Sparse Grid Numerical Integration Research Articles

Propagation algorithm for hybrid uncertainty parameters based on polynomial chaos expansion

AbstractThis article presents an uncertainty analysis method for systems with hybrid stochastic and fuzzy uncertainty parameters based on polynomial chaos expansion (PCE). Parameters in the system are described by probability boxes, interval numbers, and fuzzy sets, respectively, based on the differences in their limited stochastic knowledge. First, this method transforms the uncertain parameters into standard normal distribution and interval variables through equal probability transformation or ‐cut operations. Second, the Legendre and Hermite polynomials are used as the PCE model's primary functions, and the expansion coefficients are calculated by the Galerkin projection method based on sparse grid numerical integration. Then, the system response bounds under the pre‐defined confidence level can be obtained using a genetic algorithm to solve the optimization problem constructed based on PCE models. Finally, the feasibility and effectiveness of the method are illustrated by taking the tank bi‐directional stabilized system and the double‐pendulum‐controlled system as examples. The numerical results show that the system response bounds obtained by the PCE model optimization algorithm are consistent with the Monte Carlo simulation. Still, the computational efficiency is much higher. The proposed method effectively combines fuzzy sets and probability boxes and dramatically simplifies the analysis process of uncertain systems. The method exhibits fine precision even in high‐dimensional uncertainty analysis problems.

Simultaneous reliability and reliability-sensitivity analyses based on the information-reuse of sparse grid numerical integration

Simultaneous reliability and reliability-sensitivity analyses based on the information-reuse of sparse grid numerical integration

Dynamic reliability analysis of gear transmission system based on sparse grid numerical integration and saddle‐point approximation method

AbstractGear systems are widely used in various mechanical transmission systems. This paper aims to develop an effective and practical method for dynamic reliability analysis of gear transmission system. The proposed method can comprehensively evaluate the dynamic reliability of gear transmission system by adopting the fourth‐moment SPA method. First, a nonlinear dynamics model of a single‐stage spur gear transmission system is established, which simultaneously takes into account the nonlinear backlash, time‐varying meshing stiffness, and static transmission error. After that, a dynamic reliability model for the tooth surface contact fatigue failure of gear system is established with the uncertainty of the motion, structure, and material parameters using stress‐strength interference (SSI) theory. To be specific, the sparse grid numerical integration (SGNI) method is applied to solve the statistical characteristic parameters of the dynamic reliability of the system. The probability distribution of the performance function is obtained with the fourth‐moment SPA method. Test examples show that the results of the proposed method are consistent with the results obtained by the Monte Carlo simulation (MCS) and superior to the maximum entropy with fractional moments (ME‐FM) method, which verifies the effectiveness of this approach. Finally, the dynamic reliability of the gear transmission system with respect to load times is evaluated.

A novel decoupled time-variant reliability-based design optimization approach by improved extreme value moment method

Time-variant reliability-based design optimization (t-RBDO) is an effective tool to guarantee a high reliability of the product during the full life cycle. This paper proposes a new decoupled method for t-RBDO via an improved extreme value moment method (EVMM). In the proposed method, a weight approach based on sparse grid numerical integration (WA-SGNI) is employed to establish the local approximation of the extreme value moment. Since evaluating the failure probability functions (FPFs) using fourth-moment transformation method has certain limitations, a local approximation method for quantile functions (QFs) is proposed to consider the time-variant reliability constraints in t-RBDO. After the local approximations of QFs are established, t-RBDO problem is decoupled into a deterministic one in the design sub-region. Combining sequential approximation optimization, a unified t-RBDO framework is presented. Four numerical examples are investigated to validate the accuracy and efficiency of the proposed method.

An efficient extreme value moment method combining adaptive Kriging model for time-variant imprecise reliability analysis

For time-variant imprecise reliability problem with parameterized probability-box (p-box), determining the upper and lower bounds of time-variant failure probability is usually more of concern for engineers. To compute these two bounds, this paper proposes a new extreme value moment method (EVMM) combining adaptive Kriging model. In the proposed method, the adaptive Kriging models are first employed to approximate limit state function (LSF) responses and the predicted values are mapped into the extreme value function (EVF) counterparts. Then, the weighted approach based on sparse grid numerical integration (WA-SGNI) is developed to estimate the statistical moments of the approximated EVF responses. The resulting moments are further used to decouple the calculation of bounds of time-variant failure probability into two deterministic optimization problems. The accuracy and efficiency of the proposed method for time-variant imprecise reliability analysis are demonstrated through three numerical examples with LSFs involving non-Gaussian processes and finite element analysis.

