New variance reduction methods in Monte Carlo rare event simulation

Para sistemas que proveen algún tipo de servicio mientras están operativos y dejan de proveerlo cuando fallan, es de interés determinar parámetros como, por ejemplo, la probabilidad de encontrar el sistema en falla en un instante cualquiera, el tiempo medio transcurrido entre fallas, o cualquier medida capaz de reflejar la capacidad del sistema para proveer servicio. Las determinaciones de estas medidas de seguridad de funcionamiento se ven afectadas por diversos factores, entre ellos, el tamaño del sistema y la rareza de las fallas. En esta tesis se estudian algunos métodos concebidos para determinar estas medidas sobre sistemas grandes y altamente confiables, es decir sistemas formados por gran cantidad de componentes, en los que las fallas del sistema son eventos raros. Ya sea en forma directa o indirecta, parte de las las expresiones que permiten determinar las medidas de interés corresponden a la probabilidad de que el sistema se encuentre en algún estado de falla. De un modo u otro, estas expresiones evaluan la fracción —ponderada por la distribución de probabilidad de las configuraciones del sistema—entre el número de configuraciones en las que el sistema falla y la totalidad de las configuraciones posibles. Si el sistema es grande el cálculo exacto de estas probabilidades, y consecuentemente de las medidas de interés, puede resultar inviable. Una solución alternativa es estimar estas probabilidades mediante simulación. Uno de los mecanismos para hacer estas estimaciones es la simulación de tipo Monte Carlo, cuya versión más simple es la simulación en crudo o estándar. El problema es que si las fallas son raras, el número de iteraciones necesario para estimar estas probabilidades mediante simulación estándar con una precisión aceptable, puede resultar desmesuradamente grande. En esta tesis se analizan algunos métodos existentes para mejorar la simulación estándar en el contexto de eventos raros, se hacen análisis de varianza y se prueban los métodos sobre una variedad de modelos. En todos los casos la mejora se consigue a costa de una reducción de la varianza del estimador con respecto a la varianza del estimador estándar. Gracias a la reducción de varianza es posible estimar la probabilidad de ocurrencia de eventos raros con una precisión aceptable, a partir de un número razonable de iteraciones. Como parte central del trabajo se proponen dos métodos nuevos, uno relacionado con Spliting y otro relacionado con Monte Carlo Condicional. Splitting es un método de probada eficiencia en entornos en los que se busca evaluar desempeño y confiabilidad combinados, escasamente utilizado en la simulación de sistemas altamente confiables sobre modelos estáticos (sin evolución temporal). En vi su formulación básica Splitting hace un seguimiento de las trayectorias de un proceso estocástico a través de su espacio de estados y multiplica su número ante cada cruce de umbral, para un conjunto dado de umbrales distribuidos entre los estados inicial y final. Una de las propuestas de esta tesis es una adaptación de Splitting a un modelo estático de confiabilidad de redes. En el método propuesto se construye un proceso estocástico a partir de un tiempo ficticio en el cual los enlaces van cambiando de estado y se aplica Splitting sobre ese proceso. El método exhibe elevados niveles de precisión y robustez. Monte Carlo Condicional es un método clásico de reducción de varianza cuyo uso no está muy extendido en el contexto de eventos raros. En su formulación básica Monte Carlo Condicional evalúa las probabilidades de los eventos de interés, condicionando las variables indicatrices a eventos no raros y simples de detectar. El problema es que parte de esa evaluación incluye el cálculo exacto de algunas probabilidades del modelo. Uno de los métodos propuestos en esta tesis es una adaptación de Monte Carlo Condicional al análisis de modelos Markovianos de sistemas altamente confiables. La propuesta consiste en estimar las probabilidades cuyo valor exacto se necesita, mediante una aplicación recursiva de Monte Carlo Condicional. Se estudian algunas características de este modelo y se verifica su eficiencia en forma experimental.For systems that provide some kind of service while they are operational and stop providing it when they fail, it is of interest to determine parameters such as, for example, the probability of finding the system failed at any moment, the mean time between failures, or any measure that reflects the capacity of the system to provide service. The determination of these measures —known as dependability measures— is affected by a variety of factors, including the size of the system and the rarity of failures. This thesis studies some methods designed to determine these measures on large and highly reliable systems, i.e. systems formed by a large number of components, such that systems’ failures are rare events. Either directly or indirectly, part of the expressions for determining the measures of interest correspond to the probability that the system is in some state of failure. Somehow, this expressions evaluate the ratio —weighted by the probability distribution of the systems’ configurations— between the number of configurations in which the system fails and all possible configurations. If the system is large, the exact calculation of these probabilities, and consequently of the measures of interest, may be unfeasible. An alternative solution is to estimate these probabilities by simulation. One mechanism to make such estimation is Monte Carlo simulation, whose simplest version is crude or standard simulation. The problem is that if failures are rare, the number of iterations required to estimate this probabilities by standard simulation, with acceptable accuracy, may be extremely large. In this thesis some existing methods to improve the standard simulation in the context of rare events are analyzed, some variance analyses are made and the methods are tested empirically over a variety of models. In all cases the improvement is achieved at the expense of reducing the variance of the estimator with respect to the standard estimator’s variance. Due to this variance reduction, the probability of the occurrence of rare events, with acceptable accuracy, can be achieved in a reasonable number of iterations. As a central part of this work, two new methods are proposed, one of them related to Splitting and the other one related to Conditional Monte Carlo. Splitting is a widely used method in performance and performability analysis, but scarcely applied for simulating highly reliable systems over static models (models with no temporal evolution). In its basic formulation Splitting keeps track of the trajectories of a stochastic process through its state space and it splits or multiplies the number of them at each threshold cross, for a given set of thresholds distributed between the initial and the final state. One of the proposals of this thesis is an adaptation of Splitting to a static network reliability model. In the proposed method, a fictitious time stochastic process in which the network links keep changing their state is built, and Splitting is applied to this process. The method shows to be highly accurate and robust. Conditional Monte Carlo is a classical variance reduction technique, whose use is not widespread in the field of rare events. In its basic formulation Conditional Monte Carlo evaluates the probabilities of the events of interest, conditioning the indicator variables to not rare and easy to detect events. The problem is that part of this assessment includes the exact calculation of some probabilities in the model. One of the methods proposed in this thesis is an adaptation of Conditional Monte Carlo to the analysis of highly reliable Markovian systems. The proposal consists in estimating the probabilities whose exact value is needed, by means of a recursive application of Conditional Monte Carlo. Some features of this model are discussed and its efficiency is verified experimentally

