Circulant preconditioners for stochastic automata networks
Chan, Raymond H. and Ching, Wai Ki (2000) Circulant preconditioners for stochastic automata networks. Numerische Mathematik, 87, (1), 35-57. (doi:10.1007/s002110000173).
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Stochastic Automata Networks (SANs) are widely used in modeling communication systems, manufacturing systems and computer systems. The SAN approach gives a more compact and efficient representation of the network when compared to the stochastic Petri nets approach. To find the steady state distribution of SANs, it requires solutions of linear systems involving the generator matrices of the SANs. Very often, direct methods such as the LU decomposition are inefficient because of the huge size of the generator matrices. An efficient algorithm should make use of the structure of the matrices. Iterative methods such as the conjugate gradient methods are possible choices. However, their convergence rates are slow in general and preconditioning is required. We note that the MILU and MINV based preconditioners are not appropriate because of their expensive construction cost. In this paper, we consider preconditioners obtained by circulant approximations of SANs. They have low construction cost and can be inverted efficiently. We prove that if only one of the automata is large in size compared to the others, then the preconditioned system of the normal equations will converge very fast. Numerical results for three different SANs solved by CGS are given to illustrate the fast convergence of our method.
|Digital Object Identifier (DOI):||doi:10.1007/s002110000173|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||University Structure - Pre August 2011 > School of Mathematics > Operational Research
|Date Deposited:||20 Jul 2006|
|Last Modified:||06 Aug 2015 02:30|
|RDF:||RDF+N-Triples, RDF+N3, RDF+XML, Browse.|
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