Order Of Approximation Research Articles

P-stable abstractions of hybrid systems

Stability is a fundamental requirement of dynamical systems. Most of the works concentrate on verifying stability for a given stability region. In this paper, we tackle the problem of synthesizingP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}$$\\end{document}-stable abstractions. Intuitively, the P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}$$\\end{document}-stable abstraction of a dynamical system characterizes the transitions between stability regions in response to external inputs. The stability regions are not given—rather, they are synthesized as their most precise representation with respect to a given set of predicates P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}$$\\end{document}. A P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}$$\\end{document}-stable abstraction is enriched by timing information derived from the duration of stabilization. We implement a synthesis algorithm in the framework of Interpretation that allows different degrees of approximation. We show the representational power of P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}$$\\end{document}-stable abstractions that provide a high-level account of the behavior of the system with respect to stability, and we experimentally evaluate the effectiveness of the algorithm in synthesizing P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}$$\\end{document}-stable abstractions for significant systems.

Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes

It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As a consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic non-uniform mesh. We focus on grids mapped by a smooth function which consists in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix–vector product and detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter-free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power-graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability.

A dual number formulation to efficiently compute higher order directional derivatives

This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular cases of symmetric multilinear forms; therefore, using their symmetric properties and their coordinate representation, we implement functions to calculate mixed partial derivatives. Moreover, with directional derivatives, we deduce concise formulas for the velocity, acceleration, jerk, and jounce/snap vectors. The utility of our formulation is proved with three examples. The first example presents a comparison against the forward mode of finite differences to compute the fourth-order directional derivative of a scalar function. To this end, we have coded the finite differences method to calculate partial derivatives until the fourth order, to any order of approximation. The second example presents efficient computations of the velocity, acceleration, jerk, and jounce/snap. Finally, the third example is related to the computation of some partial derivatives. The implemented code of the proposed formulation and the finite differences method is proportioned as additional material to this article.

Approximation Theorems for Complex α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich Operators

In this paper, we introduce the complex form of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich operators. Respectively, upper quantitative estimates for the complex α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich operator and its derivatives, Voronovskaya type result and the exact order of approximation of these operators are studied.

A Generalized Family of Ratio-Product Exponential Type Estimator Using Power Transformation

Usage of Auxiliary information has been widely addressed in literature with a view of enhancing the precision of the estimators of population parameters. this paper reviewed research works of some exponential and ratio type estimators for estimating the population parameters using the population mean of an auxiliary variable. In this research, a generalized family of ratio-product type estimator of population mean as well as the coefficient of kurtosis of two auxiliary variables x and z under stratified random sampling has been proposed using power transformation and exponential techniques. The Bias and MSE of proposed estimator have been derived up to second order of approximation. The conditions in which the proposed estimator is better than existing estimators were obtained. The empirical study shows that the proposed estimator is more efficient than existing estimators in reference to the validation statistics employed.

Enhanced Variance Estimation Techniques for Addressing Non-Response and Measurement Error Situations

The use of relevant information from auxiliary variables at the estimation stage and design stage to obtain reliable and efficient estimates is a common practice in a sample survey. Several Estimators of population variance have been suggested. However, these estimators do not consider the situation of non-response due to non-availability of respondents, refusal to respond, presence of hard- core respondents or due to non-understanding of the question thereby, reducing the efficiency of the estimators and the parameters of the auxiliary variable X̄ , S2x used are sensitive to outliers or extreme values which can either lead to underestimation or overestimation. To address the aforementioned observations, classes of variance estimators under the simultaneous influence of non-response and measurement errors using outlier-free parameters as well as calibration approaches were proposed. The properties (Bias and MSE) of the modified estimators were derived up to the first order of approximation using the Taylor series approach. The efficiency conditions of the proposed estimators over the existing estimators considered in the study were established. The empirical studies were conducted using simulation and the results revealed that the proposed class of estimators have minimum MSEs and higher PREs among all the competing estimators. These imply that the proposed estimators are more efficient and can produce a better estimate of the population mean compared to other existing estimators considered in the study.

