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Sequential directional importance sampling (SDIS) is an efficient adaptive simulation method for estimating failure probabilities. It expresses the failure probability as the product of a group of integrals that are easy to estimate, wherein the first one is estimated with Monte Carlo simulation (MCS), and all the subsequent ones are estimated with directional importance sampling. In this work, we propose an enhanced SDIS method for structural reliability analysis. We discuss the efficiency of MCS for estimating the first integral in standard SDIS and propose using Subset Simulation as an alternative method. Additionally, we propose a Kriging-based active learning algorithm tailored to identify multiple roots in certain important directions within a specificed search interval. The performance of the enhanced SDIS is demonstrated through various complex benchmark problems. The results show that the enhanced SDIS is a versatile reliability analysis method that can efficiently and robustly solve challenging reliability problems

Branch and bound algorithms have been developed for reliability analysis of coherent systems. They exhibit a set of advantages; in particular, they can find a computationally efficient representation of a system failure or survival event, which can be re-used when the input probability distributions change over time or when new data is available. However, existing branch-and-bound algorithms can handle only a limited set of system performance functions, mostly network connectivity and maximum flow. Furthermore, they run redundant analyses on component vector states whose system state can be inferred from previous analysis results. This study addresses these limitations by proposing branch and bound for reliability analysis of general coherent systems} (BRC) algorithm: an algorithm that automatically finds minimal representations of failure/survival events of general coherent systems. Computational efficiency is attained by dynamically inferring importance of component events from hitherto obtained results. We demonstrate advantages of the BRC method as a real-time risk management tool by application to the Eastern Massachusetts highway benchmark network., Comment: Preprint for peer-reviewed article

We present a new surrogate model for emulating the behavior of complex nonlinear dynamical systems with external stochastic excitation. The model represents the system dynamics in state space form through a sparse Kriging model. The resulting surrogate model is termed state space Kriging (S2K) model. Sparsity in the Kriging model is achieved by selecting an informative training subset from the observed time histories of the state vector and its derivative with respect to time. We propose a tailored technique for designing the training time histories of state vector and its derivative, aimed at enhancing the robustness of the S2K prediction. We validate the performance of the S2K model with various benchmarks. The results show that S2K yields accurate prediction of complex nonlinear dynamical systems under stochastic excitation with only a few training time histories of state vector.

Global variance-based reliability sensitivity indices arise from a variance decomposition of the indicator function describing the failure event. The first-order indices reflect the main effect of each variable on the variance of the failure event and can be used for variable prioritization; the total-effect indices represent the total effect of each variable, including its interaction with other variables, and can be used for variable fixing. This contribution derives expressions for the variance-based reliability indices of systems with multiple failure modes that are based on the first-order reliability method (FORM). The derived expressions are a function of the FORM results and, hence, do not require additional expensive model evaluations. They do involve the evaluation of multinormal integrals, for which effective solutions are available. We demonstrate that the derived expressions enable an accurate estimation of variance-based reliability sensitivities for general system problems to which FORM is applicable.

To maximize its value, the design, development and implementation of Structural Health Monitoring (SHM) should focus on its role in facilitating decision support. In this position paper, we offer perspectives on the synergy between SHM and decision-making. We propose a classification of SHM use cases aligning with various dimensions that are closely linked to the respective decision contexts. The types of decisions that have to be supported by the SHM system within these settings are discussed along with the corresponding challenges. We provide an overview of different classes of models that are required for integrating SHM in the decision-making process to support management and operation and maintenance of structures and infrastructure systems. Fundamental decision-theoretic principles and state-of-the-art methods for optimizing maintenance and operational decision-making under uncertainty are briefly discussed. Finally, we offer a viewpoint on the appropriate course of action for quantifying, validating and maximizing the added value generated by SHM. This work aspires to synthesize the different perspectives of the SHM, Prognostic Health Management (PHM), and reliability communities, and deliver a roadmap towards monitoring-based decision support.

We propose a novel Deep Reinforcement Learning (DRL) architecture for sequential decision processes under uncertainty, as encountered in inspection and maintenance (I&M) planning. Unlike other DRL algorithms for (I&M) planning, the proposed +RQN architecture dispenses with computing the belief state and directly handles erroneous observations instead. We apply the algorithm to a basic I&M planning problem for a one-component system subject to deterioration. In addition, we investigate the performance of Monte Carlo tree search for the I&M problem and compare it to the +RQN. The comparison includes a statistical analysis of the two methods' resulting policies, as well as their visualization in the belief space.

