High-dimensional Reliability Problems Research Articles

An active learning method using deep adversarial autoencoder-based sufficient dimension reduction neural network for high-dimensional reliability analysis

Reliability analysis often requires time-consuming evaluations, especially when dealing with high-dimensional and nonlinear problems. To address this challenge, surrogate model methods are frequently employed. One way to improve the efficiency of surrogate model methods involves selecting informative samples that significantly enhance the accuracy of the surrogate model. This paper introduces a novel approach to facilitate the construction of surrogate models and selection of informative samples in high-dimensional reliability analysis, through an active learning method based on a deep adversarial autoencoder-based sufficient dimension reduction (AAE-SDR) neural network. The AAE-SDR neural network serves as a surrogate model, transforming complex high-dimensional variables into tractable, low-dimensional embeddings relevant to the target. These embeddings are Gaussian-distributed with a distinct latent limit state boundary. A new sampling strategy is proposed to select informative misclassified samples by iteratively identifying candidate samples near the latent limit state boundary and uniformly sampling from the candidate sample dataset based on the latent Gaussian distribution. The effectiveness of the proposed approach is demonstrated through two high-dimensional numerical examples and a cable-stayed bridge case study. Results show that the proposed method simplifies complex high-dimensional reliability problems and provides a relatively accurate estimated failure probability with a limited number of samples.

Adaptive active subspace-based metamodeling for high-dimensional reliability analysis

To address the challenges of reliability analysis in high-dimensional probability spaces, this paper proposes a new metamodeling method that couples active subspace, heteroscedastic Gaussian process, and active learning. The active subspace is leveraged to identify low-dimensional salient features of a high-dimensional computational model. A surrogate computational model is built in the low-dimensional feature space by a heteroscedastic Gaussian process. Active learning adaptively guides the surrogate model training toward the critical region that significantly contributes to the failure probability. A critical trait of the proposed method is that the three main ingredients–active subspace, heteroscedastic Gaussian process, and active learning–are coupled to adaptively optimize the feature space mapping in conjunction with the surrogate modeling. This coupling empowers the proposed method to accurately solve nontrivial high-dimensional reliability problems via low-dimensional surrogate modeling. Finally, numerical examples of a high-dimensional nonlinear function and structural engineering applications are investigated to verify the performance of the proposed method.

Generalized Stratified Sampling for Efficient Reliability Assessment of Structures against Natural Hazards

Performance-based engineering for natural hazards facilitates the design and appraisal of structures with rigorous evaluation of their uncertain structural behavior under potentially extreme stochastic loads expressed in terms of failure probabilities against stated criteria. As a result, efficient stochastic simulation schemes are central to computational frameworks that aim to estimate failure probabilities associated with multiple limit states using limited sample sets. In this work, a generalized stratified sampling scheme is proposed in which two phases of sampling are involved: the first is devoted to the generation of strata-wise samples and the estimation of strata probabilities, whereas the second phase aims at the estimation of strata-wise failure probabilities. Phase-I sampling enables the selection of a generalized stratification variable (i.e., not necessarily belonging to the input set of random variables) for which the probability distribution is not known a priori. To improve the efficiency, Markov Chain Monte Carlo Phase-I sampling is proposed when Monte Carlo simulation is deemed infeasible, and optimal Phase-II sampling is implemented based on user-specified target coefficients of variation for the limit states of interest. The expressions for these coefficients are derived with due regard to the sample correlations induced by the Markov chains and the uncertainty in the estimated strata probabilities. The proposed stochastic simulation scheme reaps the benefits of near-optimal stratified sampling for a broader choice of stratification variables in high-dimensional reliability problems with a mechanism to approximately control the accuracy of the estimators of multiple failure probabilities. The practicality and efficiency of the scheme are demonstrated using two examples involving the estimation of failure probabilities associated with highly nonlinear responses induced by wind and seismic excitations.