A copula-based uncertainty propagation method for structures with correlated parametric p-boxes

In the response analysis of uncertain structural models with limited information, probability-boxes can be effectively employed to address the aleatory and epistemic uncertainty together. This paper presents a copula-based uncertainty propagation method which can accurately perform uncertainty propagation analysis with correlated parametric probability-boxes. Firstly, the parameter estimation and Akaike information criterion analysis are utilized to select an optimal copula based on available samples, by which the joint cumulative distribution function is constructed for the correlated input variables. Then, using the obtained joint cumulative distribution function, the correlated parametric probability-boxes are transformed into independent normal variables, and an efficient method based on sparse grid numerical integration is proposed to calculate the bounds on statistical moments of a response function. Finally, numerical examples and an engineering application are provided to verify the effectiveness of the presented method.

Proficiency of statistical moment-based methods for analysis of positional accuracy reliability of industrial robots

General presence of uncertainties in geometrical parameters of industrial robots, such as link length, distance between two connecting rods, joint rotation angle and torsional angle, leads to deviations from the specified trajectory of robotic end-effector. It is of practical significance to analyze the positional accuracy reliability for industrial robots in terms of these uncertainties. Among the existing analysis methods, statistical moment-based methods are highly prioritized in evaluating the positional accuracy reliability for industrial robots due to the high accuracy and good computing efficiency. In this study, three different statistical moment-based methods, namely the sparse grid numerical integration (SGNI) method, the point estimation method (PEM) and the univariate dimension reduction method (UDRM), are applied to quantitatively evaluate the positional accuracy reliability of industrial robots. The kinematics model of industrial robots is established through the Denavit-Hartenberg method. The aforementioned three methods are then programmed to calculate the first-four order moments of the established kinematics model. The industrial robots’ positional accuracy reliability is calculated using the SGNI, PEM and UDRM under specified threshold and compared with that from Monte Carlo simulation (MCS) method. Comparison of results shows that the SGNI method performs best in terms of computational accuracy and the PEM exhibits the highest computational efficiency among the three candidate methods.

A time-variant uncertainty propagation analysis method based on a new technique for simulating non-Gaussian stochastic processes

In this paper, the time-variant uncertainty propagation analysis is defined to solve the output stochastic process of a time-variant function with uncertainty. And a time-variant uncertainty propagation analysis method is constructed with the combination of an extended orthogonal series expansion method (extended OSE) and sparse grid numerical integration (SGNI). The SGNI serving as a classical uncertainty propagation method is utilized here to solve the moments and autocorrelation function at discrete time points of the time-variant performance function. And the extended OSE is proposed to simulate the output stochastic process based on the results from SGNI. By extended OSE, a non-Gaussian stochastic process is represented as the sumof orthogonal time functions with random coefficients, and these coefficients can be directly obtained by discretization of the target process. For these coefficients are correlated and non-Gaussian, the correlated polynomial chaos expansion method (c-PCE) is presented to represent them in terms of correlated standard Gaussian variables, and then the principal component analysis (PCA) is adopted to transform them into independent ones with dimension reduction. Finally we can obtain an explicit expression to represent the non-Gaussian process whatever it is stationary or non-stationary. Three illustrative examples are used to verify the performance of the extended OSE. In addition, two engineering problems are investigated to demonstrate the effectiveness of the time-variant uncertainty propagation method.

A computational framework of kinematic accuracy reliability analysis for industrial robots

A new computational method to evaluate comprehensively the positional accuracy reliability for single coordinate, single point, multipoint and trajectory accuracy of industrial robots is proposed using the sparse grid numerical integration method and the saddlepoint approximation method. A kinematic error model of end-effector is constructed in three coordinate directions using the sparse grid numerical integration method considering uncertain parameters. The first-four order moments and the covariance matrix for three coordinates of the end-effector are calculated by extended Gauss–Hermite integration nodes and corresponding weights. The eigen-decomposition is conducted to transform the interdependent coordinates into independent standard normal variables. An equivalent extreme value distribution of response is applied to assess the reliability of kinematic accuracy. The probability density function and probability of failure for extreme value distribution are then derived through the saddlepoint approximation method. Four examples are given to demonstrate the effectiveness of the proposed method.

