Non-intrusive imprecise stochastic simulation by line sampling

The aim of this article is to study the reliability analysis, parametric reliability sensitivity analysis and global reliability sensitivity analysis of structures with extremely rare failure events. First, the global reliability sensitivity indices are restudied, and we show that the total effect index can also be interpreted as the effect of randomly copying each individual input variable on the failure surface. Second, a new method, denoted as Active learning Kriging Markov Chain Monte Carlo (AK-MCMC), is developed for adaptively approximating the failure surface with active learning Kriging surrogate model as well as dynamically updated Monte Carlo or Markov chain Monte Carlo populations. Third, the AK-MCMC procedure combined with the quasi-optimal importance sampling procedure is extended for estimating the failure probability and the parametric reliability sensitivity and global reliability sensitivity indices. For estimating the global reliability sensitivity indices, two new importance sampling estimators are derived. The AK-MCMC procedure can be regarded as a combination of the classical Monte Carlo Simulation (AK-MCS) and subset simulation procedures, but it is much more effective when applied to extremely rare failure events. Results of test examples show that the proposed method can accurately and robustly estimate the extremely small failure probability (e.g. 1e–9) as well as the related parametric reliability sensitivity and global reliability sensitivity indices with several dozens of function calls.

Global sensitivity analysis has been widely used to detect the relative contributions of input variables to the uncertainty of model output, and then more resources can be assigned to the important input variables to reduce the uncertainty of model output more efficiently. In this paper, a new kind of global sensitivity index based on Gini’s mean difference is proposed. The proposed sensitivity index is more robust than the variance-based first order sensitivity index for the cases with non-normal distributions. Through the decomposition of Gini’s mean difference, it shows that the proposed sensitivity index can be represented by the energy distance, which measures the difference between probability distributions. Therefore, the proposed sensitivity index also takes the probability distribution of model output into consideration. In order to estimate the proposed sensitivity index efficiently, an efficient Monte Carlo simulation method is also proposed, which avoids the nested sampling procedure. The test examples show that the proposed sensitivity index is more robust than the variance-based first order sensitivity index for the cases with non-normal distributions.

<p>The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters.</p> <p>The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process.</p> <p>The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.</p>

Statistical model selection and evaluation methods like Akaike information criteria (AIC) and Monte Carlo simulation (MCS) have often established efficient output for reliability analysis with large sample size. Information criterion can provide better model selection and evaluation in small sample sizes setup by considering the well-known measure of bootstrap resampling. Our purpose is to utilize the capabilities of bootstrap resampling in information criterion to check for uncertainty arising from model selection as well as statistics of interest for small sample size using reliability analysis. In this study, therefore, a unique and efficient simulation scheme is proposed which contemplates the best model selection devised from efficient bootstrap simulation or variance reduced bootstrap information criterion to be combined with reliability analysis. It is beneficial to compute the spread of reliability values as against solitary fixed values with desirable statistics of interest for uncertainty analysis. The proposed simulation scheme is verified using a number of sample size focused response functions under repetitions-centred approach with AIC-based reliability analysis for comparison and MCS for accuracy. The results show that the proposed simulation scheme aids the statistics of interest by reducing the spread and hence the uncertainty in sample size-based reliability analysis when compared with conventional methods.

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bounds of system failure probability. The second method is SA based Nataf approximation, in which SA is used to estimate the probability density function (PDF) and cumulative distribution function (CDF) for the approximately linearized performance function of each failure mode. After the PDF of each failure mode and the correlation coefficients among approximately linearized performance functions are estimated, Nataf distribution is employed to approximate the joint PDF of multiple structural system performance functions, and then the system failure probability can be estimated directly by numerical simulation using the joint PDF. The third method is SA based line sampling (LS). The standardization transformation is needed to eliminate the dimensions of variables firstly in this case. Then LS method can express the system failure probability as an arithmetic average of a set of failure probabilities of the linear performance functions, and the probabilities of the linear performance functions can be estimated by the SA in the non-normal variables space. By comparing basic concepts, implementations and results of illustrations, the following conclusions can be drawn: (1) The first method can only obtain the bounds of system failure probability and it is only acceptable for the linear limit state function; (2) the second method can give the estimation of system failure probability, and its error mostly results from the approximation of Nataf distribution for the joint PDF of the structural system performance functions and the linearization of the performance functions; (3) the SA based LS method can obtain the estimator of system failure probability, which converges to the actual value along with the increase of sample size. The SA based LS method considers the influence of nonlinearity of limit state function on the failure probability, and it is acceptable for the structural system both with a single failure mode and with multiple failure modes, therefore it has the widest applicability.