Investigation on a novel reliability analysis approach integrating adaptive space division and direction sampling

Multitudes of algorithms have been proposed to evaluate the failure probability of components and systems. Among all these algorithms, direction sampling is a promising one. However, a major computational effort involved in direction sampling is that for each direction sample a root-searching process needs to be conducted to obtain the distance between the origin and the failure surface, which can be computationally expensive. In addition, the failure probability along a direction is hard to obtain when the direction vector intersects the failure surface multiple times. This paper proposes a novel approach to obtain densely populated points locating on the failure surface and utilizing these points to conduct reliability analysis with direction sampling. The process of obtaining the population consists in dividing the initial hypercube space into multiple lattice grids, identifying the grids crossing the failure surface, and dividing these grids into smaller ones. The iterative division process stops when the grids cut across by the failure surface are small enough so that the centers of these grids can be considered located on the failure surface. Several numerical tests are performed to validate the applicability of the proposed approach. Through these tests, it can be concluded that for the situation where the intersection between the direction vector and the failure surface is unique, the proposed approach will lead, with high accuracy, to a failure probability. For the situation where the intersection is not unique, satisfactory estimation of failure probabilities can also be achieved using approximation methods. For high-dimensional reliability problems, clustering techniques can be utilized to reduce the computational cost.

High dimensional reliability analysis based on combinations of adaptive Kriging and dimension reduction technique

AbstractHigh dimensional reliability analysis is unavoidable if a structural system involving stochastic process because Karhunen–Loève (K‐L) expansion is commonly used. Reliability analysis for structural systems with computationally intensive numerical models and high dimensions is challenging. In this study, an effective high dimensional reliability analysis method is proposed based on principal components and active subspace (PCAS) and active Kriging, and is termed as PCAS‐AK. The proposed method can address two shortcomings existing in PCAS: the latter select training samples randomly and cannot deal with parallel processing. The proposed method, however, selects training samples adaptively, subsequently, the training samples that have high contribution to probability of failure are selected, which can improve computational efficiency. To reduce overall computational time and allow parallelization, K‐weighted‐means combined with learning function are used to select multiple samples at each iteration. After the dimension reduction of both input and output spaces, the samples selected by adaptive method are used to construct Kriging model in the latent low‐dimensional space. Three examples are used to demonstrate the applicability and accuracy of the proposed method. Results show that the proposed method can deal with parallel processing and is, generally, more effective than PCAS for high dimensional reliability problems.

Time-Dependent Reliability of Aging Structures: Overview of Assessment Methods

Reliability assessment of engineered structures is a powerful and useful concept to estimate the structural capacity of withstanding hazardous events during their service lives. Taking into account the time variation of both structural resistance and the external load processes, the structural safety level is dependent on the duration of service period of interest, due to the accumulation of hazards by exposure in time. This paper presents an overview on the nonempirical assessment methods for time-dependent reliability of deteriorating structures. Generally, these methods can be classified into two types, namely simulation-based and analytical methods. The former is usually brute, and is especially suitable for solving high-dimensional reliability problems. Conversely, analytical solutions may improve the calculation efficiency significantly, and offer insights into the reliability problem that otherwise could be difficult to achieve through Monte Carlo simulation. Both the simulation-based and analytical methods will be reviewed in this paper. Furthermore, the application of time-dependent reliability methods in practical engineering is discussed. Recommendations for future research efforts are also presented.

Efficient reliability analysis based on deep learning-enhanced surrogate modelling and probability density evolution method

An improved method, termed as the AL-DLGPR-PDEM, is presented to address high-dimensional reliability problems. The novelty of this work lies in developing a complete framework for combining the deep learning (DL) architectures, serving as the utility of dimension reduction, and the Gaussian process regression (GPR), resulting in the so-called DLGPR model. First, the parameters of both the DL and the GPR are inferred using a joint-optimization scheme, rather than the traditional two-step, separate-training scheme. Second, the network configuration of the DLGPR is optimally determined by using a grid-search procedure involving cross-validation, instead of an empirical setting manner. On this basis, the DLGPR is adaptively refined via an active learning (AL)-based sampling strategy, so as to gain the desired DLGPR using as fewer training samples as possible. Eventually, the finalized DLGPR is evaluated at the whole representative point set, thereby the probability density evolution method (PDEM) is conducted accordingly. Two numerical examples are investigated. The first one tackles with the static reliability analysis of a planner steel frame, where the case of small failure probabilities is also considered; the second one addresses the dynamic reliability analysis of the steel frame under fully non-stationary stochastic seismic excitation. Comparisons against other existing reliability methods are conducted as well. Results demonstrate that the proposed AL-DLGPR-PDEM achieves a fair tradeoff between accuracy and efficiency for dealing with high-dimensional reliability problems in both static and dynamic analysis examples.

