Ideal flow of Markov Chain

Ideal flow network is a strongly connected network with flow, where the flows are in steady state and conserved. The matrix of ideal flow is premagic, where vector, the sum of rows, is equal to the transposed vector containing the sum of columns. The premagic property guarantees the flow conservation in all nodes. The scaling factor as the sum of node probabilities of all nodes is equal to the total flow of an ideal flow network. The same scaling factor can also be applied to create the identical ideal flow network, which has from the same transition probability matrix. Perturbation analysis of the elements of the stationary node probability vector shows an insight that the limiting distribution or the stationary distribution is also the flow-equilibrium distribution. The process is reversible that the Markov probability matrix can be obtained from the invariant state distribution through linear algebra of ideal flow matrix. Finally, we show that recursive transformation [Formula: see text] to represent [Formula: see text]-vertices path-tracing also preserved the properties of ideal flow, which is irreducible and premagic.

Convergence Analysis of Surrogate Assisted (1+1) Evolutionary Algorithm

This paper analyzes the convergence deviation of surrogate assisted (1+1)EA. A model of surrogate assisted (1+1)EA can be built by the finite markov chain, then we got the transition matrix of this algorithm. The deviation of surrogate model can be expressed by the perturbation of transition matrix. So we can estimate the convergence deviation with the method of matrix perturbation analysis. Analyzing of the convergence changes brought by surrogate model’s deviations can help us to have a better select of the surrogate model.

Improved Calculation of Eigenvector Sensitivities Using Matrix Perturbation Analysis

Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. Based on the concept of matrix perturbation, this paper presents a new method for the improved calculation of eigenvector derivatives in the case where only few of the lower modes of a system under study have been computed. By using this new proposed method, considerable improvement on the accuracy of the estimation of eigenvector derivatives can be achieved at the expense of very tiny extra computational effort since only few matrix vector operations are required. Convergency criterion of the method has been established and the required accuracy can be controlled by including more higher order terms. Numerical results from practical finite element model have demonstrated the practicality of the proposed method. Further, the proposed method can be easily incorporated into commercial finite element packages to improve the accuracy of eigenderivatives needed for practical applications.