Aggregation Methods for Nearly Uncoupled Systems

ABSTRACT

The underlying application is to analyze and compute the steady state behaviour of large evolutionary systems comprised of many discrete states which naturally divide into clusters such that within each cluster the states are strongly coupled, but the clusters themselves are only weakly coupled to each other. Such systems are common encountered in the analysis of queueing networks and computer systems, discrete economic models, and in many stochastic models found in biological and social science.

The strategy is to examine the closely coupled clusters in isolation, and then somehow couple this information together to deduce the global steady state behaviour. This is accomplished by so-called aggregation/disaggregation methods based on the Simon-Ando theory for nearly uncoupled systems. This amounts to first uncoupling a particular eigenvector problem followed by a coupled eigenvector problem reminiscent of an unsymmetric divide-and-conquer scheme involving an unsymmetric Rayleigh-Ritz like step.

The presentation will include a survey and intuitive overview of the Simon-Ando theory followed by and intuitive derivation of the basic aggregation/disaggregation process. It will conclude with a short error analysis. The development will be in the context of linear algebra together with some elementary finite Markov chain techniques.

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