scholarly journals ANALYSIS OF MULTI-THREADED MARKOV SYSTEMS

2021 ◽  
Vol 5 (4) ◽  
pp. 70-78
Author(s):  
Lev Raskin ◽  
Larysa Sukhomlyn ◽  
Dmytro Sagaidachny ◽  
Roman Korsun
Keyword(s):  
Markov Property ◽  
Computational Method ◽  
High Dimensional ◽  
Algebraic Equations ◽  
Parametric Synthesis ◽  
Markov Systems ◽  
Linear Algebraic Equations ◽  
Computational Implementation ◽  
System States ◽  
Analytical Relations

Known technologies for analyzing Markov systems use a well-operating mathematical apparatus based on the computational implementation of the fundamental Markov property. Herewith the resulting systems of linear algebraic equations are easily solved numerically. Moreover, when solving lots of practical problems, this numerical solution is insufficient. For instance, both in problems of structural and parametric synthesis of systems, as well as in control problems. These problems require to obtain analytical relations describing the dependences of probability values of states of the analyzed system with the numerical values of its parameters. The complexity of the analytical solution of the related systems of linear algebraic equations increases rapidly along with the increase in the system dimensionality. This very phenomenon manifests itself especially demonstratively when analyzing multi-threaded queuing systems.  Accordingly, the objective of this paper is to develop an effective computational method for obtaining analytical relations that allow to analyze high-dimensional Markov systems. To analyze such systems this paper provides for a decomposition method based on the idea of phase enlargement of system states. The proposed and substantiated method allows to obtain analytical relations for calculating the distribution of Markov system states.  The method can be effectively applied to solve problems of analysis and management in high-dimensional Markov systems. An example has been considered

scholarly journals Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

2021 ◽  
Vol 17 (1) ◽  
pp. 33
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Triangular System ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of  linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is  O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

A General Computational Method for Converting Normal Spectra into Derivative Spectra

2005 ◽  
Vol 59 (5) ◽  
pp. 584-592 ◽  
Author(s):  
Y. Leong Yeow ◽  
Y. K. Leong
Keyword(s):  
Regularization Parameter ◽  
Mathematical Problem ◽  
Computational Method ◽  
Second Derivative ◽  
Derivative Spectrum ◽  
Algebraic Equations ◽  
Derivative Spectra ◽  
Linear Algebraic Equations ◽  
Normal Spectrum ◽  
Second Derivative Spectra

The mathematical problem of converting a normal spectrum into the corresponding first- and second-derivative spectra is formulated as an integral equation of the first kind. Tikhonov regularization is then applied to solve the spectral conversion problem. The end result is a set of linear algebraic equations that takes in as input the original spectrum and produces as output the second-derivative spectrum, which is then integrated to yield the first-derivative spectrum. Noise amplification is kept under control by adjusting the regularization parameter (guided by generalized cross-validation) in the algebraic equations. The performance of this procedure is demonstrated by applying it to different types of spectral data taken from the literature.

scholarly journals SECTIONING OF HIGH DIMENSIONAL BANDED MATRICES

2014 ◽  
pp. 14-21
Author(s):  
Dmytro Fedasyuk ◽  
Pavlo Serdyuk ◽  
Yuriy Semchyshyn
Keyword(s):  
Linear Equations ◽  
Computing Time ◽  
Mathematical Physics ◽  
High Dimensional ◽  
Systems Of Equations ◽  
Algebraic Equations ◽  
Banded Matrices ◽  
Linear Algebraic Equations ◽  
Systems Of Linear Equations ◽  
The Subject

Solving high dimensional systems of linear algebraic equations is of use to many problems of mathematical physics, in particular, it is one of the main subgoals at solving systems of equations in partial derivatives. Distributed solving of high dimensional systems of linear equations allows to reduce computing time, especially in cases when these matrices can not be kept in one computer's RAM. The subject of this study is the search of optimal high dimensional matrices sectioning algorithms for distributed solving systems of linear algebraic equations.

scholarly journals A new computational method based on the minimum lithostatic deviation (MLD) principle to analyse slope stability in the frame of the 2-D limit-equilibrium theory

2008 ◽  
Vol 8 (4) ◽  
pp. 671-683 ◽  
Author(s):  
S. Tinti ◽  
A. Manucci
Keyword(s):  
Cross Section ◽  
Limit Equilibrium ◽  
Computational Method ◽  
Bottom Surface ◽  
Average Weight ◽  
Vertical Force ◽  
Algebraic Equations ◽  
Local Weight ◽  
Limit Equilibrium Methods ◽  
Linear Algebraic Equations

Abstract. The stability of a slope is studied by applying the principle of the minimum lithostatic deviation (MLD) to the limit-equilibrium method, that was introduced in a previous paper (Tinti and Manucci, 2006; hereafter quoted as TM2006). The principle states that the factor of safety F of a slope is the value that minimises the lithostatic deviation, that is defined as the ratio of the average inter-slice force to the average weight of the slice. In this paper we continue the work of TM2006 and propose a new computational method to solve the problem. The basic equations of equilibrium for a 2-D vertical cross section of the mass are deduced and then discretised, which results in cutting the cross section into vertical slices. The unknowns of the problem are functions (or vectors in the discrete system) associated with the internal forces acting on the slice, namely the horizontal force E and the vertical force X, with the internal torque A and with the pressure on the bottom surface of the slide P. All traditional limit-equilibrium methods make very constraining assumptions on the shape of X with the goal to find only one solution. In the light of the MLD, the strategy is wrong since it can be said that they find only one point in the searching space, which could provide a bad approximation to the MLD. The computational method we propose in the paper transforms the problem into a set of linear algebraic equations, that are in the form of a block matrix acting on a block vector, a form that is quite suitable to introduce constraints on the shape of X, but also alternatively on the shape of E or on the shape of X. We test the new formulation by applying it to the same cases treated in TM2006 where X was expanded in a three-term sine series. Further, we make different assumptions by taking a three-term cosine expansion corrected by the local weight for X, or for E or for A, and find the corresponding MLDs. In the illustrative applications given in this paper, we find that the safety factors associated with the MLD resulting from our computations may differ by some percent from the ones computed with the traditional limit-equilibrium methods.

scholarly journals A Galerkin-Parameterization Method for the Optimal Control of Smart Microbeams

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Marwan Abukhaled ◽  
Ibrahim Sadek
Keyword(s):  
Optimal Control ◽  
Operational Matrix ◽  
Performance Measure ◽  
Point Of View ◽  
Computational Method ◽  
Algebraic Equations ◽  
State Control ◽  
Lumped Parameter ◽  
Linear Algebraic Equations ◽  
Quadratic Cost Function

A proposed computational method is applied to damp out the excess vibrations in smart microbeams, where the control action is implemented using piezoceramic actuators. From a mathematical point of view, we wish to determine the optimal boundary actuators that minimize a given energy-based performance measure. The minimization of the performance measure over the actuators is subjected to the full motion of the structural vibrations of the micro-beams. A direct state-control parametrization approach is proposed where the shifted Legendre polynomials are employed to solve the optimization problem. Legendre operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the lumped parameter systems with respect to the quadratic cost function by solving linear algebraic equations. Numerical examples are provided to demonstrate the applicability and efficiency of the proposed approach.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

Sign in / Sign up

Export Citation Format

Share Document