An Aggregation/Disaggregation Algorithm for Stochastic Automata Networks

1997 ◽  
Vol 11 (2) ◽  
pp. 229-253 ◽  
Author(s):  
Peter Buchholz
Keyword(s):  
Markov Chains ◽  
Stationary Solution ◽  
Solution Method ◽  
Compact Representation ◽  
Generator Matrix ◽  
Iterative Techniques ◽  
Stochastic Automata ◽  
Loosely Coupled ◽  
Automata Networks ◽  
Solution Methods

Stochastic automata networks (SANs) have recently received much attention in the literature as a means to analyze complex Markov chains in an efficient way. The main advantage of SANs over most other paradigms is that they allow a very compact description of the generator matrix by means of much smaller matrices for single automata. This representation can be exploited in different iterative techniques to compute the stationary solution. However, the set of applicable solution methods for SANs is restricted, because a solution method has to respect the specific representation of the generator matrix to exploit the compact representation. In particular, aggregation/disaggregation (a/d) methods cannot be applied in their usual realization for SANs without losing the possibility to exploit the compact representation of the generator matrix.In this paper, a new a/d algorithm for SANs is introduced. The algorithm differs significantly from standard a/d methods because the parts to be aggregated are defined in a completely different way, exploiting the structure of the generator matrix of a SAN. Aggregation is performed with respect to single automata or sets of automata, which are the basic parts generating a SAN. It is shown that the new algorithm is efficient even if the automata are not loosely coupled.

scholarly journals Multipreconditioned GMRES for simulating stochastic automata networks

2018 ◽  
Vol 16 (1) ◽  
pp. 986-998
Author(s):  
Chun Wen ◽  
Ting-Zhu Huang ◽  
Xian-Ming Gu ◽  
Zhao-Li Shen ◽  
Hong-Fan Zhang ◽  
...  
Keyword(s):  
Steady State ◽  
Probability Distribution ◽  
Linear Systems ◽  
Iterative Methods ◽  
Communication Systems ◽  
Queueing Systems ◽  
Steady State Probability ◽  
Stochastic Automata ◽  
Automata Networks ◽  
Generator Matrices

AbstractStochastic Automata Networks (SANs) have a large amount of applications in modelling queueing systems and communication systems. To find the steady state probability distribution of the SANs, it often needs to solve linear systems which involve their generator matrices. However, some classical iterative methods such as the Jacobi and the Gauss-Seidel are inefficient due to the huge size of the generator matrices. In this paper, the multipreconditioned GMRES (MPGMRES) is considered by using two or more preconditioners simultaneously. Meanwhile, a selective version of the MPGMRES is presented to overcome the rapid increase of the storage requirements and make it practical. Numerical results on two models of SANs are reported to illustrate the effectiveness of these proposed methods.

An extension of a convergence theorem for Markov chains arising in population genetics

2016 ◽  
Vol 53 (3) ◽  
pp. 953-956 ◽  
Author(s):  
Martin Möhle ◽  
Morihiro Notohara
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Convergence Theorem ◽  
Transition Matrix ◽  
Initial Distribution ◽  
Generator Matrix ◽  
Finite Dimensional ◽  
Finite State ◽  
Initial Distributions ◽  
Finite State Space

AbstractAn extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer N let (XN(r))r be a Markov chain with the same finite state space S and transition matrix ΠN=I+dNBN, where I is the unit matrix, Q a generator matrix, (BN)N a sequence of matrices, limN℩∞cN= limN→∞dN=0 and limN→∞cN∕dN=0. Suppose that the limits P≔limm→∞(I+dNQ)m and G≔limN→∞PBNP exist. If the sequence of initial distributions PXN(0) converges weakly to some probability measure μ, then the finite-dimensional distributions of (XN([t∕cN))t≥0 converge to those of the Markov process (Xt)t≥0 with initial distribution μ, transition matrix PetG and limN→∞(I+dNQ+cNBN)[t∕cN]

scholarly journals Iterative disaggregation for a class of lumpable discrete-time stochastic automata networks

2003 ◽  
Vol 53 (1) ◽  
pp. 43-69 ◽  
Author(s):  
Oleg Gusak ◽  
Tuğrul Dayar ◽  
Jean-Michel Fourneau
Keyword(s):  
Discrete Time ◽  
Stochastic Automata ◽  
Automata Networks

Graphs and Stochastic Automata Networks

1995 ◽  
pp. 217-235 ◽  
Author(s):  
Jean-Michel Fourneau ◽  
Franck Quessette
Keyword(s):  
Stochastic Automata ◽  
Automata Networks

scholarly journals Solution of Contact Problems for Nonlinear Gao Beam and Obstacle

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
J. Machalová ◽  
H. Netuka
Keyword(s):  
Solution Method ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Contact Problems ◽  
Beam Model ◽  
Contact Conditions ◽  
Beam Elements ◽  
Solution Methods ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

Contact problem for a large deformed beam with an elastic obstacle is formulated, analyzed, and numerically solved. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao, while the obstacle is considered as the elastic foundation of Winkler’s type in some distance under the beam. The problem is static without a friction and modeled either using Signorini conditions or by means of normal compliance contact conditions. The problems are then reformulated as optimal control problems which is useful both for theoretical aspects and for solution methods. Discretization is based on using the mixed finite element method with independent discretization and interpolations for foundation and beam elements. Numerical examples demonstrate usefulness of the presented solution method. Results for the nonlinear Gao beam are compared with results for the classical Euler-Bernoulli beam model.

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