS FOR A CLASS OF STOCHASTIC LINEAR ELASTICITY EQUATIONS

We consider a class of stochastic linear elasticity problems whose elastic moduli depend linearly on a countable set of random variables. The stochastic equation is studied via a deterministic parametric problem on an infinite-dimensional parameter space. We first study the best N-term approximation of the generalized polynomial chaos (gpc) expansion of the solution to the displacement formula by considering a Galerkin projection onto the space obtained by truncating the gpc expansion. We provide sufficient conditions on the coefficients of the elastic moduli's expansion so that a rate of convergence for this approximation holds. We then consider two classes of stochastic and parametric mixed elasticity problems. The first one is the Hellinger–Reissner formula for approximating directly the gpc expansion of the stress. For isotropic problems, the multiplying constant of the best N-term convergence rate for the displacement formula grows with the ratio of the Lamé constants. We thus consider stochastic and parametric mixed problems for nearly incompressible isotropic materials whose best N-term approximation rate is uniform with respect to the ratio of the Lamé constants.

Adaptive stochastic Galerkin FEM

A framework for residual-based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic PDEs with random coefficients is presented. A parametric, deterministic elliptic boundary value problem on an infinite-dimensional parameter space is discretized by means of a Galerkin projection onto finite generalized polynomial chaos (gpc) expansions, and by discretizing each gpc coefficient by a FEM in the physical domain.An anisotropic residual-based a posteriori error estimator is developed. It contains bounds for both contributions to the overall error: the error due to gpc discretization and the error due to Finite Element discretization of the gpc coefficients in the expansion. The reliability of the residual estimator is established.Based on the explicit form of the residual estimator, an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic Galerkin discretization, and, embedded in the gpc adaptation loop, also the Finite Element mesh refinement of the gpc coefficients in the physical domain. Asynchronous mesh adaptation for different gpc coefficients is permitted, subject to a minimal compatibility requirement on meshes used for different gpc coefficients.Details on the implementation with the open-source software framework ALEA are presented; it is generic, and is based on available stiffness and mass matrices of a FEM for the deterministic, nonparametric nominal problem evaluated in the FEniCS environment.Preconditioning of the resulting matrix equation and iterative solution are discussed. Numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented. They indicate substantial savings in total computational complexity due to FE mesh coarsening in high gpc coefficients.