Kernel principal component analysis-based Gaussian process regression modelling for high-dimensional reliability analysis

An efficient reliability method is presented to address the challenge inherent in the high-dimensional reliability analysis. The critical contribution is an elegant implementation of combining the kernel principal component analysis (KPCA)-based nonlinear dimension reduction and the Gaussian process regression (GPR) surrogate model by introducing a nonintrusive, joint-training scheme. This treatment leads to an optimal KPCA-based subspace in which an accurate low-dimensional GPR model, denoted as the KPCA-GPR model, can be readily achieved. Then, the KPCA-GPR model is combined with the active learning (AL) -based sampling strategy and the Monte Carlo simulation (MCS). In this regard, the newly-added sample at each iteration can be employed simultaneously to both update the estimated KPCA-based subspace and refine the GPR model built on that subspace, which alleviates the 'curse of dimensionality' to some extent and improves progressively the failure probability estimation provided by the low-dimensional GPR model. In order to demonstrate the effectiveness of the proposed method, two numerical examples are investigated, involving both the analysis of high-dimensional, nonlinear, explicit performance functions and the static/dynamic reliability analysis of a truss structure with implicit performance functions. Numerical results indicate that the proposed method is of both accuracy and efficiency for high-dimensional reliability problems.

First-order reliability method based on Harris Hawks Optimization for high-dimensional reliability analysis

The first-order reliability method (FORM) is a prevalent method in the structural reliability community. However, when solving the high-dimensional problem with a highly nonlinear limit state function, FORM usually encounters non-convergence or divergence. In this study, an improved FORM combining Harris Hawks Optimization (HHO-FORM) is presented for high-dimensional reliability analysis. HHO is a meta-heuristic algorithm mimicking the predatory behavior of Harris hawks, and efficient in finding the global optimum of high-dimensional problems. In HHO-FORM, the reliability index is firstly formulated as the solution of a constrained optimization problem according to the FORM theory. Then, the constraints are handled with the exterior penalty function method. In addition, the optimal reliability index is determined by the Harris Hawks Optimization that accelerates the convergence by the population-based mechanism and the strategy of Levy Flight. The HHO-FORM does not require the derivatives of the limit state functions that reduce the computational burden for high-dimensional problems. So the simplicity of HHO-FORM greatly improves the efficiency in solving high-dimensional reliability problems. The HHO-FORM is firstly tested on three challenging numerical high-dimensional problems and then applied to two high-dimensional engineering problems to verify its performance. Four gradient-based FORM algorithms and four heuristic-based FORM algorithms are also compared with the proposed method. The experimental results demonstrate that HHO-FORM provides good accuracy and efficiency for high-dimensional reliability problems.

A novel fractional moments-based maximum entropy method for high-dimensional reliability analysis

• High-dimensional structural reliability problem is properly tackled. • The maximum entropy method with fractional moments is used to derive the PDF. • A new method is proposed for highly efficient fractional moments assessment. • Numerical examples are investigated to demonstrate the efficacy of the proposed method. High-dimensional reliability analysis is still an open challenge in structural reliability community. To address this problem, a new sampling approach, named the good lattice point method based partially stratified sampling is proposed in the fractional moments-based maximum entropy method. In this approach, the original sample space is first partitioned into several orthogonal low-dimensional sample spaces, say 2 and 1 dimensions. Then, the samples in each low-dimensional sample space are generated by the good lattice point method, which are deterministic points and possess the property of large variance reduction. Finally, the samples in the original space can be obtained by randomly pairing the samples in low-dimensions, which may also significantly reduce the variance in high-dimensional cases. Then, this sampling approach is applied to evaluate the low-order fractional moments in the maximum entropy method with the tradeoff of efficiency and accuracy for high-dimensional reliability problems. In this regard, the probability density function of the performance function involving a large number of random inputs can be derived accordingly, where the reliability can be straightforwardly evaluated by a simple integral over the probability density function. Numerical examples are studied to validate the proposed method, which indicate the proposed method is of accuracy and efficiency for high-dimensional reliability analysis.

