|Ph.D Student||Khazen Michael|
|Subject||A General Transport Equation and its Solution by the Monte|
Carlo Approach with Phase Space Splitting in
Systems Engineering and Reliability
|Department||Department of Quality Assurance and Reliability||Supervisor||Professor A. Dubi|
Estimation of the probabilities of rare events with significant consequences, e.g. disasters, is the most difficult problem in Monte Carlo applications to Systems Engineering and Reliability. The Bernoulli type estimator used by the Analog Monte Carlo method is characterized by extremely high Variance when applied to estimation of rare event probability. Variance reduction methods (VRM) are, therefore, of importance in this field.
The neutral particles transport, which is one of the subjects in the field of Nuclear Engineering, often faces a similar problem of high variance. A variety of Variance Reduction methods based either on Biasing or on Geometric Splitting have been developed in that field during the last 30 years. Although the transport formalism can be applied to a system, a direct implementation of these methods is impossible due to the following reasons. The existing Biasing methods implement the Importance Sampling technique. Efficiency of this technique depends on non-analog probability measures that should be introduced as part of the method. The criteria developed in Nuclear Engineering use properties that are unique to the transport of neutral particles in a media and can not be transferred into the realm of systems. Also the other method - Geometric Splitting - is based on metric, namely distance and direction that have no parallel definitions in the phase space of a system.
The present work suggests a parametric non-analog probability measure based on the superposition of Transition Biasing and Forced Events Biasing. The Cluster-Event model is developed providing an effective and reliable approximation for the Second Moment and the Benefit along with a methodology of selecting near optimal biasing parameters. Numerical examples show a considerable benefit when the method is applied to problems of particular difficulty for the Analog Monte Carlo method.
A formal approach enabling application of the Geometric Splitting to complex systems is suggested introducing the Shortest-Cut metric in a complex system phase space. The phase space splitting is applied to problems with single and multiple splitting surfaces with the suggested metric. A problem of combination of splitting and biasing is considered demonstrating a benefit of the combination over each method when applied separately. Multiple numeric examples show agreement between the theory and the experiment for all the developed approaches.
In its final part, this work suggests a formalism enabling development of the explicit form of the General Transport Equation when applied to a problem in which the order of events is of importance.