A novel structural reliability analysis method via improved maximum entropy method based on nonlinear mapping and sparse grid numerical integration

Abstract This paper proposes an improved maximum entropy method for reliability analysis (i-MEM), in which the limit state function is transformed by a nonlinear mapping to predict the failure probability accurately. Through the nonlinear mapping, more statistical information can be obtained by the first-four statistical moments, and the truncation error originating from numerical integration is solved by the bounded limit state function after the nonlinear mapping, therefore the i-MEM can capture the tail information of the real probability distribution. In order to calculate the statistical moments in i-MEM with accuracy and efficiency, an improved sparse grid numerical integration method (i-SGNI) is developed on the basis of the normalized moment-based quadrature rule. Combining the i-MEM and i-SGNI, a novel reliability analysis method is proposed. To illustrate the accuracy, efficiency and numerical stability of the proposed method, six numerical examples and one engineering example are presented, compared with some common reliability analysis methods. The results show that the proposed method, with the combination of i-MEM and i-SGNI, can achieve a good balance between accuracy and efficiency for structural reliability analysis.

Probabilistic Distribution Modeling of Crosstalk in Multi-Conductor Transmission Lines via Maximum Entropy Method

A novel statistical method, referred to as the maximum entropy (MaxEnt) method is proposed in this paper for effective probabilistic modeling of crosstalk in multi-conductor transmission lines (MTLs). The principle of the MaxEnt method states that for the given information constraints, the probability distribution that has the MaxEnt is considered to be the most unbiased and reasonable one. As for low- and high-dimensional spaces composed of the input random variables in MTL, the statistical moments of crosstalk required by the MaxEnt method as its information constraints are obtained through full-factorial numerical integration and sparse grid numerical integration, respectively. Then, a probability distribution model of crosstalk in MTL is established based on the MaxEnt method. Compared with the conventional Monte Carlo method, the proposed method can not only accurately predict the probability distribution and statistics of crosstalk in MTL but also considerably increases the computational efficiency. Therefore, the MaxEnt method is effective in modeling the probability distribution of crosstalk in MTL.

Uncertainty propagation analysis using sparse grid technique and saddlepoint approximation based on parameterized p-box representation

Uncertainty propagation analysis, which assesses the impact of the uncertainty of input variables on responses, is an important component in risk assessment or reliability analysis of structures. This paper proposes an uncertainty propagation analysis method for structures with parameterized probability-box (p-box) representation, which could efficiently compute both the bounds on statistical moments and also the complete probability bounds of the response function. Firstly, based on the sparse grid numerical integration (SGNI) method, an optimized SGNI (OSGNI) is presented to calculate the bounds on the statistical moments of the response function and the cumulants of the cumulant generating function (CGF), respectively. Then, using the bounds on the first four cumulants, an optimization procedure based on the saddlepoint approximation is proposed to obtain the whole range of probability bounds of the response function. Through using the saddlepoint approximation, the present approach can achieve a good accuracy in estimating the tail probability bounds of a response function. Finally, two numerical examples and an engineering application are investigated to demonstrate the effectiveness of the proposed method.

Uncertainty propagation analysis by an extended sparse grid technique

In this paper, an uncertainty propagation analysis method is developed based on an extended sparse grid technique and maximum entropy principle, aiming at improving the solving accuracy of the high-order moments and hence the fitting accuracy of the probability density function (PDF) of the system response. The proposed method incorporates the extended Gauss integration into the uncertainty propagation analysis. Moreover, assisted by the Rosenblatt transformation, the various types of extended integration points are transformed into the extended Gauss-Hermite integration points, which makes the method suitable for any type of continuous distribution. Subsequently, within the sparse grid numerical integration framework, the statistical moments of the system response are obtained based on the transformed points. Furthermore, based on the maximum entropy principle, the obtained first four-order statistical moments are used to fit the PDF of the system response. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed method, which includes two mathematical problems with explicit expressions and an engineering application with a black-box model.

General validation and decomposition of the variance-based measures for models with correlated inputs

To clearly explore the contributions by the correlated inputs to the variance of model output, the general expressions of the variance contributions of the correlated inputs are firstly derived in this paper, which provide a general validation for the accurate connotations of these variance contributions. The universal expressions of the components included in the variance contributions of the correlated inputs are also derived. These components not only allow the engineer to distinguish between the independent variance contribution and the correlated one of the input, but also provide information for differentiating the independent contribution by the input itself from that by the interaction between the input and the others. All the general expressions of the variance contributions in this paper are derived based on the high dimensional model representation (HDMR) of the output. Thus, they are independent of the form of the computational model. Compared with the existing interpretation and decomposing methods for the variance contributions of the correlated inputs, the proposed method has wider applicability. Based on the form of the general expressions of the various variance contributions, their efficient solutions are built by using the advantages of the sparse grid numerical integration (SGI). Numerical and engineering tests show the effectiveness and applicability of the derived variance contributions for the correlated inputs, as well as the efficiency and accuracy of the established SGI based method.

