Uncertainty quantification of high‐dimensional complex systems by multiplicative polynomial dimensional decompositions

The central theme of this paper is multiplicative polynomial dimensional decomposition (PDD) methods for solving high‐dimensional stochastic problems. When a stochastic response is dominantly of multiplicative nature, the standard PDD approximation, predicated on additive function decomposition, may not provide sufficiently accurate probabilistic solutions of a complex system. To circumvent this problem, two multiplicative versions of PDD, referred to as factorized PDD and logarithmic PDD, were developed. Both versions involve a hierarchical, multiplicative decomposition of a multivariate function, a broad range of orthonormal polynomial bases for Fourier‐polynomial expansions of component functions, and a dimension‐reduction or sampling technique for estimating the expansion coefficients. Three numerical problems involving mathematical functions or uncertain dynamic systems were solved to corroborate how and when a multiplicative PDD is more efficient or accurate than the additive PDD. The results show that indeed, both the factorized and logarithmic PDD approximations can effectively exploit the hidden multiplicative structure of a stochastic response when it exists. Since a multiplicative PDD recycles the same component functions of the additive PDD, no additional cost is incurred. Finally, the random eigensolutions of a sport utility vehicle comprising 40 random variables were evaluated, demonstrating the ability of the new methods to solve industrial‐scale problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A hybrid polynomial dimensional decomposition for uncertainty quantification of high-dimensional complex systems

This paper presents a novel hybrid polynomial dimensional decomposition (PDD) method for stochastic computing in high-dimensional complex systems. When a stochastic response does not possess a strongly additive or a strongly multiplicative structure alone, then the existing additive and multiplicative PDD methods may not provide a sufficiently accurate probabilistic solution of such a system. To circumvent this problem, a new hybrid PDD method was developed that is based on a linear combination of an additive and a multiplicative PDD approximation, a broad range of orthonormal polynomial bases for Fourier-polynomial expansions of component functions, and a dimension-reduction or sampling technique for estimating the expansion coefficients. Two numerical problems involving mathematical functions or uncertain dynamic systems were solved to study how and when a hybrid PDD is more accurate and efficient than the additive or the multiplicative PDD. The results show that the univariate hybrid PDD method is slightly more expensive than the univariate additive or multiplicative PDD approximations, but it yields significantly more accurate stochastic solutions than the latter two methods. Therefore, the univariate truncation of the hybrid PDD is ideally suited to solving stochastic problems that may otherwise mandate expensive bivariate or higher-variate additive or multiplicative PDD approximations. Finally, a coupled acoustic-structural analysis of a pickup truck subjected to 46 random variables was performed, demonstrating the ability of the new method to solve large-scale engineering problems.     

Novel computational methods for high‐dimensional stochastic sensitivity analysis

This paper presents three new computational methods for calculating design sensitivities of statistical moments and reliability of high‐dimensional complex systems subject to random input. The first method represents a novel integration of the polynomial dimensional decomposition (PDD) of a multivariate stochastic response function and score functions. Applied to the statistical moments, the method provides mean‐square convergent analytical expressions of design sensitivities of the first two moments of a stochastic response. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD‐saddlepoint approximation (SPA) or PDD‐SPA method, entailing SPA and score functions; and the PDD‐Monte Carlo simulation (MCS) or PDD‐MCS method, utilizing the embedded MCS of the PDD approximation and score functions. For all three methods developed, the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Numerical examples, including a 100‐dimensional mathematical problem, indicate that the new methods developed provide not only theoretically convergent or accurate design sensitivities, but also computationally efficient solutions. A practical example involving robust design optimization of a three‐hole bracket illustrates the usefulness of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

A polynomial dimensional decomposition for stochastic computing

This article presents a new polynomial dimensional decomposition method for solving stochastic problems commonly encountered in engineering disciplines and applied sciences. The method involves a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for Fourier‐polynomial expansion of component functions, and an innovative dimension‐reduction integration for calculating the coefficients of the expansion. The new decomposition method does not require sample points as in the previous version; yet, it generates a convergent sequence of lower‐variate estimates of the probabilistic characteristics of a generic stochastic response. The results of five numerical examples indicate that the proposed decomposition method provides accurate, convergent, and computationally efficient estimates of the tail probability of random mathematical functions or the reliability of mechanical systems. Copyright © 2008 John Wiley & Sons, Ltd.     

Stochastic multiscale fracture analysis of three-dimensional functionally graded composites

A new moment-modified polynomial dimensional decomposition (PDD) method is presented for stochastic multiscale fracture analysis of three-dimensional, particle-matrix, functionally graded materials (FGMs) subject to arbitrary boundary conditions. The method involves Fourier-polynomial expansions of component functions by orthonormal polynomial bases, an additive control variate in conjunction with Monte Carlo simulation for calculating the expansion coefficients, and a moment-modified random output to account for the effects of particle locations and geometry. A numerical verification conducted on a two-dimensional FGM reveals that the new method, notably the univariate PDD method, produces the same crude Monte Carlo results with a five-fold reduction in the computational effort. The numerical results from a three-dimensional, edge-cracked, FGM specimen under a mixed-mode deformation demonstrate that the statistical moments or probability distributions of crack-driving forces and the conditional probability of fracture initiation can be efficiently generated by the univariate PDD method. There exist significant variations in the probabilistic characteristics of the stress-intensity factors and fracture-initiation probability along the crack front. Furthermore, the results are insensitive to the subdomain size from concurrent multiscale analysis, which, if selected judiciously, leads to computationally efficient estimates of the probabilistic solutions.     

