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 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 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

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

Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model

2019 ◽  
Vol 17 (01) ◽  
pp. 1850061 ◽  
Author(s):  
Dana Černá
Keyword(s):  
Boundary Conditions ◽  
Option Pricing ◽  
Cubic Spline ◽  
Dirichlet Boundary Conditions ◽  
Vanishing Moments ◽  
Spline Wavelet ◽  
Spline Wavelets ◽  
Jump Diffusion Model ◽  
The Stability ◽  
Homogeneous Dirichlet Boundary Conditions

The paper is concerned with the construction of a cubic spline wavelet basis on the unit interval and an adaptation of this basis to the first-order homogeneous Dirichlet boundary conditions. The wavelets have four vanishing moments and they have the shortest possible support among all cubic spline wavelets with four vanishing moments corresponding to B-spline scaling functions. We provide a rigorous proof of the stability of the basis in the space [Formula: see text] or its subspace incorporating boundary conditions. To illustrate the applicability of the constructed bases, we apply the wavelet-Galerkin method to option pricing under the double exponential jump-diffusion model and we compare the results with other cubic spline wavelet bases and with other methods.

Unsteady flow in a supersonic cascade with strong in-passage shocks

1977 ◽  
Vol 83 (3) ◽  
pp. 569-604 ◽  
Author(s):  
M. E. Goldstein ◽  
Willis Braun ◽  
J. J. Adamczyk
Keyword(s):  
Unsteady Flow ◽  
Aircraft Engine ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Trailing Edges ◽  
Linearized Theory ◽  
Closed Form Analytical Solution ◽  
Supersonic Region ◽  
Shock Motion ◽  
The Stability

Linearized theory is used to study the unsteady flow in a supersonic cascade with in-passage shock waves. We use the Wiener–Hopf technique to obtain a closed-form analytical solution for the supersonic region. To obtain a solution for the rotational flow in the subsonic region we must solve an infinite set of linear algebraic equations. The analysis shows that it is possible to correlate quantitatively the oscillatory shock motion with the Kutta condition at the trailing edges of the blades. This feature allows us to account for the effect of shock motion on the stability of the cascade.Unlike the theory for a completely supersonic flow, the present study predicts the occurrence of supersonic bending flutter. It therefore provides a possible explanation for the bending flutter that has recently been detected in aircraft-engine compressors at higher blade loadings.

Unrestricted Harmonic Balance

1980 ◽  
Vol 35 (10) ◽  
pp. 1054-1061 ◽  
Author(s):  
Friedrich Franz Seelig
Keyword(s):  
Linear Systems ◽  
Harmonic Balance ◽  
Open Systems ◽  
Periodic Structures ◽  
Simulation Method ◽  
Algebraic Equations ◽  
Good Convergence ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
The Stability

Abstract Periodic structures in chemical kinetic systems can be evaluated by an extension of the well-known method of harmonic balance, which yields very simple expressions in the case of linear systems containing only zero and first order reactions. The far more interesting non-linear systems containing e.g. second order reactions which in case of open systems far from thermodynamic equilibrium give rise to non-classical phenomena like oscillations, chemical waves, excitability, hysteresis, multistability, dissipative structures etc. can be treated in a similar way by introducing new pseudo-linear quantities utilizing certain group properties of harmonic expansions. The resulting complicated implicit non-linear algebraic equations are solved by a method developed by Powell and show good convergence. Since this method - in contrast to the conventional method of simulation - is independent from the stability of the periodic structure to be evaluated it can even be applied to unstable cases where the simulation method necessarily fails. An evaluation of the stability is included in the developed computer program.

scholarly journals Stability of orthotropic plates

2020 ◽  
Vol 166 ◽  
pp. 06004
Author(s):  
Mykola Surianinov ◽  
Dina Lazarieva ◽  
Iryna Kurhan
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Boundary Conditions ◽  
Critical Loads ◽  
Orthotropic Plate ◽  
Algebraic Equations ◽  
Stability Equation ◽  
Element Analysis ◽  
Linear Algebraic Equations ◽  
The Stability

The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green’s functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author’s method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.

scholarly journals Common Lyapunov Function Based on Kullback–Leibler Divergence for a Switched Nonlinear System

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Omar M. Abou Al-Ola ◽  
Ken'ichi Fujimoto ◽  
Tetsuya Yoshinaga
Keyword(s):  
Lyapunov Function ◽  
Switched Systems ◽  
Switched System ◽  
Algebraic Equations ◽  
Common Lyapunov Function ◽  
Arbitrary Switching ◽  
Switched Nonlinear System ◽  
Linear Algebraic Equations ◽  
Leibler Divergence ◽  
The Stability

Many problems with control theory have led to investigations into switched systems. One of the most urgent problems related to the analysis of the dynamics of switched systems is the stability problem. The stability of a switched system can be ensured by a common Lyapunov function for all switching modes under an arbitrary switching law. Finding a common Lyapunov function is still an interesting and challenging problem. The purpose of the present paper is to prove the stability of equilibrium in a certain class of nonlinear switched systems by introducing a common Lyapunov function; the Lyapunov function is based on generalized Kullback–Leibler divergence or Csiszár'sI-divergence between the state and equilibrium. The switched system is useful for finding positive solutions to linear algebraic equations, which minimize theI-divergence measure under arbitrary switching. One application of the stability of a given switched system is in developing a new approach to reconstructing tomographic images, but nonetheless, the presented results can be used in numerous other areas.

Sign in / Sign up

Export Citation Format

Share Document