scholarly journals Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets

2021 ◽  
Vol 20 ◽  
pp. 717-728
Author(s):  
Boris M. Shumilov
Keyword(s):  
Band System ◽  
Cubic Spline ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Diagonal Dominance ◽  
Sweep Method ◽  
Spline Wavelets ◽  
Linear Algebraic Equations ◽  
Initial Scale ◽  
Strict Diagonal Dominance

In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

On splitting for cubic spline wavelets with four zero moments on an interval

2021 ◽  
pp. 72-87
Author(s):  
Борис Михайлович Шумилов
Keyword(s):  
Cubic Spline ◽  
Spline Space ◽  
Dirichlet Boundary Conditions ◽  
Second Derivative ◽  
Algebraic Equations ◽  
Splitting Algorithm ◽  
Spline Wavelets ◽  
Wavelet Space ◽  
Linear Algebraic Equations ◽  
The Stability

В пространстве кубических сплайнов построены вейвлеты, удовлетворяющие однородным граничным условиям Дирихле и обнулению первых четырех моментов. Получены неявные соотношения, связывающие сплайн-коэффициенты разложения на начальном уровне со сплайн-коэффициентами и вейвлет-коэффициентами на вложенном уровне ленточной системой линейных алгебраических уравнений с невырожденной матрицей. После расщепления на четные и нечетные уравнения матрица преобразования имеет пять (вместо трех в случае двух нулевых моментов) диагоналей. Доказано наличие строгого диагонального доминирования по столбцам. Для сравнения использованы вейвлеты с двумя нулевыми моментами и интерполяционные кубические сплайновые вейвлеты. Результаты численных экспериментов показывают, что схема с четырьмя нулевыми моментами точнее при аппроксимации функций, но грубее при аппроксимации второй производной. The article examines the problem of constructing a splitting algorithm for cubic spline wavelets. First, a cubic spline space is constructed for splines with homogeneous Dirichlet boundary conditions. Then, using the first four zero moments, the corresponding wavelet space is constructed. The resulting space consists of cubic spline wavelets that satisfy the orthogonality conditions for all thirddegree polynomials. The originality of the research lies in obtaining implicit relations connecting the coefficients of the spline expansion at the initial level with the spline coefficients and wavelet coefficients at the embedded level by a band system of linear algebraic equations with a nondegenerate matrix. Excluding the even rows of the system, the resulting transformation algorithm is obtained as a solution to a sequence of band systems of linear algebraic equations with five (instead of three in the case of two zero moments) diagonals. The presence of strict diagonal dominance over the columns is proved, which confirms the stability of the computational process. For comparison, we adopt the results of calculations using wavelets orthogonal to first-degree polynomials and interpolating cubic spline wavelets with the property of the best mean-square approximation of the second derivative of the function being approximated. The results of numerical experiments show that the scheme with four zero moments is more accurate in the approximation of functions, but becomes inferior in accuracy to the approximation of the second derivative.

scholarly journals On the seven-diagonals splitting for the cubic spline wavelets with six vanishing moments on an interval

2021 ◽  
Vol 2099 (1) ◽  
pp. 012016
Author(s):  
B M Shumilov
Keyword(s):  
Cubic Spline ◽  
Dirichlet Boundary Conditions ◽  
Algebraic Equations ◽  
Splitting Algorithm ◽  
Discrete Function ◽  
Vanishing Moments ◽  
Spline Wavelets ◽  
Linear Algebraic Equations ◽  
Comparative Results ◽  
The Stability

Abstract This study uses a zeroing property of the first six moments for constructing a splitting algorithm for the cubic spline wavelets. First, we construct a system of cubic basic spline-wavelets, realizing orthogonal conditions to all polynomials up to any degree. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the orthogonality to all polynomials up to the fifth degree on the closed interval. The originality of the study consists of obtaining implicit finite relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, the resulting transformation matrix has seven diagonals, instead of five as in the previous case with four zero moments. A modification of the system is performed, which ensures a strict diagonal dominance, and, consequently, the stability of the calculations. The comparative results of numerical experiments on approximating and calculating the derivatives of a discrete function are presented.

scholarly journals Application Of The Interlaced Sweep Method For The Solution Of Problems In Field Theory

2015 ◽  
Vol 3 ◽  
pp. 170
Author(s):  
Aloizs Ratnieks ◽  
Marina Uhanova
Keyword(s):  
Field Theory ◽  
Finite Element Method ◽  
Finite Element ◽  
Direct Methods ◽  
System Of Equations ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Sweep Method ◽  
Linear Algebraic Equations ◽  
Separate Block