An efficient approach for high-dimensional structural reliability analysis

Abstract This paper presents a new method to address the challenge of high-dimensional reliability analysis based on a small number of samples. The method is established based on the maximum entropy method (MEM) with the low-order fractional moments as constraints. A coordinate transformation is first implemented since a positive random variable is required for fractional operations. Then, an estimator-corrector scheme is employed to obtain the probability density function (PDF) of the performance function. This scheme first promptly provides an estimated fractional orders and Lagrange multipliers by solving a linear system of equations as initial values, and then searches the more accurate solutions around the initial values to recover the PDF. Besides, the local optimum can be avoided by using the estimator-corrector scheme. The centered L2 (CL2) discrepancy oriented sequential Latin-hyper cube simulation is proposed to evaluate the low-order fractional moments involved in MEM, which is of critical importance to the efficiency and accuracy for high-dimensional reliability analysis. Typical numerical examples are investigated to validate the proposed method. The results show that the proposed method is of efficacy for high-dimensional reliability problems, even with very low failure probabilities.

Cross entropy-based importance sampling using Gaussian densities revisited

The computation of the probability of a rare (failure) event is a common task in structural reliability analysis. In most applications, the numerical model defining the rare event is nonlinear and the resulting failure domain often multimodal. One strategy for estimating the probability of failure in this context is the importance sampling method. The efficiency of importance sampling depends on the choice of the importance sampling density. A near-optimal sampling density can be found through application of the cross entropy method. The cross entropy method is an adaptive sampling approach that determines the sampling density through minimizing the Kullback-Leibler divergence between the theoretically optimal importance sampling density and a chosen parametric family of distributions. In this paper, we investigate the suitability of the multivariate normal distribution and the Gaussian mixture model as importance sampling densities within the cross entropy method. Moreover, we compare the performance of the cross entropy method to sequential importance sampling, another recently proposed adaptive sampling approach, which uses the Gaussian mixture distribution as a proposal distribution within a Markov Chain Monte Carlo algorithm. For the parameter updating of the Gaussian mixture within the cross entropy method, we propose a modified version of the expectation-maximization algorithm that works with weighted samples. To estimate the number of distributions in the mixture, the density-based spatial clustering of applications with noise (DBSCAN) algorithm is adapted to the use of weighted samples. We compare the performance of the different methods in several examples, including component reliability problems, system reliability problems and reliability in varying dimensions. The results show that the cross entropy method using a single Gaussian outperforms the cross entropy method using Gaussian mixture and that both distribution types are not suitable for high dimensional reliability problems.

RM5Tree: Radial basis M5 model tree for accurate structural reliability analysis

The surrogate models-based prediction of performance functions is an efficient and accurate methodology in structural reliability analyses. In this paper, the M5 model tree (M5Tree) is improved based on radial basis training data set and it is named as Radial basis M5Tree (RM5Tree). To predict the performance function, the random input variables are transferred from ordinal space to radial space using several effective points for nonlinear calibrated model of RM5Tree. The input datasets are controlled using the radial dataset for high-dimensional reliability problems to reduce computational efforts to evaluate the performance function. The abilities of RM5Tree using Monte Carlo Simulation (MCS) with respect to accuracy and efficiency are investigated through five nonlinear reliability problems. The results indicate that the proposed RM5Tree performs superior manner in accuracy and efficiency compared to the M5Tree, response surface method (RSM) and first order reliability method.