A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems

In this paper, a new computational framework based on the topology derivative concept is presented for evaluating stochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with high dimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculate topology sensitivities of the first three stochastic moments which are often required in robust topology optimization (RTO). On another hand, it offers embedded Monte Carlo Simulation (MCS) and finite difference formulations to estimate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases, the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochastic analysis. Moreover, an original example of two random variables is developed for the first time to obtain analytical solutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructed for analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity of failure probabilities in order to verify the accuracy and efficiency of the proposed method for high-dimensional scenarios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivities of existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achieves better accuracies for stochastic topology sensitivities than for the stochastic quantities themselves.     

Reduced chaos expansions with random coefficientsin reduced‐dimensional stochastic modeling of coupled problems

We address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions. The proposed method entails a partitioned iterative solution algorithm that relies on a reduced‐dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems. The proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients. The representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling. After laying out the theoretical framework, we apply the proposed method to a multiphysics problem from nuclear engineering. Copyright © 2013 John Wiley & Sons, Ltd.     

Iterative Polynomial Dimensional Decomposition approach towards solution of structural mechanics problems with material randomness

One of the major difficulties in solving stochastic mechanics problems is the curse of dimensionality, where an exponential increase in the dimension of the problem is encountered with the increase in the number of random variables and/or order of expansion considered in any approximation. A prominent method in addressing the curse of dimensionality is ANOVA dimension Decomposition (ADD), which represents a mathematical function with multiple lower variate functions. These lower variate functions are represented using orthogonal polynomials, which yields Polynomial Dimensional Decomposition (PDD). In recent articles, the authors proposed an Iterative Polynomial Chaos (ImPC) based method for the solution of structural mechanics problems, where computational efficacy of ImPC was demonstrated against Polynomial Chaos (PC). In ImPC, the problems are solved iteratively using smaller sizes of PC expansions. Thus, it reduces the curse of dimensionality of PC expansion. The PDD reduces the size of the system matrix by considering a fewer number of random variables at a time, while ImPC can be considered to solve each components of PDD iteratively so that a converged solution can be achieved without increasing the order of expansion, which is termed as iterative PDD in the present study. Thus, the overall convergence can be achieved with a lesser size of the system matrix, which enables to perform analyses with a lesser computational facility. Further, the stiffness matrix size can be reduced by considering the random field at Gauss points instead of the mid point. Numerical studies with both Gaussian and non-Gaussian random field of Young’s modulus are conducted, and computational efficiency of the iterative PDD is compared with that of PDD, ImPC, and first order perturbation method. The iterative PDD is observed to be computationally less demanding and exhibits reduced dimensional curse.     

稀疏偏最小二乘回归-多项式混沌展开代理模型方法

为解决传统多项式混沌展开方法在高维全局灵敏度和结构可靠度分析当中存在的维数灾难与多重共线性问题,该文提出一种稀疏偏最小二乘回归-多项式混沌展开代理模型方法。该方法首先采用偏最小二乘回归技术得到多项式混沌展开系数的初步估计,然后根据回归误差阈值允许下的最大稀疏度原则,采用带有惩罚的矩阵分解技术自适应地保留与结构响应相关性强的多项式,并采用偏最小二乘回归得到多项式混沌展开系数的更新估计。通过对展开系数进行简单后处理即可得到Sobol灵敏度指数。在此基础上保留重要输入变量并按新方法重新进行回归可实现对代理模型的精简,从而在不增加计算代价的情况下实现高精度结构可靠度分析。算例结果表明在保证精度的情况下,采用新方法进行全局灵敏度和结构可靠度分析比传统方法在计算效率方面有显著优势。     

Stochastic sensitivity analysis by dimensional decomposition and score functions

This article presents a new class of computational methods, known as dimensional decomposition methods, for calculating stochastic sensitivities of mechanical systems with respect to probability distribution parameters. These methods involve a hierarchical decomposition of a multivariate response function in terms of variables with increasing dimensions and score functions associated with probability distribution of a random input. The proposed decomposition facilitates univariate and bivariate approximations of stochastic sensitivity measures, lower-dimensional numerical integrations or Lagrange interpolations, and Monte Carlo simulation. Both the probabilistic response and its sensitivities can be estimated from a single stochastic analysis, without requiring performance function gradients. Numerical results indicate that the decomposition methods developed provide accurate and computationally efficient estimates of sensitivities of statistical moments or reliability, including stochastic design of mechanical systems. Future effort includes extending these decomposition methods to account for the performance function parameters in sensitivity analysis.     