<p class="R-AbstractKeywords"><span lang="EN-US">For solution of problems in field theory the method of sweep is very popular. The authors suggest a very effective method of interlaced sweep. The essence of the interlaced sweep method lies in the fact that matrix of the linear algebraic equations system is broken into parts and the solution of separate blocks is sought by direct methods. Usually for each line of the grid a separate block is created. The system of block equations has a tridiagonal matrix where only elements of the main diagonal and two neighboring diagonals are different from zero. The system of equations with such a matrix is easily solved by the method of elimination of unknowns.</span></p><p class="R-AbstractKeywords"><span lang="EN-US">By solving the problems by the finite element method, the nodes of touching of neighboring elements can be placed on curved lines, and the sweep on these lines can be performed observing the principle of interlaced sweep. By following this method, the neighboring lines should not belong to the same half-step.</span></p>

scholarly journals On Five-Diagonal Splitting for Cubic Spline Wavelets with Six Vanishing Moments on a Segment

2021 ◽  
Vol 17 ◽  
pp. 156-165
Author(s):  
Boris Shumilov
Keyword(s):  
Cubic Spline ◽  
Dirichlet Boundary Conditions ◽  
Algebraic Equations ◽  
Splitting Algorithm ◽  
Discrete Function ◽  
Vanishing Moments ◽  
Spline Wavelets ◽  
Orthogonality Conditions ◽  
Wavelet Space ◽  
Linear Algebraic Equations

In this study, we use the vanishing property of the first six moments for constructing a splitting algorithm for cubic spline wavelets. First, we construct the corresponding wavelet space that satisfies the orthogonality conditions for all fifth-degree polynomials. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the closed interval. The originality of the study consists in obtaining implicit relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, in contrast to the case with two zero moments, the resulting transformation matrix has five (instead of three) diagonals. The results of numerical experiments on calculating the derivatives of a discrete function are presented.

scholarly journals Particular properties of estimation of partial capacitances of insulation of three core power cables by applying aggregate measurements

2021 ◽  
pp. 15-20
Author(s):  
Ivan Kostiukov
Keyword(s):  
Root Mean Square Error ◽  
Mean Square Error ◽  
Root Mean Square ◽  
Least Squares Method ◽  
Power Cable ◽  
System Of Equations ◽  
Power Cables ◽  
Mean Square ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

This paper presents a description of specific properties of determining the values of partial capacitances of insulation gaps in power cables with paper insulation for various ways of forming and solving the system of linear algebraic equations. Possible ways of inspection the insulation of three core power cables for the estimation of values of partial capacitances by applying aggregate measurements which are based on various ways of connection of emittance meter to tested sample of power cable are given. Estimation of partial capacitances by the direct solution of a system of linear algebraic equations, by minimizing the root mean square error of solving an overdetermined system of equations by the least squares method, as well as by finding a normal solution of an indefinite system of equations by the pseudo-inverse matrix, is also considered. It is shown that minimization of the root mean square error by the least squares method and the direct solution of system of equations show quite similar results for the case of estimation of partial capacitances by means of aggregate measurements, at the same time the solution of an indefinite system of equations by the method of a pseudo-inverted matrix allows to reproduce rather accurately only 3 out of 6 values of partial capacitances. The uneven effect of frequency on the electrical capacitance of the insulation gaps between the cores of the power cable and between its cores and the sheath is shown. It was proposed to use the frequency dependence of the electrical capacitance of insulation gaps as an informative parameter about the technical state of insulating gaps between the cores of the power cable and between its cores and its sheath.   Keywords: root mean square error; least squares method; system of linear algebraic equations; dielectric losses; dielectric permittivity.

scholarly journals Construction Fragile Digital Watermark Used by a Team of Authors in Executable Files

2020 ◽  
Vol 18 (1) ◽  
pp. 65-73
Author(s):  
I. V. Nechta
Keyword(s):  
Unique Solution ◽  
Digital Watermark ◽  
The Other ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Statistical Research ◽  
Other Hand ◽  
Linear Algebraic Equations ◽  
Software Product ◽  
The One

According to statistical research, a violation of license agreements annually causes huge losses to software companies. On the one hand, illegal copies of the software product are created, on the other hand, some fragments of the programs are used by third parties unauthorized. Another important problem is the violation of the program integrity, for example, in terms of blocking functions of the license key checking. In this regard, the task of construction methods for protecting intellectual property in software applications is highly relevant. Previously known methods solve this problem by means of fragile digital watermarks. Below is presented a method for constructing a fragile digital watermark used in executable files. A model of a developers team creating software product protected by DWM is considered. The application of this method will allow to reveal the fact of the container integrity violation, on the one hand, and, on the other hand, will allow the author, if it is necessary, to confirm his participation in the development and embedding of the DWM. In this method we use mathematical properties of systems of linear algebraic equations, digital signature and cryptographic hash functions. The scheme is based on the Kronecker – Capelli theorem. To find the group password the co-author who is in the group finds one root of the consistent system of linear algebraic equations. The indicated system consists of n equations and contains n + 1 variables. For an outsider who is not in the group, the system of equations does not have a unique solution. The co-author of the group is able to calculate one variable in system based on their passport data. Therefore, the system of equations for such co-author has a unique solution. Various attacks on a protected by the new method are analyzed, and it is shown its efficiency. The constructed method can be applied in companies with a large team of developers.