Hyper-spherical extrapolation method (HEM) for general high dimensional reliability problems

To tackle challenges in low-probability, high-dimensional reliability analysis, this paper proposes hyper-spherical extrapolation method to estimate the failure probabilities efficiently and accurately. The extrapolation method employs hyper-spherical formulations of reliability problems developed based on geometric insights of high dimensional standard normal space. The proposed method can extrapolate the low probability region of interest using failure probabilities obtained from high/median probability region. Owing to the generality of the formulation, the proposed method is expected to work for general, component and system reliability problems defined in high-dimensional space of random variables. Using different presumptions on extrapolation, two slightly different versions of the extrapolation method are developed. Numerical examples with analytical limit-state functions and those concerning nonlinear random vibration of a hysteretic oscillator are investigated to test and demonstrate the performance of the proposed method. Finally, to facilitate an in-depth understanding of the proposed method and further developments, insights gained from the development of the method are also provided. The supporting source code and data are available for download at https://github.com/ziqidwang/Hyper-spherical-extrapolation-method.

Bayesian post-processor and other enhancements of Subset Simulation for estimating failure probabilities in high dimensions

Estimation of small failure probabilities is one of the most important and challenging computational problems in reliability engineering. The failure probability is usually given by an integral over a high-dimensional uncertain parameter space that is difficult to evaluate numerically. This paper focuses on enhancements to Subset Simulation (SS), proposed by Au and Beck, which provides an efficient algorithm based on MCMC (Markov chain Monte Carlo) simulation for computing small failure probabilities for general high-dimensional reliability problems. First, we analyze the Modified Metropolis algorithm (MMA), an MCMC technique, which is used in SS for sampling from high-dimensional conditional distributions. The efficiency and accuracy of SS directly depends on the ergodic properties of the Markov chains generated by MMA, which control how fast the chain explores the parameter space. We present some observations on the optimal scaling of MMA for efficient exploration, and develop an optimal scaling strategy for this algorithm when it is employed within SS. Next, we provide a theoretical basis for the optimal value of the conditional failure probability p 0, an important parameter one has to choose when using SS. We demonstrate that choosing any p 0 ∈ [0.1, 0.3] will give similar efficiency as the optimal value of p 0. Finally, a Bayesian post-processor SS+ for the original SS method is developed where the uncertain failure probability that one is estimating is modeled as a stochastic variable whose possible values belong to the unit interval. Simulated samples from SS are viewed as informative data relevant to the system’s reliability. Instead of a single real number as an estimate, SS+ produces the posterior PDF of the failure probability, which takes into account both prior information and the information in the sampled data. This PDF quantifies the uncertainty in the value of the failure probability and it may be further used in risk analyses to incorporate this uncertainty. To demonstrate SS+, we consider its application to two different reliability problems: a linear reliability problem and reliability analysis of an elasto-plastic structure subjected to strong seismic ground motion. The relationship between the original SS and SS+ is also discussed.

Applications of asymptotic sampling on high dimensional structural dynamic problems

The paper represents application of the asymptotic sampling on various structural models subjected to random excitations. A detailed study on the effect of different distributions of the so-called support points is performed. This study shows that the distribution of the support points has considerable effect on the final estimations of the method, in particular on the coefficient of variation of the estimated failure probability. Based on these observations, a simple optimization algorithm is proposed which distributes the support points so that the coefficient of variation of the method is minimized. Next, the method is applied on different cases of linear and nonlinear systems with a large number of random variables representing the dynamic excitation. The results show that asymptotic sampling is capable of providing good approximations of low failure probability events for very high dimensional reliability problems in structural dynamics.

The Horseracing Simulation algorithm for evaluation of small failure probabilities