We employ the Bayesian improved cross entropy (BiCE) method for rare event estimation in static networks and choose the categorical mixture as the parametric family to capture the dependence among network components. At each iteration of the BiCE method, the mixture parameters are updated through the weighted maximum a posteriori (MAP) estimate, which mitigates the overfitting issue of the standard improved cross entropy (iCE) method through a novel balanced prior, and we propose a generalized version of the expectation-maximization (EM) algorithm to approximate this weighted MAP estimate. The resulting importance sampling distribution is proved to be unbiased. For choosing a proper number of components $K$ in the mixture, we compute the Bayesian information criterion (BIC) of each candidate $K$ as a by-product of the generalized EM algorithm. The performance of the proposed method is investigated through a simple illustration, a benchmark study, and a practical application. In all these numerical examples, the BiCE method results in an efficient and accurate estimator that significantly outperforms the standard iCE method and the BiCE method with the independent categorical distribution.

Rare event simulation and rare event probability estimation are important tasks within the analysis of systems subject to uncertainty and randomness. Simultaneously, accurately estimating rare event probabilities is an inherently difficult task that calls for dedicated tools and methods. One way to improve estimation efficiency on difficult rare event estimation problems is to leverage gradients of the computational model representing the system in consideration, e.g., to explore the rare event faster and more reliably. We present a novel approach for estimating rare event probabilities using such model gradients by drawing on a technique to generate samples from non-normalized posterior distributions in Bayesian inference - the Stein variational gradient descent. We propagate samples generated from a tractable input distribution towards a near-optimal rare event importance sampling distribution by exploiting a similarity of the latter with Bayesian posterior distributions. Sample propagation takes the shape of passing samples through a sequence of invertible transforms such that their densities can be tracked and used to construct an unbiased importance sampling estimate of the rare event probability - the Stein variational rare event estimator. We discuss settings and parametric choices of the algorithm and suggest a method for balancing convergence speed with stability by choosing the step width or base learning rate adaptively. We analyze the method's performance on several analytical test functions and two engineering examples in low to high stochastic dimensions ($d = 2 - 869$) and find that it consistently outperforms other state-of-the-art gradient-based rare event simulation methods.

Partially Observable Markov Decision Processes (POMDPs) can model complex sequential decision-making problems under stochastic and uncertain environments. A main reason hindering their broad adoption in real-world applications is the lack of availability of a suitable POMDP model or a simulator thereof. Available solution algorithms, such as Reinforcement Learning (RL), require the knowledge of the transition dynamics and the observation generating process, which are often unknown and non-trivial to infer. In this work, we propose a combined framework for inference and robust solution of POMDPs via deep RL. First, all transition and observation model parameters are jointly inferred via Markov Chain Monte Carlo sampling of a hidden Markov model, which is conditioned on actions, in order to recover full posterior distributions from the available data. The POMDP with uncertain parameters is then solved via deep RL techniques with the parameter distributions incorporated into the solution via domain randomization, in order to develop solutions that are robust to model uncertainty. As a further contribution, we compare the use of transformers and long short-term memory networks, which constitute model-free RL solutions, with a model-based/model-free hybrid approach. We apply these methods to the real-world problem of optimal maintenance planning for railway assets.

Reliability sensitivity analysis is concerned with measuring the influence of a system's uncertain input parameters on its probability of failure. Statistically dependent inputs present a challenge in both computing and interpreting these sensitivity indices; such dependencies require discerning between variable interactions produced by the probabilistic model describing the system inputs and the computational model describing the system itself. To accomplish such a separation of effects in the context of reliability sensitivity analysis we extend on an idea originally proposed by Mara and Tarantola (2012) for model outputs unrelated to rare events. We compute the independent (influence via computational model) and full (influence via both computational and probabilistic model) contributions of all inputs to the variance of the indicator function of the rare event. We compute this full set of variance-based sensitivity indices of the rare event indicator using a single set of failure samples. This is possible by considering $d$ different hierarchically structured isoprobabilistic transformations of this set of failure samples from the original $d$-dimensional space of dependent inputs to standard-normal space. The approach facilitates computing the full set of variance-based reliability sensitivity indices with a single set of failure samples obtained as the byproduct of a single run of a sample-based rare event estimation method. That is, no additional evaluations of the computational model are required. We demonstrate the approach on a test function and two engineering problems.