scholarly journals Shifted Cubic Spline Wavelets with Two Vanishing Moments on the Interval and a Splitting Algorithm

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Dirichlet Boundary ◽  
Linear Equations ◽  
Cubic Spline ◽  
Dirichlet Boundary Conditions ◽  
System Of Linear Equations ◽  
Splitting Algorithm ◽  
Vanishing Moments ◽  
Spline Wavelets ◽  
Strict Diagonal Dominance ◽  
Homogeneous Dirichlet Boundary Conditions

This paper deals with the use of the first two vanishing moments for constructing cubic spline-wavelets orthogonal to polynomials of the first degree. A decrease in the supports of these wavelets is shown in comparison with the classical semiorthogonal wavelets. For splines with homogeneous Dirichlet boundary conditions of the second order, an algorithm of the shifted wavelet transform is obtained in the form of a solution of a tridiagonal system of linear equations with a strict diagonal dominance

scholarly journals PARALLEL IMPLEMENTATION OF THE YANENKO METHOD FOR SOLVING THE HEAT EQUATION

2021 ◽  
pp. 127-135
Author(s):  
A. N. Semyatova ◽  
E. G. Kenzhebek
Keyword(s):  
Heat Equation ◽  
Parallel Implementation ◽  
Algebraic Equations ◽  
Sweep Method ◽  
Differential Problem ◽  
Computing Systems ◽  
One Dimensional ◽  
Parallel Data ◽  
Linear Algebraic Equations ◽  
Inter Process Communication

In this article, we will consider the parallel implementation of the Yanenko algorithm for the two-dimensional heat equation, and the sweep method was used to numerically solve  the heat equation. The implementation of the sequential  program is carried out simply in two-part steps by the longitudinal-transverse run, however, parallelization of two fractional  steps with an indefinite scheme is difficult due to the creation of inter-process communication of data. In the course of the study, a parallel data distribution with one-dimensional decompositions is shown in the application of the Yanenko method for calculating heat conductivity. The results of parallelization of this task using the 1D decomposition were obtained and acceleration and efficiency images were analyzed in order to evaluate the parallel program. Currently, modeling of processes by numerical solution of differential equations is widely used in every field of Science, the most common methods bring the differential problem to a system of linear algebraic equations, methods that solve such systems include various startup options. The emergence and development of computing systems using Multi-Core processors and graphics accelerators make the problem of startup parallelization relevant; the results of the study are used for teaching in research institutes and universities.

scholarly journals NUMERICAL SOLUTION OF THE BOUNDARY PROBLEM FOR PARABOLIC EQUATION WITH NON-SELF-ADJOINT OPERATOR AND RELATED BOUNDARY CONDITIONS

2020 ◽  
pp. 7-11
Author(s):  
B. Dovgiy ◽  
L. Vakal ◽  
E. Vakal
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Diffusion Processes ◽  
Boundary Value ◽  
Adjoint Operator ◽  
Second Order ◽  
Algebraic Equations ◽  
Sweep Method ◽  
Linear Algebraic Equations

A boundary value problem for a second-order parabolic equation with a non-self-adjoint operator is considered. Such problems are mathematicalmodels for a number of problems, describing convective-diffusion processes of matter transfer, breakdown mechanisms of laser activity in plasma, etc. While studying the physics of breakdown, one should take into account the avalanche-like increase in the number of free electrons due to multiphoton ionization processes under the influence of optical pulses. This requires the inclusion of related boundary conditions in the problem formulation. An important circumstance that must be taken into account when developing a method for solving the problem is fulfillment of a certain conservation law for its solution. To solve the boundary value problem an approach based on the finite difference method is proposed. The approximation of the equation and boundary conditions is constructed so that the difference scheme is completely conservative. It approximates the original problem with the second order in the spatial variable and in time, and it has the second order of convergence. To effectively solve a system of linear algebraic equations at each time layer, the sweep method for complex systems in combination with the non-monotonic sweep method for systems with a tridiagonal matrix is used. Software based on computer mathematics MATLAB is developed to perform numerical calculations. It is obtained an approximate solution of an applied problem for different instants of time, as well as values of an absorption coefficient, the change in sign of which determines the transition of the plasma in a laser-active state.

scholarly journals Perturbation theory for the self-consistent field

1941 ◽  
Vol 178 (975) ◽  
pp. 499-505 ◽  
Keyword(s):  
Perturbation Theory ◽  
The Self ◽  
System Of Equations ◽  
Dirac System ◽  
Algebraic Equations ◽  
Consistent Field ◽  
Linear Algebraic Equations ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Perturbation Energy

The perturbation theory has been applied to the Fock-Dirac system of equations. To obtain the perturbed Fock functions, it is necessary to solve a system of linear algebraic equations. To obtain the perturbation energy of the i th order, the perturbed Fock functions up to the ( i - 1)th order are needed.

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