Advanced Bayesian networks for reliability and risk analysis in geotechnical engineering

The stability and deformation problems of soil have been a research topic of great concern since the past decades. The potential catastrophic events are induced by various complex factors, such as uncertain geotechnical conditions, external environment, and anthropogenic influence, etc. To prevent the occurrence of disasters in geotechnical engineering, the main purpose of this study is to enhance the Bayesian networks (BNs) model for quantifying the uncertainty and predicting the risk level in solving the geotechnical problems. The advanced BNs model is effective for analyzing the geotechnical problems in the poor data environment. The advanced BNs approach proposed in this study is applied to solve the stability of soil slopes problem associated with the specific-site data. When probabilistic models for soil properties are adopted, enhanced BNs approach was adopted to cope with continuous input parameters. On the other hand, Credal networks (CNs), developed on the basis of BNs, are specially used for incomplete input information. In addition, the probabilities of slope failure are also investigated for different evidences. A discretization approach for the enhanced BNs is applied in the case of evidence entering into the continuous nodes. Two examples implemented are to demonstrate the feasibility and predictive effectiveness of the BNs model. The results indicate the enhanced BNs show a precisely low risk for the slope studied. Unlike the BNs, the results of CNs are presented with bounds. The comparison of three different input information reveals the more imprecision in input, the more uncertainty in output. Both of them can provide the useful disaster-induced information for decision-makers. According to the information updating in the models, the position of the water table shows a significant role in the slope failure, which is controlled by the drainage states. Also, it discusses how the different types of BNs contribute to assessing the reliability and risk of real slopes, and how new information could be introduced in the analysis. The proposed models in this study illustrate the advanced BN model is a good diagnosis tool for estimating the risk level of the slope failure. In a follow-up study, the BNs model is developed based on its potential capability for the information updating and importance measure. To reduce the influence of uncertainty, with the proposed BN model, the soil parameters are updated accurately during the excavation process, and besides, the contribution of epistemic uncertainty from geotechnical parameters to the potential disaster can be characterized based on the developed BN model. The results of this study indicate the BNs model is an effective and flexible tool for risk analysis and decision making support in geotechnical engineering

How to effectively compute the reliability of a thermal-hydraulic nuclear passive system

International audienceThe computation of the reliability of a thermal-hydraulic (T-H) passive system of a nuclear power plant can be obtained by (i) Monte Carlo (MC) sampling the uncertainties of the system model and parameters, (ii) computing, for each sample, the system response by a mechanistic T-H code and (iii) comparing the system response with pre-established safety thresholds, which define the success or failure of the safety function. The computational effort involved can be prohibitive because of the large number of (typically long) T-H code simulations that must be performed (one for each sample) for the statistical estimation of the probability of success or failure. The objective of this work is to provide operative guidelines to effectively handle the computation of the reliability of a nuclear passive system. Two directions of computation efficiency are considered: from one side, efficient Monte Carlo Simulation (MCS) techniques are indicated as a means to performing robust estimations with a limited number of samples: in particular, the Subset Simulation (SS) and Line Sampling (LS) methods are identified as most valuable; from the other side, fast-running, surrogate regression models (also called response surfaces or meta-models) are indicated as a valid replacement of the long-running T-H model codes: in particular, the use of bootstrapped Artificial Neural Networks (ANNs) is shown to have interesting potentials, including for uncertainty propagation.The recommendations drawn are supported by the results obtained in an illustrative application of literature