Prognostic Health Management aims to predict the Remaining Useful Life (RUL) of degrading components/systems utilizing monitoring data. These RUL predictions form the basis for optimizing maintenance planning in a Predictive Maintenance (PdM) paradigm. We here propose a metric for assessing data-driven prognostic algorithms based on their impact on downstream PdM decisions. The metric is defined in association with a decision setting and a corresponding PdM policy. We consider two typical PdM decision settings, namely component ordering and/or replacement planning, for which we investigate and improve PdM policies that are commonly utilized in the literature. All policies are evaluated via the data-based estimation of the long-run expected maintenance cost per unit time, using monitored run-to-failure experiments. The policy evaluation enables the estimation of the proposed metric. We employ the metric as an objective function for optimizing heuristic PdM policies and algorithms' hyperparameters. The effect of different PdM policies on the metric is initially investigated through a theoretical numerical example. Subsequently, we employ four data-driven prognostic algorithms on a simulated turbofan engine degradation problem, and investigate the joint effect of prognostic algorithm and PdM policy on the metric, resulting in a decision-oriented performance assessment of these algorithms.

Structural Health Monitoring (SHM) describes a process for inferring quantifiable metrics of structural condition, which can serve as input to support decisions on the operation and maintenance of infrastructure assets. Given the long lifespan of critical structures, this problem can be cast as a sequential decision making problem over prescribed horizons. Partially Observable Markov Decision Processes (POMDPs) offer a formal framework to solve the underlying optimal planning task. However, two issues can undermine the POMDP solutions. Firstly, the need for a model that can adequately describe the evolution of the structural condition under deterioration or corrective actions and, secondly, the non-trivial task of recovery of the observation process parameters from available monitoring data. Despite these potential challenges, the adopted POMDP models do not typically account for uncertainty on model parameters, leading to solutions which can be unrealistically confident. In this work, we address both key issues. We present a framework to estimate POMDP transition and observation model parameters directly from available data, via Markov Chain Monte Carlo (MCMC) sampling of a Hidden Markov Model (HMM) conditioned on actions. The MCMC inference estimates distributions of the involved model parameters. We then form and solve the POMDP problem by exploiting the inferred distributions, to derive solutions that are robust to model uncertainty. We successfully apply our approach on maintenance planning for railway track assets on the basis of a "fractal value" indicator, which is computed from actual railway monitoring data.

We propose a modification of the improved cross entropy (iCE) method to enhance its performance for network reliability assessment. The iCE method performs a transition from the nominal density to the optimal importance sampling (IS) density via a parametric distribution model whose cross entropy with the optimal IS is minimized. The efficiency and accuracy of the iCE method are largely influenced by the choice of the parametric model. In the context of reliability of systems with independent multi-state components, the obvious choice of the parametric family is the categorical distribution. When updating this distribution model with standard iCE, the probability assigned to a certain category often converges to 0 due to lack of occurrence of samples from this category during the adaptive sampling process, resulting in a poor IS estima tor with a strong negative bias. To circumvent this issue, we propose an algorithm termed Bayesian improved cross entropy method (BiCE). Thereby, the posterior predictive distribution is employed to update the parametric model instead of the weighted maximum likelihood estimation approach employed in the original iCE method. A set of numerical examples illustrate the efficiency and accuracy of the proposed method.

In inverse problems, the parameters of a model are estimated based on observations of the model response. The Bayesian approach is powerful for solving such problems; one formulates a prior distribution for the parameter state that is updated with the observations to compute the posterior parameter distribution. Solving for the posterior distribution can be challenging when, e.g., prior and posterior significantly differ from one another and/or the parameter space is high-dimensional. We use a sequence of importance sampling measures that arise by tempering the likelihood to approach inverse problems exhibiting a significant distance between prior and posterior. Each importance sampling measure is identified by cross-entropy minimization as proposed in the context of Bayesian inverse problems in Engel et al. (2021). To efficiently address problems with high-dimensional parameter spaces we set up the minimization procedure in a low-dimensional subspace of the original parameter space. The principal idea is to analyse the spectrum of the second-moment matrix of the gradient of the log-likelihood function to identify a suitable subspace. Following Zahm et al. (2021), an upper bound on the Kullback-Leibler-divergence between full-dimensional and subspace posterior is provided, which can be utilized to determine the effective dimension of the inverse problem corresponding to a prescribed approximation error bound. We suggest heuristic criteria for optimally selecting the number of model and model gradient evaluations in each iteration of the importance sampling sequence. We investigate the performance of this approach using examples from engineering mechanics set in various parameter space dimensions., Comment: 31 pages, 12 figures

