•An elaborate algorithm of analyzing the moment-independent sensitivity is derived.•The proposed algorithm avoids the probability density function estimation.•The proposed algorithm obtains ...sensitivity indices of all inputs simultaneously.•The computational burden of the proposed algorithm does not depend on the dimensionality of model inputs.
Borgonovo moment-independent sensitivity index (BMSI) was proposed to measure the sensitivity of model inputs according to the whole distribution of model output not only a specific moment. The main computational difficulty of the BMSI is to estimate the unconditional probability density function (PDF) and the conditional PDF of the model output. Generally, the estimation of cumulative distribution function (CDF) is easier than that of the PDF, but CDF-based method needs to calculate the extreme points of the differences between the unconditional PDF and the conditional PDF of model output. In addition, the computational cost of the existing CDF-based method also depends on the dimensionality of model inputs. To avoid these accessional computations, this paper derives a new formula by innovatively combining the law of total expectation in the successive intervals without overlapping and the Bayes theorem. The proposed new formula can obtain every input's BMSI only by one group of unconditional model inputs-output samples and does not need to estimate the PDF and the extreme points, which greatly reduces the computational difficulty of the BMSI by replacing the PDF estimation with the probability estimation. Four case studies are analyzed, and the results demonstrate the effectiveness of the proposed algorithm for estimating the BMSI.
Most of the existing methods for estimating time-dependent failure credibility (TDFC) are based on optimization algorithms, which may result in a heavy burden on the computational cost and accuracy ...issue related to the local optimization. In order to achieve the best compromise between computational accuracy and cost, an efficient method is proposed in this work by embedding a single-loop adaptive Kriging model (S-AK) into the dichotomy searching algorithm (S-AK-DSA). The proposed S-AK-DSA can be regarded as a double-loop procedure. In the inner loop, the Kriging model of the real time-dependent performance function (TD-PF) is updated iteratively to accurately predict the signs of the upper/lower boundary of the TD-PF minimum with respect to the time variable at the given membership level. Based on the inner loop, the outer loop searches TDFC by continuously dichotomizing the searching interval of the TDFC. The advantages of S-AK-DSA are mainly manifested in two aspects. Firstly, S-AK-DSA converts the problem of optimizing the exact value of the upper/lower boundary into the problem of accurately identifying their signs, which can avoid the use of optimization algorithms. Secondly, at different membership levels, the S-AK-DSA method chooses the candidate sample pool and continuously updates the current Kriging model of TD-PF, and the adaptive learning function as well as appropriate stopping criterion can effectively reduce the cost of predicting the signs of the upper/lower boundary and improve the computational accuracy. Four case studies are introduced to demonstrate the feasibility and superiority of the proposed S-AK-DSA approach.
Time-dependent global reliability sensitivity can quantify the effect of input variables in their whole distribution ranges on the time-dependent failure probability. To efficiently estimate it to ...help researchers control the time-dependent failure probability, a novel method is proposed. The proposed method transforms the estimation of unconditional-conditional time-dependent failure probabilities into that of the unconditional-conditional probability density functions (PDFs) of the minimum of time-dependent performance function. Firstly, the minimum of time-dependent performance function is evaluated by adaptive Kriging, and its unconditional-conditional fractional moments are estimated by multiplicative dimensional reduction method (M-DRM). Then, the maximum entropy (MaxEnt) constrained by these fractional moments is used to estimate the unconditional-conditional PDFs, on which the unconditional-conditional time-dependent failure probabilities can be obtained. Finally, the one-dimensional Gaussian quadrature is applied to estimate the time-dependent global reliability sensitivity indices. Due to the high efficiency of adaptive Kriging for estimating the minimum of time-dependent performance function, the avoidance of dimensional curse by M-DRM, and the high efficiency of MaxEnt constrained by fractional moments for estimating PDF, the proposed method can reduce the computational cost dramatically.
The Kriging model with numerical simulation can analyze reliability efficiently, but its extension in multi-mode system is troubled by the hard quantification of correlations among Kriging models of ...multiple limit state functions. For this issue, a new learning strategy (NLS) is proposed by considering correlation effect. Firstly, NLS accurately derives the cumulative distribution function (CDF) boundary of the system Kriging model, and it considers the correlations among Kriging models of all system modes. By this CDF boundary, NLS derives the upper bound probability of the system Kriging model misjudging candidate sample state, on which the most contributive sample is selected to improve the capability of system Kriging model judging system state. Secondly, NLS only adds most contributive sample to the training set of the most easily identified mode to avoid computational cost on updating the Kriging models of unimportant modes. Thirdly, by employing the upper bound of expected relative error of failure probability estimated by prediction and prediction mean of system Kriging model, a convergence criterion is used to improve efficiency under acceptable accuracy. The superiorities, including in selecting training point, updating mode and convergence criterion, of NLS over the up-to-date methods for analyzing the system reliability are demonstrated by examples.
