Advanced Monte Carlo Methods and Applications

The special collection on Advanced Monte Carlo Methods and Applications is available in the ASCE Library at http://ascelibrary.org/page/ajrua6/monte_carlo_methods_applications.

In the reliability and risk analyses of engineering systems, mathematical models are built and subsequently encoded into computer codes that are used to simulate the response of a system under various operational transients and accident scenarios. In practice, not all the characteristics of the system under analysis can be fully captured in the model due to the complexity and/or the incomplete knowledge of the phenomena involved (such as lack of experimental results and limited operational experience).

Thus, uncertainties arise either with regards to the future occurrences of some events of interest (e.g., component degradation, failures, or more generally, stochastic transitions between different performance states), or to the mathematical and computational tools used to model the consequent system behaviors (e.g., the values of the model input parameters and/or on the hypotheses supporting the model structure itself).

A proper treatment of these uncertainties is then a fundamental task for the quantitative assessment of the reliability of a system and, consequently, of the potential risk associated. Within a probabilistic framework, the typical approaches to address this issue are those based on Monte Carlo (MC) simulation, where several realizations (or simulations) of the system dynamics are generated by randomly sampling from suitable probability distributions describing the uncertainties involved, so that it is possible to estimate the distributions of the outcomes of interest by any available statistical tools.

However, because the design of real engineering systems is such that the occurrences of undesired events are rare, standard MC-based approaches would require a large number of simulations and, possibly, prohibitive computational expenses. This problem is particularly relevant for cases in which the computer codes used to model the system behavior are computationally very intensive, so that it is mandatory to resort to strategies for improving the efficiency of sampling-based MC methods.

In this context, the present special collection aims at presenting recent achievements with regards to the use of efficient MC-based methods for treating uncertainties in the reliability and risk analyses of complex engineering systems, with particular reference to advanced variance-reduction techniques [e.g., importance sampling (IS), subset simulation (SS), and orthogonal plain sampling (OPS)], metamodels (e.g., kriging and polynomial expansions), or adaptive combinations of the two. The applications include the probabilistic risk assessment of industrial and structural systems.

In more detail, this special collection contains four papers. In de Angelis et al. (2016), a general methodology for the efficient solution of a time-variant reliability-based maintenance optimization problem is proposed, where the solution is a trade-off between the costs associated with inspection and repair activities and the benefits related to the faultless operation of the system or infrastructure of interest. The methodology exploits the concept of forced MC simulation to maximize the computational efficiency, without reducing its accuracy. In particular, the strategy is based on the calculation of weights that emulate the outcomes of the maintenance process. By so doing, it is capable of assessing the reliability of the system at any given time and for any number of maintenance inspections, performing only one full reliability analysis, whose results can subsequently be used, with nearly no additional computational effort. This represents an impressive advantage with respect to a plain, standard MC approach to the problem, which is demonstrated by means of an example involving a fatigue-prone weld in a bridge girder.

Wang and Der Kiureghian (2016) introduce a modified OPS for high-dimensional reliability analysis. In OPS, sample points are generated on a plane that is orthogonal to the directional vector of the design point in the standard normal space. The failure probability estimate is obtained in terms of distances from the plane to the limit-state surface. Wang and Der Kiureghian develop a modified version of the OPS method for high-dimensional reliability problems by fixing the uncertain space to a hypersphere and picking highly regular points on the hypersphere (instead of random points like in traditional OPS). Numerical results indicate the performances of the original and modified OPS are not sensitive to the dimension of the problem and to the order of the failure probability, but they are mildly influenced by the nonlinearity of the limit-state surface. In addition, it is shown that this modified method is at least as accurate as the conventional OPS, but generally much more efficient (i.e., in terms of higher precision and lower computational cost).

Within the context of stochastic analysis of engineering systems, Jia et al. (2015) address instead the problem of efficiently generating samples according to a target probability distribution of interest, while reducing as much as possible the number of evaluations of the (possibly long-running) system model. In this light, they introduce a novel adaptive method for rejection sampling that uses optimized adaptive kernel sampling densities (AKSDs) as proposal distributions within the context of an iterative implementation. Such an implementation can be encountered, for example, within robust design applications, where the iterations correspond to different iterations of the numerical optimization algorithm searching for the optimal solution, or in sampling from nested sequences of subsets, as in the SS method, where the sequence corresponds to the intermediate conditional failure events generated by SS. Within the considered iterative implementation, the proposed formulation relies on having available a small number of (1) samples from the target density to approximate, as well as (2) evaluations of the (computationally expensive) system model (involved in the target density definition) over some sample set. Based on this information, new samples are simulated. An implementation of the proposed adaptive stochastic sampling is demonstrated within the context of SS for generating independent conditional failure samples and calculating (small) probabilities of rare failure events. These ideas are illustrated in two examples, involving a simply supported beam and a floor isolation system for protection of computer servers against near-fault earthquake excitations, respectively. The efficiency improvement provided by the proposed AKSD is demonstrated and various characteristics for the AKSD are investigated, including applicability in high-dimensional problems. Overall, it is shown that the combination of SS and AKSD provides robust and efficient estimation for small failure probabilities by sharing information across the levels (iterations) of SS.

Finally, Schöbi et al. (2016) combine adaptive MC simulation methods and metamodels for the estimation of small probabilities of failure and extreme quantiles. Actually, rare events estimation is often related to the concept of experimental design enrichment (also known as adaptive experimental design). The core idea is to start calibrating a metamodel with a limited set of computational runs and then to add new model evaluations iteratively to increase the prediction accuracy of the rare event statistics of interest. In this view, Schöbi et al. propose a new method based on the recently developed polynomial-chaos kriging (PC-kriging) approach coupled with an active learning algorithm known as adaptive kriging Monte Carlo simulation (AK-MCS). They focus on the best choice of additional samples in the experimental design as well as on the option of adding multiple samples in each iteration of the algorithm; adding multiple samples in each iteration also allows one to take advantage of distributed computing facilities. The application of the resulting framework is demonstrated on a low-dimensional analytical function and on two realistic (8- and 10-dimensional) structural engineering applications: in all cases, failure probabilities up to ${10}^{-5}$ are estimated and accurate results are obtained with very few (i.e., approximately 100) runs of the original model.

Because this Journal aims to serve as a medium for dissemination of research findings, best practices, and concerns, and for the discussion and debate on risk and uncertainty in civil and mechanical engineering, it has been natural to present this special collection reporting on recent achievements with respect to the use of advanced MC-based methods for propagating uncertainties in the models of complex engineering systems for reliability and risk assessment purposes. This will meet the needs of the researchers and engineers to efficiently address risk- and failure-related challenges due to many sources and types of uncertainty in planning, design, analysis, construction, manufacturing, operation, utilization, and lifecycle management of existing and new engineering systems.