Data-informed predictive maintenance planning largely relies on stochastic deterioration models. Monitoring information can be utilized to update sequentially the knowledge on time-invariant deterioration model parameters either within an off-line (batch) or an on-line (recursive) Bayesian framework. With a focus on the quantification of the full parameter uncertainty, we review, adapt and investigate selected Bayesian filters for parameter estimation: an on-line particle filter, an on-line iterated batch importance sampling filter, which performs Markov chain Monte Carlo (MCMC) move steps, and an off-line MCMC-based sequential Monte Carlo filter. A Gaussian mixture model is used to approximate the posterior distribution within the resampling process in all three filters. Two numerical examples serve as the basis for a comparative assessment of off-line and on-line Bayesian estimation of time-invariant deterioration model parameters. The first case study considers a low-dimensional, nonlinear, non-Gaussian probabilistic fatigue crack growth model that is updated with sequential crack monitoring measurements. The second high-dimensional, linear, Gaussian case study employs a random field to model corrosion deterioration across a beam, which is updated with sequential measurements from sensors. The numerical investigations provide insights into the performance of off-line and on-line filters in terms of the accuracy of posterior estimates and the computational cost, when applied to problems of different nature, increasing dimensionality and varying sensor information amount. Importantly, they show that a tailored implementation of the on-line particle filter proves competitive with the computationally demanding MCMC-based filters. Suggestions on the choice of the appropriate method in function of problem characteristics are provided.

The difficulty in quantifying the benefit of Structural Health Monitoring (SHM) for decision support is one of the bottlenecks to an extensive adoption of SHM on real-world structures. In this paper, we present a framework for such a quantification of the value of vibration-based SHM, which can be flexibly applied to different use cases. These cover SHM-based decisions at different time scales, from near-real time diagnostics to the prognosis of slowly evolving deterioration processes over the lifetime of a structure. The framework includes an advanced model of the SHM system. It employs a Bayesian filter for the tasks of sequential joint deterioration state-parameter estimation and structural reliability updating, using continuously identified modal and intermittent visual inspection data. It also includes a realistic model of the inspection and maintenance decisions throughout the structural life-cycle. On this basis, the Value of SHM is quantified by the difference in expected total life-cycle costs with and without the SHM. We investigate the framework through application on a numerical model of a two-span bridge system, subjected to gradual and shock deterioration, as well as to changing environmental conditions, over its lifetime. The results show that this framework can be used as an a-priori decision support tool to inform the decision on whether or not to install a vibration-based SHM system on a structure, for a wide range of SHM use cases.

To date, the analysis of high-dimensional, computationally expensive engineering models remains a difficult challenge in risk and reliability engineering. We use a combination of dimensionality reduction and surrogate modelling termed partial least squares-driven polynomial chaos expansion (PLS-PCE) to render such problems feasible. Standalone surrogate models typically perform poorly for reliability analysis. Therefore, in a previous work, we have used PLS-PCEs to reconstruct the intermediate densities of a sequential importance sampling approach to reliability analysis. Here, we extend this approach with an active learning procedure that allows for improved error control at each importance sampling level. To this end, we formulate an estimate of the combined estimation error for both the subspace identified in the dimension reduction step and surrogate model constructed therein. With this, it is possible to adapt the design of experiments so as to optimally learn the subspace representation and the surrogate model constructed therein. The approach is gradient-free and thus can be directly applied to black box-type models. We demonstrate the performance of this approach with a series of low- (2 dimensions) to high- (869 dimensions) dimensional example problems featuring a number of well-known caveats for reliability methods besides high dimensions and expensive computational models: strongly nonlinear limit-state functions, multiple relevant failure regions and small probabilities of failure.

We propose and discuss sensitivity metrics for reliability analysis, which are based on the value of information. These metrics are easier to interpret than other existing sensitivity metrics in the context of a specific decision and they are applicable to any type of reliability assessment, including those with dependent inputs. We develop computational strategies that enable efficient evaluation of these metrics, in some scenarios without additional runs of the deterministic model. The metrics are investigated by application to numerical examples.