Over the past decade, the civil engineering community has ever more realized the importance and perspective of reliability-based design optimization (RBDO). Since then several advanced stochastic simulation algorithms for computing small failure probabilities encountered in reliability analysis of engineering systems have been developed: Subset Simulation (Au and Beck (2001) [2]), Line Sampling (Schuëller et al. (2004) [3]), The Auxiliary Domain Method (Katafygiotis et al. (2007) [4]), ALIS (Katafygiotis and Zuev (2007) [5]), etc. In this paper we propose a novel advanced stochastic simulation algorithm for solving high-dimensional reliability problems, called Horseracing Simulation (HRS). The key idea behind HS is as follows. Although the reliability problem itself is high-dimensional, the limit-state function maps this high-dimensional parameter space into a one-dimensional real line. This mapping transforms a high-dimensional random parameter vector, which may represent the stochastic input load as well as any uncertain structural parameters, into a random variable with unknown distribution, which represents the uncertain structural response. It turns out that the corresponding cumulative distribution function (CDF) of this random variable of interest can be accurately approximated by empirical CDFs constructed from specially designed samples. The generation of samples is governed by a process of “racing” towards the failure domain, hence the name of the algorithm. The accuracy and efficiency of the new method are demonstrated with a real-life wind engineering example.

Geometric insight into the challenges of solving high-dimensional reliability problems

In this paper we adopt a geometric perspective to highlight the challenges associated with solving high-dimensional reliability problems. Adopting a geometric point of view we highlight and explain a range of results concerning the performance of several well-known reliability methods. We start by investigating geometric properties of the N -dimensional Gaussian space and the distribution of samples in such a space or in a subspace corresponding to a failure domain. Next, we discuss Importance Sampling (IS) in high dimensions. We provide a geometric understanding as to why IS generally does not work in high dimensions [Au SK, Beck JL. Importance sampling in high dimensions. Structural Safety 2003;25(2):139–63]. We furthermore challenge the significance of “design point” when dealing with strongly nonlinear problems. We conclude by showing that for the general high-dimensional nonlinear reliability problems the selection of an appropriate fixed IS density is practically impossible. Next, we discuss the simulation of samples using Markov Chain Monte Carlo (MCMC) methods. Firstly, we provide a geometric explanation as to why the standard Metropolis–Hastings (MH) algorithm does “not work” in high-dimensions. We then explain why the modified Metropolis–Hastings (MMH) algorithm introduced by Au and Beck [Au SK, Beck JL. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics 2001;16(4):263–77] overcomes this problem. A study of the correlation of samples obtained using MMH as a function of different parameters follows. Such study leads to recommendations for fine-tuning the MMH algorithm. Finally, the MMH algorithm is compared with the MCMC algorithm proposed by Katafygiotis and Cheung [Katafygiotis LS, Cheung SH. Application of spherical subset simulation method and auxiliary domain method on a benchmark reliability study, Structural Safety 2006 (in print)] in terms of the correlation of samples they generate.

Application of spherical subset simulation method and auxiliary domain method on a benchmark reliability study

This paper addresses a benchmark study designed to evaluate the performance of various methods in calculating the reliability of large systems. In particular, this paper focuses on evaluating two reliability methods recently proposed by the authors, referred to as spherical subset simulation (S 3) and auxiliary domain method (ADM). S 3 is based on dividing the failure domain into a number of appropriately selected subregions and calculating the failure probability as a sum of the probabilities associated with each of these subregions. The probability of each subregion is calculated as a product of factors. These factors can be estimated accurately by a relatively small number of samples generated according to the conditional distribution corresponding to the particular subregion. The generation of such samples is achieved through Markov Chain Monte Carlo (MCMC) simulations using a MCMC algorithm proposed by the authors. The proposed method is very robust and is suitable for treating general high-dimensional problems such as the given benchmark problems. ADM is applicable to reliability problems involving deterministic dynamic systems subjected to stochastic excitation. The first step in ADM involves the determination of an auxiliary failure domain (AFD). The choice of the AFD is based on preliminary MCMC simulations in the target failure domain. It must be noted that although the AFD is chosen to be specified as a union of linear failure domains, the method does not assume any restriction with respect to the target failure domain, which is assumed to be generally non-linear. Once the AFD is determined, the ADM proceeds with a modified subset simulation procedure where the first step involves the direct simulation of points in the AFD. This is in contrast to standard subset simulation (SSM) where the first step involves standard Monte Carlo Simulations. The number of steps and the computational effort required by ADM, assuming an appropriate AFD is chosen, can be smaller than that required by SSM. Results for the benchmark problems show that both S 3 and ADM are efficient for treating high dimensional reliability problems.