Failure credibility is popular in measuring safety degree of structure under fuzzy uncertainty due to its excellent property of self-duality. Existing methods for estimating failure credibility can ...be mainly divided into two categories, i.e., the simulation-based methods and the optimization-based methods. The simulation-based methods are universal and robust, but time-consuming in dealing with high-dimensional problems. The optimization-based methods are efficient and flexible, but the precision of the failure credibility estimate will be impacted by the local optimum solution resulted from some optimization techniques. Thus, in order to overcome the disadvantages of existing methods, an efficient method by combining genetic algorithm and active learning Kriging (AK-GA) is proposed to estimate failure credibility in this paper. Firstly, by use of the fuzzy simulation, the estimation of failure credibility is transformed into finding for the failure/safety sample that processes the maximum joint membership function (MF) from the fuzzy simulation pool, which makes the credibility estimation less difficult to be computed. Secondly, the genetic algorithm is used to find the failure/safety sample with maximum joint MF. Thirdly, to drastically improve the computational efficiency of the proposed method, active learning Kriging is embedded to accurately and efficiently distinguish the failure and safety sample in computing the fitness function of genetic algorithm. Four mathematical and engineering test examples are used to demonstrate the accuracy and efficiency of the proposed AK-GA for estimating failure credibility.
•Two support vector regression models are developed through Bayesian inference.•The developed regression models provide probabilistic prediction as Kriging model.•Two adaptive algorithms are ...presented to assess small and large failure probability.•Support vector regression outperforms Kriging for medium-dimensional problems.
In this paper, Bayesian support vector regression (SVR) model is developed for structural reliability analysis adaptively. Two SVR models, namely, least-square SVR and ε-SVR, are constructed under the Bayesian inference framework with a square loss function and a ε-insensitive square one respectively. In this framework, a Gaussian process prior is assigned to the regression function, and maximum posterior estimate results in a SVR problem. The proposed Bayesian SVR models provide point-wise probabilistic prediction while keeps the structural risk minimization principle, and it allows us to determine the optimal hyper-parameters by maximizing Bayesian model evidence. Two active learning algorithms are presented based on the Bayesian SVR models to estimate large and small failure probability of complex structure with limited model evaluations respectively. Four benchmark examples are employed to validate the performance of the presented method.
Validating models with correlated multivariate outputs involves the comparison of multiple stochastic quantities. Considering both uncertainty and correlations among multiple responses from model and ...physical observations imposes challenges. Existing marginal comparison methods and the hypothesis testing-based methods either ignore correlations among responses or only reach Boolean conclusions (yes or no) without accounting for the amount of discrepancy between a model and the underlying reality. A new validation metric is needed to quantitatively characterize the overall agreement of multiple responses considering correlations among responses and uncertainty in both model predictions and physical observations. In this paper, by extending the concept of “area metric” and the “u-pooling method” developed for validating a single response, we propose new model validation metrics for validating correlated multiple responses using the multivariate probability integral transformation (PIT). One new metric is the PIT area metric for validating multi-responses at a single validation site. The other is the t-pooling metric that allows for pooling observations of multiple responses observed at multiple validation sites to assess the global predictive capability. The proposed metrics have many favorable properties that are well suited for validation assessment of models with correlated responses. The two metrics are examined and compared with the direct area metric and the marginal u-pooling method respectively through numerical case studies and an engineering example to illustrate their validity and potential benefits.
•Ensemble of surrogate models is investigated for reliability analysis;•An active learning algorithm is developed to enrich the experiment design adaptively.•The proposed method is more efficient ...than the traditional single surrogate method.
Assessing the failure probability of complex structure is a difficult task in presence of various uncertainties. In this paper, a new adaptive approach is developed for reliability analysis by ensemble learning of multiple competitive surrogate models, including Kriging, polynomial chaos expansion and support vector regression. Ensemble of surrogates provides a more robust approximation of true performance function through a weighted average strategy, and it helps to identify regions with possible high prediction error. Starting from an initial experimental design, the ensemble model is iteratively updated by adding new sample points to regions with large prediction error as well as near the limit state through an active learning algorithm. The proposed method is validated with several benchmark examples, and the results show that the ensemble of multiple surrogate models is very efficient for estimating failure probability (>10−4) of complex system with less computational costs than the traditional single surrogate model.
There are various uncertain parameters in the techno-economic assessments (TEAs) of biodiesel production, including capital cost, interest rate, feedstock price, maintenance rate, biodiesel ...conversion efficiency, glycerol price and operating cost. However, fewer studies focus on the influence of these parameters on TEAs. This paper investigated the effects of these parameters on the life cycle cost (LCC) and the unit cost (UC) in the TEAs of biodiesel production. The results show that LCC and UC exhibit variations when involving uncertain parameters. Based on the uncertainty analysis, three global sensitivity analysis (GSA) methods are utilized to quantify the contribution of an individual uncertain parameter to LCC and UC. The GSA results reveal that the feedstock price and the interest rate produce considerable effects on the TEAs. These results can provide a useful guide for entrepreneurs when they plan plants.
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