Non-destructive evaluation (NDE) inspections are an integral part of asset integrity management. The relationship between the condition of interest and the quantity measured by NDE is described with probabilistic models such as PoD or ROC curves. These models are used to assess the quality of the information provided by NDE systems, which is affected by factors such as the experience of the inspector, environmental conditions, ease of access, and the precision of the measurement device. In this paper, we review existing probabilistic models of NDE andshow how they are connected within a unifying framework. This frameworkprovides insights into how these models should be learned, calibrated, and applied. We investigate and highlighthow the choice of the model can affect the maintenance decisions taken on the basis of NDE results. In addition, we analyze the impact of experimental design on the performance of a given NDE system in a decision-making context., Comment: Title changed to accurately reflect content. Details added to Sections 1, 2 and 3

Quantifying the value of the information extracted from a structural health monitoring (SHM) system is an important step towards convincing decision makers to implement these systems. We quantify this value by adaptation of the Bayesian decision analysis framework. In contrast to previous works, we model in detail the entire process of data generation to processing, model updating and reliability calculation, and investigate it on a deteriorating bridge system. The framework assumes that dynamic response data are obtained in a sequential fashion from deployed accelerometers, subsequently processed by an output-only operational modal analysis scheme for identifying the system's modal characteristics. We employ a classical Bayesian model updating methodology to sequentially learn the deterioration and estimate the structural damage evolution over time. This leads to sequential updating of the structural reliability, which constitutes the basis for a preposterior Bayesian decision analysis. Alternative actions are defined and a heuristic-based approach is employed for the life-cycle optimization. By solving the preposterior Bayesian decision analysis, one is able to quantify the benefit of the availability of long-term SHM vibrational data. Numerical investigations show that this framework can provide quantitative measures on the optimality of an SHM system in a specific decision context.

Optimizing inspection and maintenance (I&M) plans for a large deteriorating structure is a computationally challenging task, in particular if one considers interdependences among its components. This is due to the sheer number of possible decision alternatives over the lifetime of the structure and the uncertainty surrounding the deterioration processes, the structural performance and the outcomes of inspection and maintenance actions. To address this challenge, Luque and Straub (2019) proposed a heuristic approach in which I&M plans for structural systems are defined through a set of simple decision rules. Here, we formalize the optimization of these decision rules and extend the approach to enable adaptive planning. The initially optimal I&M plan is successively adapted throughout the service life, based on past inspection and monitoring results. The proposed methodology uses stochastic deterioration models and accounts for the interdependence among structural components. The heuristic-based adaptive planning is illustrated for a structural frame subjected to fatigue.

The estimation of rare event or failure probabilities in high dimensions is of interest in many areas of science and technology. We consider problems where the rare event is expressed in terms of a computationally costly numerical model. Importance sampling with the cross-entropy method offers an efficient way to address such problems provided that a suitable parametric family of biasing densities is employed. Although some existing parametric distribution families are designed to perform efficiently in high dimensions, their applicability within the cross-entropy method is limited to problems with dimension of O(1e2). In this work, rather than directly building sampling densities in high dimensions, we focus on identifying the intrinsic low-dimensional structure of the rare event simulation problem. To this end, we exploit a connection between rare event simulation and Bayesian inverse problems. This allows us to adapt dimension reduction techniques from Bayesian inference to construct new, effectively low-dimensional, biasing distributions within the cross-entropy method. In particular, we employ the approach in [47], as it enables control of the error in the approximation of the optimal biasing distribution. We illustrate our method using two standard high-dimensional reliability benchmark problems and one structural mechanics application involving random fields.

The challenges for non-intrusive methods for Polynomial Chaos modeling lie in the computational efficiency and accuracy under a limited number of model simulations. These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, we present a novel sparse Bayesian learning technique for obtaining sparse Polynomial Chaos expansions which is based on a Relevance Vector Machine model and is trained using Variational Inference. The methodology shows great potential in high-dimensional data-driven settings using relatively few data points and achieves user-controlled sparse levels that are comparable to other methods such as compressive sensing. The proposed approach is illustrated on two numerical examples, a synthetic response function that is explored for validation purposes and a low-carbon steel plate with random Young's modulus and random loading, which is modeled by stochastic finite element with 38 input random variables., Comment: Submitted to Journal of Computational Physics