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Méthodes numériques pour la solution de systèmes Markoviens à grand espace d'états

Abstract : This thesis develops techniques for optimizing the numerical evaluation of Markovian models. These techniques are applied to models in which the transition matrix of the associated Markov chain is stored in a tensor format. A stochastic automata network formalism is used to describe the models. The performance index of interest is the stationary distribution of the Markov chain. We define a generalized tensor algebra and we prove a number of theorems that allow us to lay the foundations for the algorithms needed to solve the Markov chains. The main objective in this thesis is the efficiency of iterative algorithms for solving Markov chains. This objective has two aspects: the reduction of the computational cost of each iteration step; and the reduction of the number of iterations needed for convergence. The basic operation in each iteration step is the product of a vector by a matrix stored in a tensor format, the so-called descriptor. The efficiency of such operations is our first goal. We propose an approach (vector-descriptor product) which reduces memory requirements without increasing the computational complexity of the usual standard sparse matrix approach. The second goal is the adaptation of the power, Arnoldi and GMRES methods in their standard and preconditioned versions. These implementations attempt to reduce the number of iterations needed for convergence, while avoiding an increase in the iteration step costs. In addition, our technique does not need to explicitly generate the matrix corresponding to the descriptor to provide a preconditioner. All the concepts developed in this thesis are implemented in the PEPS 2.0 software package. Several practical examples of stochastic automata network models have been tested with PEPS 2.0 in order to illustrate the results of this thesis.
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Paulo Fernandes. Méthodes numériques pour la solution de systèmes Markoviens à grand espace d'états. Modélisation et simulation. Institut National Polytechnique de Grenoble - INPG, 1998. Français. ⟨tel-00004886⟩

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