In this work, we attempt to combine the tensor auto-regression (TAR) model with multilinear systems, expanding the randomized Gauss-Seidel (RGS) method for the parameters estimation and prediction of the TAR model. Inspired by the Count Sketch maximal weighted residual Kaczmarz method, we introduce a Count Column Sketch method with the randomized Gauss-Seidel (CCS-RGS) method for the parameters estimation. In addition, we present the convergence properties of the CCS-RGS method and optimize the upper bound of the convergence rate of the CCS-RGS method with the Simulated Annealing (SA) algorithm. In numerical experiments, we apply the CCS-RGS method to optimize and predict the L-TAR model. The results show that both the CCS-RGS method with the SA (SA-CCS-RGS) method and the CCS-RGS method are effective, they require less training time and iteration steps compared to the RGS method.
By incorporating the Euler-extrapolated Hermitian/skew-Hermitian splitting (EHS) method as an internal iteration for solving large-scale and sparse nonlinear systems with complex symmetric Jacobian matrix, we introduce the King-EHS method in this paper. A rigorous theoretical framework for the methods is established under the conditions of Hölder continuity. The local convergence of King-EHS method is also presented. Moreover, numerical results are given to validate the feasibility and superior effectiveness on a specific subset of nonlinear systems solved by our methods.
In order to reduce the sensitivity to perturbations of closed-loop system, we consider robust partial pole assignment problem for the damped gyroscopic systems in this paper. Based on the condition number of the closed-loop eigenvector matrix, we give the robust measurement of closed-loop system. Then we propose an algebraic method to solve robust partial pole assignment problem by using system matrices and orthogonality relationship. Our method ensures the robustness of the closed-loop system as much as possible. Numerical results exhibit the efficiency of our method.
We consider a structure-fluid interaction of oscillating piezodevice and acoustic liquid. Such problem requires finding resonance or anti-resonance frequencies of the solid structure to use them in the latter frequency-response analysis step. After finite element discretization of the issue, we need to solve a partial generalized eigenvalue problem. The algorithms which exploit the characterization of the eigenvalues as successive minima of the Rayleigh quotient are used. Some preconditioners for gradient-based methods are compared in terms of efficiency. The results of the numerical experiments are presented.
To study changes in the dynamics of the bearing capacity of the soil around the emergency support of a railway bridge in the region of permafrost, a mathematical model is proposed to assess the influence of various climatic factors on the thermo-physical parameters of the soil. Using the developed numerical method that describes the propagation of thermal fields in the soil, it is shown solar radiation affects the reduction in the bearing capacity of the soil around railway support consisting of large-diameter piles. The developed algorithm considers climate change in a specific region over the past forty years and uses data from temperature sensors in exploration wells around the given railway support to calibrate the model and verify the obtained numerical results. The results of numerical calculations showed that sun protective installation over the southern part of the railway support makes it possible to increase the bearing capacity of the soil. The proposed model of the thermal behavior of a pile structure and the developed software product allow us to give a long-term forecast of changes in the bearing capacity of the soil around the railway bridge support, taking into account specific climatic conditions and soil lithology.