Importance analysis for models with dependent input variables by sparse grids

To get a better understanding on the output uncertainty contributed by an individual variable as well as the correlated variables of models with dependent inputs, a method for decomposing Sobol’s first-order effect indices into uncorrelated variations and correlated variations is investigated. Instead of using Monte Carlo simulation or full tensor product-based numerical integration approaches, a new sparse grid numerical integration method is proposed for estimating Sobol’s main effect indices as well as the two decomposed sensitivity measures. Before conducting the sparse grid numerical integration-based algorithm, an orthogonal transformation is used to transform the dependent input variables and model performance function into independent space as the joint probability density function of the correlated variables cannot be written as the product of univariate density functions. An obvious advantage of the sparse grid numerical integration-based method is that it can decrease the computational cost of the conventional methods significantly while keeping the accuracy level controllable, particularly for high-dimensional problems. The proposed approach is compared with other alternative approaches through theoretical and applied numerical experiments to demonstrate its efficiency, accuracy and high-dimensional adaptivity.

Importance analysis for models with correlated variables and its sparse grid solution

For structural models involving correlated input variables, a novel interpretation for variance-based importance measures is proposed based on the contribution of the correlated input variables to the variance of the model output. After the novel interpretation of the variance-based importance measures is compared with the existing ones, two solutions of the variance-based importance measures of the correlated input variables are built on the sparse grid numerical integration (SGI): double-loop nested sparse grid integration (DSGI) method and single loop sparse grid integration (SSGI) method. The DSGI method solves the importance measure by decreasing the dimensionality of the input variables procedurally, while SSGI method performs importance analysis through extending the dimensionality of the inputs. Both of them can make full use of the advantages of the SGI, and are well tailored for different situations. By analyzing the results of several numerical and engineering examples, it is found that the novel proposed interpretation about the importance measures of the correlated input variables is reasonable, and the proposed methods for solving importance measures are efficient and accurate.

Computationally Efficient Imprecise Uncertainty Propagation

Modeling uncertainty through probabilistic representation in engineering design is common and important to decision making that considers risk. However, representations of uncertainty often ignore elements of “imprecision” that may limit the robustness of decisions. Furthermore, current approaches that incorporate imprecision suffer from computational expense and relatively high solution error. This work presents a method that allows imprecision to be incorporated into design scenarios while providing computational efficiency and low solution error for uncertainty propagation. The work draws on an existing method for representing imprecision and integrates methods for sparse grid numerical integration, resulting in the computationally efficient imprecise uncertainty propagation (CEIUP) method. This paper presents details of the method and demonstrates the effectiveness on both numerical case studies, and a thermocouple performance problem found in the literature. Results for the numerical case studies, in most cases, demonstrate improvements in both computational efficiency and solution accuracy for varying problem dimension and variable interaction when compared to optimized parameter sampling (OPS). For the thermocouple problem, similar behavior is observed when compared to OPS. The paper concludes with an overview of design problem scenarios in which CEIUP is the preferred method and offers opportunities for extending the method.

A Derivative-Free Optimization Algorithm Using Sparse Grid Integration

We present a new derivative-free optimization algorithm based on the sparse grid numerical integration. The algorithm applies to a smooth nonlinear objective function where calculating its gradient is impossible and evaluating its value is also very expensive. The new algorithm has: 1) a unique starting point strategy; 2) an effective global search heuristic; and 3) consistent local convergence. These are achieved through a uniform use of sparse grid numerical integration. Numerical experiment result indicates that the algorithm is accurate and efficient, and benchmarks favourably against several state-of-art derivative free algorithms.

Sparse pseudospectral approximation method

Multivariate global polynomial approximations – such as polynomial chaos or stochastic collocation methods – are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials.By reexamining Smolyak’s algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.

A new sparse grid based method for uncertainty propagation

Current methods for uncertainty propagation suffer from their limitations in providing accurate and efficient solutions to high-dimension problems with interactions of random variables. The sparse grid technique, originally invented for numerical integration and interpolation, is extended to uncertainty propagation in this work to overcome the difficulty. The concept of Sparse Grid Numerical Integration (SGNI) is extended for estimating the first two moments of performance in robust design, while the Sparse Grid Interpolation (SGI) is employed to determine failure probability by interpolating the limit-state function at the Most Probable Point (MPP) in reliability analysis. The proposed methods are demonstrated by high-dimension mathematical examples with notable variate interactions and one multidisciplinary rocket design problem. Results show that the use of sparse grid methods works better than popular counterparts. Furthermore, the automatic sampling, special interpolation process, and dimension-adaptivity feature make SGI more flexible and efficient than using the uniform sample based metamodeling techniques.