Efficient Probabilistic Analysis Methods for Complex Numerical Models with Large Numbers of Random Variables, 20-9190

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Principal Investigators
Sitakanta Mohanty
Michael P. Enright
Y-T (Justin) Wu (consultant)

Inclusive Dates: 04/01/00 - Current

Background - Physics-based probabilistic reliability and risk analysis of engineered and natural system is emerging as a less expensive addition to field and laboratory-test-based methods. However, new challenges exist because highly complicated physics-based models are computationally intensive and have a large number of variables required to conduct accurate system-level failure analyses. The Total-System Performance Assessment code developed at the Center for Nuclear Waste Regulatory Analyses (a division of the Southwest Research Institute) to evaluate a high-level radioactive waste (HLW) disposal facility and the codes for reliability-based motor vehicle design involving hundreds of input variables are typical examples. These problems cannot be solved efficiently and accurately by the existing methods that have been successfully used for problems with a small number of variables.

Approach - The objective of the project is to develop and demonstrate a fast and accurate method to tackle risk and reliability problems involving computationally intensive and numerically complex models with large numbers of parameters. The framework of the method features a hybrid approach that combines the sampling approach that explores the parameter space and the advanced reliability methods that focus the analysis at the tail of the probability distribution. The use of probabilistic sensitivity measures that can effectively identify the most important random variables is being developed. The method is being extended to combine the low-probability high-consequence scenarios with the most likely scenarios to obtain a single performance measure. An efficient reliability analysis method and an efficient reliability-based design optimization method are being developed. The demonstration examples include nuclear waste management, oil and gas exploration and transportation, and automotive applications.

Accomplishments - A formal and comprehensive study of the method was pursued by investigating two sampling-based sensitivity measures, (∂p/p)/(∂µi /σi) and (∂p/p)/(∂σi /σi), where p, µ, and σ are the CDF function, the mean, and the standard deviation, respectively. Analytical solutions were derived for linear response functions with Gaussian variables and numerical solutions for second-order response surfaces with normal or lognormal variables. The analytical solutions suggest that, in general, the (∂p/p)/(∂µi /σi) measure has a better discriminating power in identifying influential variables than the (∂p/p)/(∂σi /σi) measure; however, both measures have good discriminating power at the tails of the distribution. The method and its application to an HLW problem were published recently in the Reliability Engineering and System Safety peer-reviewed journal (volume 73, p 167-176, 2001). Two new sensitivity measures developed since then are particularly relevant to HLW disposal regulatory criteria. These measures are referred to as performance-mean-based sensitivity measures, ∂µY/∂µXi  and ∂µY/∂σXi , where Y is the model output, Xi are input parameters, and µY is the peak expected dose. Based on ∂µY/∂µXi , eleven out of 246 variables are identified as having significant sensitivities at a 95-percent acceptance limit. Also, a close relationship between (∂p/p)/(∂σi /σi ) and ∂µY/∂µXi  has been discovered. Based on the calculated ∂µY/∂µXi , eleven variables are identified as having significant sensitivities at a 95-percent acceptance limit. The top four variables are in the same rank as those identified by ∂µY/∂µXi . It appears that fewer than 1,000 samples may be sufficient to identify influential variables, which is very encouraging with respect to the computational burden typically associated with large problems. Application of the sensitivity analysis approach is currently underway for an oil and gas recovery problem.

Important variables identified by the µY/∂µXi  sensitivity measure for an example problem with 246 random variables. The illustration clearly differentiates between significant and insignificant variables with a 95-percent accuracy or acceptance level.

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