Rare events are of particular interest in synthetic biology because rare biochemical events may be catastrophic to a biological system by, for example, triggering irreversible events such as off-target drug delivery. To estimate the probability of rare events efficiently, several weighted stochastic simulation methods have been developed. Under optimal parameters and model conditions, these methods can greatly improve simulation efficiency in comparison to traditional stochastic simulation. Unfortunately, the optimal parameters and conditions cannot be deduced a priori. This paper presents a critical survey of weighted stochastic simulation methods. It shows that the methods considered here cannot consistently, efficiently, and exactly accomplish the task of rare event simulation without resorting to a computationally expensive calibration procedure, which undermines their overall efficiency. The results suggest that further development is needed before these methods can be deployed for general use in biological simulations.

Congestion revenue right (CRR) is a financial tool designed to manage congestion risk in electricity markets. In this paper, a new method for valuation of CRR -stochastic simulation of transmission shadow prices- is introduced. First, operational and functional aspects associated with electricity markets are presented. The concept of CRR and its importance as a tool for managing congestion risk is presented. Second, a stochastic simulation model for valuation of CRR is developed. The applicability of it to different system conditions is analyzed. The performance of the methodology is tested in numerical examples. Third, details of the underlying random numbers generator are discussed. The possibility of using different measures for co-dependence is explored. The estimation of transmission shadow prices using different stochastic processes, in particular extreme value theory and regime switching process are mentioned. Finally, conclusions of the paper are presented.

In stochastic simulation, we construct mathematical models to imitate the behavior of real systems, use computers to sample behavioral histories (sample paths) of these models, and exploit those samples to improve decision making with the real system. The imitation part can be very challenging, in particular for modeling uncertainty. Fitting univariate probability distribution to data is far from sufficient. Modeling the dependence is very important and much more challenging. It involves multivariate distributions, copulas, stochastic processes, and other complicated stochastic objects. Simulating the model on a computer also involves an imitation game, to simulate the realizations of random variables and stochastic processes with deterministic algorithms on a computer. Random number generation involves writing deterministic computer programs that can imitate simple probabilistic models such as independent uniform random variables uniformly distributed over the interval (0, 1). An "exact" algorithmic implantation of such models is theoretically impossible, so we settle for a reasonable fake. The talk will give snapshots and expose ideas collected from the author's journey thought stochastic simulation. The tour will start with random number generation and visit some challenging problems such as stochastic modeling, simulation-based optimization, rare events, simulation on parallel processors, and future challenges.

The sampling intensive mining and environmental industries share a common need for representative data but differ in the motivations to accomplish this objective. The desire to obtain representative sample data for “commodities” in the mining industry is driven by anticipated economic gain and the exploitation of natural resources. The desire to obtain representative sample data for “contaminants” in the environmental industry is driven by anticipated social gain and the protection of natural resources. In terms of obtaining reliably representative data, motivation driven by economic gain has thus far been the clear winner. Theory of Sampling concepts are well established and tested in the mining industry. The environmental industry, in contrast, has traditionally been plagued by scientifically unsound sampling practices and data that are not reliably representative of conditions in the field. This has significant implications for topics ranging from the efficient identification and remediation of contaminated industrial lands to the accurate assessment of risk to human health and the environment. This paper explores the nature and cause of this dichotomy and presents a methodical approach for application of Theory of Sampling concepts to environmental testing of soil, water and air. Much of the problem is tied to a general recognition of compositional and distributional heterogeneity in contaminated media but unawareness of a method to control it or an understanding of the magnitude of potential error. As a result, published regulatory guidance focused on classical sampling and statistical methods appropriate for testing of“finite element” media. A lone exception is testing of indoor air, where concepts of “Decision Units” and sampling methods more appropriate for testing of “infinite element” media have long been employed to control and represent heterogeneity. The solutions are, in hindsight, relatively simple. Pushback from affected parties and even scientists and environmental agencies can be significant, however. This is primarily due to a lack of training of environmental professionals in the Theory of Sampling and the common absence of clear evidence of erroneous or misleading sample data in the field. Reluctance to change is also tied in some cases to implications regarding liability for past and ongoing projects. The need for more reliable, efficient and science-based methods to assess and address risk posed by environmental contamination is clear, however. Progress will be made by countries like China that are beginning to tackle legacies of early development and are able to learn from the successes as well as the mistakes of countries that have been addressing environmental contamination for several decades. Training of environmental workers as well as pressure from liability-savvy responsible parties, attorneys and financial institutions will continue to force the industry to evolve, to the benefit of the environment as well as stakeholders on all sides.

