Post by Steve Yenisch on May 13, 2009 23:23:52 GMT -5
Paper: www.eng.utoledo.edu/~enikolai/research/ISSMO-2009-Imprecise-Probability-PCE.pdf
Presentation: www.eng.utoledo.edu/~enikolai/research/ISSMO-2009-PCE.ppt
ABSTRACT
In reliability design, often, there is scarce data for constructing probabilistic models. It is particularly challenging to model uncertainty in variables when the type of their probability distribution is unknown. Moreover, it is expensive to estimate the upper and lower bounds of the reliability of a system involving such variables. A method for modeling uncertainty by using Polynomial Chaos Expansion is presented. The method requires specifying bounds for statistical summaries such as the first four moments and credible intervals. A constrained optimization problem, in which decision variables are the coefficients of the Polynomial Chaos Expansion approximation, is formulated and solved in order to estimate the minimum and maximum values of a system’s reliability. This problem is solved efficiently by employing a probabilistic re-analysis approach to approximate the system reliability as a function of the moments of the random variables.
The method is demonstrated on an example where a designer must select the type of material for a rod subjected to a tensile stress with unknown PDF. The method calculates efficiently and accurately bounds for the probability of failure of the rod. It is observed that comparing two alternative designs in terms of their failure probabilities could lead to indecision because the intervals in which the probabilities of these designs lie often overlap. A designer may be able to break the tie by calculating bounds for the difference of the failure probabilities instead of calculating bounds for the individual probabilities.
Presentation: www.eng.utoledo.edu/~enikolai/research/ISSMO-2009-PCE.ppt
Imprecise Reliability Assessment When the Type of the Probability Distribution of the Random Variables is Unknown
Efstratios Nikolaidis
Mechanical Industrial and Manufacturing Engineering Department
The University of Toledo
Toledo, OH 43606
email: enikolai@eng.utoledo.edu
http://www.eng.utoledo.edu~enikolai
Zissimos P. Mourelatos
Oakland University
Rochester, MI 48309
Phone: 248-370-2686
Fax: 248-370-4416
mourelat@oakland.edu
Efstratios Nikolaidis
Mechanical Industrial and Manufacturing Engineering Department
The University of Toledo
Toledo, OH 43606
email: enikolai@eng.utoledo.edu
http://www.eng.utoledo.edu~enikolai
Zissimos P. Mourelatos
Oakland University
Rochester, MI 48309
Phone: 248-370-2686
Fax: 248-370-4416
mourelat@oakland.edu
ABSTRACT
In reliability design, often, there is scarce data for constructing probabilistic models. It is particularly challenging to model uncertainty in variables when the type of their probability distribution is unknown. Moreover, it is expensive to estimate the upper and lower bounds of the reliability of a system involving such variables. A method for modeling uncertainty by using Polynomial Chaos Expansion is presented. The method requires specifying bounds for statistical summaries such as the first four moments and credible intervals. A constrained optimization problem, in which decision variables are the coefficients of the Polynomial Chaos Expansion approximation, is formulated and solved in order to estimate the minimum and maximum values of a system’s reliability. This problem is solved efficiently by employing a probabilistic re-analysis approach to approximate the system reliability as a function of the moments of the random variables.
The method is demonstrated on an example where a designer must select the type of material for a rod subjected to a tensile stress with unknown PDF. The method calculates efficiently and accurately bounds for the probability of failure of the rod. It is observed that comparing two alternative designs in terms of their failure probabilities could lead to indecision because the intervals in which the probabilities of these designs lie often overlap. A designer may be able to break the tie by calculating bounds for the difference of the failure probabilities instead of calculating bounds for the individual probabilities.