Error Estimation and Error Reduction in Separable Monte-Carlo Method

Abstract

Reliability-based design often uses the Monte-Carlo method as a sampling procedure for predicting failure. The combination of designing for very small failure probabilities (10 8 10 6 ) and using computationally expensive finite element models, makes Monte-Carlo simulations very expensive. This paper uses an improved sampling procedure for calculating the probability of failure, called separable Monte-Carlo method. The separable MonteCarlo method can improve the accuracy of the traditional crude Monte-Carlo when response and capacity are independent. In previous research, accuracy of separable Monte-Carlo for a simple limit state was estimated via expectation calculus for a simple form of the limit state. In this paper, error estimates for a general limit state are developed through bootstrapping, and it is demonstrated that the estimates are reasonably accurate. Separable Monte-Carlo allows us to choose different sample sizes of the response and capacity in the limit state, and the paper demonstrates that bootstrapping may be used to estimate the contribution of the response and capacity to the total error. When the accuracy of the probability of failure is not good enough, the paper proposes reformulation of the limit state as another way to reduce uncertainty associated with the expensive random variable (usually the response).

Comparison of Methods for Calculating B-Basis Crack Growth Life Using Limited Tests

“…1) The normal TI method (even though the underlying distribution is not normal or lognormal) gave better coverage probabilities than the bootstrap methods (i.e., about 93%) for smaller sample sizes (8)(9)(10)(11)(12)(13)(14)(15)(16). So, the normal TI method may be a good choice if one is willing to trade off slight underestimation of coverage probability (i.e., 93% instead of 95%) with significantly better B-basis values or lower relative mean margins in comparison to nonparametric method (e.g., about 24-15% for normal TI vs 47-29% for nonparametric).…”

“…McDonald et al [10] used a jackknife resampling (the bootstrap method is an improvement on jackknife) technique to approximate the sampling uncertainty in the parameters of the Johnson probability distribution due to sparse data. Ravishankar et al [11] found that bootstrap resampling provided reasonable estimates of the epistemic uncertainty in the separable Monte Carlo estimates of the probability of failure. Pieracci [12] considered a problem of estimating the parameters of a Weibull distribution from limited experimental data for durability analysis and used the bootstrap method to approximate the confidence intervals for the probability of crack exceedance at a given time.…”

How Adaptively Constructed Reduced Order Models Can Benefit Sampling-Based Methodsfor Reliability Analyses

“…Several methods have been developed in the past allowing to decrease the number of samples required for a given accuracy on the probability of failure estimate. Such methods, that we refer to as advanced Monte Carlo approaches, include importance sampling [3] [4], separable Monte Carlo [5] [6], Markov chain Monte Carlo [7] [8].…”

“…Illustration (from[6]) of the difference between (a) Standard MC and (b) Separable MC Separable MC allows different sample sizes for response and capacity, which is very advantageous when working with limited computational budget, since a smaller sample size for the computationally expensive calculation (usually the response) can be somewhat compensated by more samples of the computationally cheap calculation (usually the capacity).…”

Bootstrapping and Separable Monte Carlo Simulation Methods Tailored for Efficient Assessment of Probability of Failure of Structural Systems

“…MCS may not be capable of providing the desired accuracy when sampling is limited due to expensive structural analysis especially when the calculation of the limit state function involves Finite Element Analysis (FEA) of large-scale models. Smarslok et al [2006Smarslok et al [ , 2008Smarslok et al [ , 2010 and Ravishankar [2010] developed a sampling method named Separable Monte Carlo (SMC) simulation. When the performance function of the system can be written in terms of functions with no common variables, SMC can take advantage of this property to increase the sampling domain.…”