Study On Several Multivariate Markov Chain Models And Saddle Point Problem

Markov problem is an important topic focused by many scholars and applied in a wide range of applications, such as: queuing theory of Markov chain models, Internet, remanufacturing systems, inventory systems, DNA sequencing and gene networks. Markov problems can be divided into two categories discrete-time Markov chain problems and continuous-time Markov chain problems. A great number of important results have been reported for discrete-time Markov chain problems, e.g., multivariate Markov chain models, developments of hidden multivariate Markov chain models, applications of the aggregation methods in Markov chain problems. The main work of this dissertation on Markov problem is discussing some multivariate Markov chain models and some higher-order multivariate Markov chain models. In a lot of practice, such as: electromagnetic field calculations, fluid dynamics can be transformed into saddle point problems. Improving the preconditioner of saddle point problems is one of the content in this article.To speed up the convergence rate and reduce the scale of the problem, a new convergence condition of the improved multivariate Markov chain model is presented. Moreover, a new improved multivariate Markov chain model is proposed and its convergence property is also analyzed. In order to obtain more precise prediction results, the new convergence condition with variables is presented. The prediction accuracies of different models in different convergence conditions are reported in numerical experiments.Improved multivariate Markov chain model is popularized in higher-order cases,called as improved higher-order multivariate Markov chain model. Two convergence conditions of the improved higher-order multivariate Markov chain model are presented. Numerical results illustrate that the improved higher-order multivariate Markov chain model is more efficient than the improved multivariate Markov chain model. The improved higher-order multivariate Markov chain model under the new convergence condition performs better than the one under the original convergence condition in reducing the scale of the problem. At the same time, a new improved higher-order multivariate Markov chain model is proposed. The convergence property of the model is also analyzed. Numerical results illustrate that the new improved higher-order multivariate Markov chain model performs better than the improved higher-order multivariate Markov chain model.A simplified higher-order multivariate Markov chain model is reported. With this simplified Markov chain model, the number of the parameters is reduced, computing resources are used effectively. With the theory of the special matrices, the convergence of the simplified Markov chain model is proved. Certainly, the idea of this model can also be applied to the improved higher-order multivariate Markov chain model.There is a problem that when the number of sequences increases the scale of the problem is growing quickly. A multivariate Markov chain model for adding a new data sequence is presented as a screening model for saving the computational cost. With the results of a multivariate Markov chain model, the multivariate Markov chain model for adding a new data sequence is obtained and reveals the relations of the original sequences and the new sequence. Numerical results show that the new model performs better than the multivariate Markov chain model in saving computational cost.Preconditioning techniques are very useful in numerical solutions of the saddle point problem which are originated from many real-world applications, e.g., management, economic models, circuit analysis, power system network and so on. A novel product preconditioner has been proposed. The distribution of the eigenvalues and the form of the eigenvectors in preconditioned matrix and are analyzed. The upper bounds on the degree of the minimal polynomial in preconditioned matrix are given. Numerical experiments with preconditioned Krylov subspace method on several problems show that the proposed product preconditioner is relatively more efficient.