The reliability of a simple turbomachinery model was calculated to demonstrate the application of a newly developing system integration tool, Probabilistic Design and Analysis Framework(PRODAF), along with efficient probabilistic methods using a response surface method. The model represents a system consisting of hypothetical turbine components. The parts include a blade, disk, and shaft with an applied angular velocity. All the components were modeled with the properties of the nickel alloy, Inconel 718. A response surface was calculated for the system of components to improve probabilistic computational efficiency. In addition, a fast probability integration method, Advanced First Order Reliability Method (AFORM), was used for the probabilistic analysis in order to provide an efficient analysis as possible. Geometric dimensions, the applied load, and material yield strength were varied for this study. The probability of failure was determined using the maximum first principal stress response and the material yield strength. A simple G function using the difference between strength and loading stress was used to determine failure limits. The probabilistic sensitivity of the failure response relative to the individual variables was determined also with material yield strength having the greatest influence. The model was recreated with every iteration of the probabilistic analysis in order to vary the geometry. As a result, the response surface method has a significant impact on improving computational efficiency and enabling reliability analysis with rapid turnaround.

Extreme quantiles are important measures in reliability analysis. At the system design stage, quantiles are often estimated via stochastic simulations. This article aims to quantify quantile estimation uncertainties by constructing confidence intervals using importance sampling when quantiles are estimated via stochastic computer models. We validate the asymptotic normality for the importance sampling quantile estimator and construct a theoretically valid confidence interval in a closed form. A drawback of the theoretical confidence interval is that it needs to consistently estimate a variance parameter. To resolve the limitation of the theoretical confidence interval, we present batching-based approaches that are also built upon the asymptotic normality of the quantile estimator. We compare the estimation performance of studied methods and other alternative methods using numerical examples and wind turbine case study.

A new perspective on the simulation of cross-correlated random fields

The seismic safety of reinforced concrete containment building can be evaluated by probabilistic analysis considering randomness of earthquake, which is more rational than deterministic analysis. In the safety assessment of earthquake-resistant structures by the deterministic theory, it is not easy to consider the effects of random variables but the reliability theory and random vibration theory are useful to assess the seismic safety with considering random effects. The reliability assessment of reinforced concrete containment building subjected to earthquake load includes the structural analysis considering random variables such as load, resistance and analysis method, the definition of limit states and the reliability analysis. The reliability analysis procedure requires much time and labor and also needs to get the high confidence in results. In this study, random vibration analysis of containment building is performed with random variables as earthquake load, concrete compressive strength, modal damping ratio. The seismic responses of critical elements of structure are approximated at the most probable failure point by the response surface method. The response surface method helps to figure out the quantitative characteristics of structural response variability. And the limit state is defined as the failure surface of concrete under multi-axial stress, finally the limit state probability of failure can be obtained simply by first-order second moment method. The reliability analysis for the multiaxial strength limit state and the uniaxial strength limit state is performed and the results are compared with each other. This study concludes that the multiaxial failure criterion is a likely limit state to predict concrete failure strength under combined state of stresses and the reliability analysis results are compatible with the fact that the maximum compressive strength of concrete under biaxial compression state increases.

Variance-based reliability sensitivity with dependent inputs using failure samples

A new framework for multi-site stochastic rainfall generator based on empirical orthogonal function analysis and Hilbert-Huang transform