An extended stochastic pseudo-spectral Galerkin finite element method (XS-PS-GFEM) for elliptic equations with hybrid uncertainties

Nouy and Clement introduced the stochastic extended finite element method to solve linear elasticity problem defined on random domain. The material properties and boundary conditions were assumed to be deterministic. In this work, we extend this framework to account for multiple independent input uncertainties, namely, material, geometry, and external force uncertainties. The stochastic field is represented using the polynomial chaos expansion. The challenge in numerical integration over multidimensional probabilistic space is addressed using the pseudo-spectral Galerkin method. Thereafter, a sensitivity analysis based on Sobol indices using the derived stochastic extended Finite Element Method solution is presented. The efficiency and accuracy of the proposed novel framework against conventional Monte Carlo methods is elucidated in detail for a few one and two dimensional problems.     

A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition

In this paper, a non‐intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine‐scale analysis. To validate the developed reduced‐order model, the method is implemented to: (1) the stochastic steady‐state heat diffusion in a square slab; (2) the incompressible, two‐dimensional laminar boundary‐layer over a flat plate with uncertainties in free‐stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi‐random sequence is used to generate the sample points. The numerical results of the three test cases show that the non‐intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non‐intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.     

Sparse grids‐based stochastic approximations with applications to aerodynamics sensitivity analysis

This work compares sample‐based polynomial surrogates, well suited for moderately high‐dimensional stochastic problems. In particular, generalized polynomial chaos in its sparse pseudospectral form and stochastic collocation methods based on both isotropic and dimension‐adapted sparse grids are considered. Both classes of approximations are compared, and an improved version of a stochastic collocation with dimension adaptivity driven by global sensitivity analysis is proposed. The stochastic approximations efficiency is assessed on multivariate test function and airfoil aerodynamics simulations. The latter study addresses the probabilistic characterization of global aerodynamic coefficients derived from viscous subsonic steady flow about a NACA0015 airfoil in the presence of geometrical and operational uncertainties with both simplified aerodynamics model and Reynolds‐Averaged Navier‐Stokes (RANS) simulation. Sparse pseudospectral and collocation approximations exhibit similar level of performance for isotropic sparse simulation ensembles. Computational savings and accuracy gain of the proposed adaptive stochastic collocation driven by Sobol' indices are patent but remain problem‐dependent. Copyright © 2015 John Wiley & Sons, Ltd.     

Neumann enriched polynomial chaos approach for stochastic finite element problems

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach.     

基于同伦分析方法的随机结构静力响应求解

该文提出了一种基于同伦分析方法的求解含随机参数结构的静力响应的新方法。该方法将随机静力平衡方程重新进行同伦构造,利用含随机变量和趋近函数的同伦级数展式来表示结构的随机静力位移响应,该同伦级数的各阶确定性系数和趋近函数可通过对一系列的变形方程求解得到。由于趋近函数的引入,该同伦级数解相较于传统的摄动法有更大的收敛范围,对于含较大变异性随机参数的结构也能获得不错的求解精度。同时,该文提出了一种降维策略来提高该方法的计算效率。数值算例表明,与目前广泛应用的广义正交多项式展开法（GPC）相比,从计算精度上看,该文方法的3阶展开与GPC 2阶展开相当,该文方法的6阶展开与GPC 4阶展开相当,而计算时间上前者均明显少于后者。此外,该文方法也可以方便地应用到随机结构的几何非线性分析当中,并具有较好的计算精度和计算效率。     

基于正交展开方法的Duffing振子随机最优控制

基于多维Hermite多项式的经典均相混沌展开,考察了Duffing振子随机最优多项式控制的正交展开方法,阐明了多项式系数演化与振子系统反应、最优控制力概率特性之间的联系.系统输入采用Karhunen-Loève展开表现的随机地震动.为降低混求解规模,引入位移-速度范数准则,发展了自适应混沌多项式展开策略.同时,基于Lyapunov稳定条件设计控制器的控制增益参数.数值算例分析表明,受控后系统位移和加速度的均方特征得到改善、振子系统的非线性程度减小,基于混沌多项式展开的最优控制方法能明显降低系统的随机涨落和显著改善系统的非线性反应性态.     

Global sensitivity analysis using polynomial chaos expansions

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.     

Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains

We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non‐intrusively computed either at the element or the global level. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three‐dimensional gas turbine blade model with uncertainty in the cooling core geometry. The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high‐quality approximations for the first two statistical moments at modest computational effort. Copyright © 2010 John Wiley & Sons, Ltd.     

Gaussian process emulators for the stochastic finite element method

This paper explores a method to reduce the computational cost of stochastic finite element codes. The method, known as Gaussian process emulation, consists of building a statistical approximation to the output of such codes based on few training runs. The incorporation of emulation is explored for two aspects of the stochastic finite element problem. First, it is applied to approximating realizations of random fields discretized via the Karhunen–Loève expansion. Numerical results of emulating realizations of Gaussian and lognormal homogeneous two‐dimensional random fields are presented. Second, it is coupled with the polynomial chaos expansion and the partitioned Cholesky decomposition in order to compute the response of the typical sparse linear system that arises due to the discretization of the partial differential equations that govern the response of a stochastic finite element problem. The advantages and challenges of adopting the proposed coupling are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions

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 © 2007 John Wiley & Sons, Ltd.