General approach on the best fitted linear operator and basis function for homotopy methods and application to strongly nonlinear oscillators

In this study, we propose a universal mathematical guideline for the selection of auxiliary linear operator L(u) and the basis function u0(x), which are the principal parts of Homotopy methods: Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). We assume the coefficients of derivatives involved in L(u) as a function of auxiliary roots of L(u)=0. Based on the residual error minimization we compute unknown roots and thereby obtain the optimal linear operator at an order of approximation. Also, an effort is made to explore the role of the number of unknowns embedded into L(u) (for different choices of real roots of L(u)=0) on the rate of converges of the solution and CPU uses. Hence the best fitted linear-operator and corresponding basis function are proposed for a particular problem. Also, we propose to discretize the exact square residual using Simpson’s 1/3 algorithms, and compare its accessibility in comparison to other existing residual calculation methods. Initially, we applied our algorithms to three IVPs (Logistic, Stiff system and Riccati system) and three singular BVPs, and then finally to a strongly nonlinear oscillator (cubic-quintic Duffing equation). We compare our technique’s accuracy, efficiency and simplicity with other contemporary analytical and numerical methods. It is demonstrated that our optimal linear operator, which is updated at every order of approximation, is much more efficient, important (than the artificial parameters and functions of developed HAM), and self-sufficient for the convergence of series solutions. Our approach is more effective, straightforward and easy to use when applied to many nonlinear problems arising in mechanics and engineering. It is seen for the initial value problems that our technique provides the globally convergent series solution, while the solutions by HPM and optimal HAM are locally convergent. This novel development will make homotopy methods more powerful.

Stability and analysis of the vibrating motion of a four degrees-of-freedom dynamical system near resonance

This article investigates the impact of three harmonically external moments on the motion of a four-degrees-of-freedom (DOF) nonlinear dynamical system composed of a double rigid pendulum connected to a nonlinear spring with linear damping. In light of the system’s generalized coordinates, the governing system (GS) of motion is derived using Lagrange’s equations. With the use of the multiple-scales method (MSM), the approximate solutions (AS) of the equations of this system are obtained up to a higher order of approximation, maybe the third order. Within the framework of the absence of secular terms, the conditions of solvability are obtained. Accordingly, the different resonance cases are categorized, and three of them are investigated simultaneously. Thus, these conditions are updated in preparation for achieving the modulation equations (ME) for the examined system. The numerical solutions (NS) of the GS are achieved using the algorithms of fourth-order Runge–Kutta (4RK), which are compared with the AS, which displays their high degree of consistency and demonstrates the precision of the MSM. The motion’s time history, steady-state solutions, and resonance curves are graphed to demonstrate the influence of various system physical parameters. All relevant fixed points at steady-state situations are identified and graphed in accordance with the Routh–Hurwitz criteria (RHC). Therefore, the zones of stability/instability of are checked and analyzed. Numerous real-world applications in disciplines like engineering and physics attest to the significance of the nonlinear dynamical system under study such as in shipbuilding, automotive devices, structure vibration, developing robots, and analysis of human walking.

Реализация бикомпактной схемы для HOLO алгоритма решения задач переноса излучения в среде

Bicompact schemes for the HOLO algorithm for solving the transport equation were applied to solve model problems of radiation transport in a medium (the first and the second Fleck problems). The main idea of HOLO algorithms is to accelerate the convergence of iterative processes due to the joint solving of high order (HO) and low order (LO) kinetic equations. The schemes are constructed by the method of lines on a minimal two-point stencil and have the fourth order of approximation in space. The Runge-Kutta method of the third order of approximation was used for integration over time. Verification and validation of the schemes were carried out. Solutions of the first and the second Fleck problems by the proposed method coincided well with solutions obtained by other methods. A modification of the second Fleck problem was proposed to investigate the dependence of the velocity of the thermal wave front on the absorption coefficient in the optically thick region.

NONLINEAR EFFECTS IN VISCOELASTIC DROP SHAPE OSCILLATIONS

A study of axisymmetric shape oscillations of viscoelastic drops in a vacuum is conducted, using the method of weakly nonlinear analysis. The motivation is the relevance of the shape oscillations for transport processes across the drop surface, as well as fundamental interest. The study is performed for, but not limited to, the two-lobed mode of initial drop deformation. The Oldroyd-B model is used for characterizing the liquid rheological behavior. The method applied yields a set of governing equations, as well as boundary and initial conditions, for different orders of approximation. In the present paper, the equations and solutions up to second order are presented, together with the characteristic equation for the viscoelastic drop. The characteristic equation has an infinite number of roots, which determine the time dependency of the oscillations. Solutions of the characteristic equation are validated against experiments on acoustically levitated individual viscoelastic aqueous polymer solution drops. Experimental data consist of decay rate and oscillation frequency of free damped drop shape oscillations. With these data, solutions of the characteristic equation dominating the oscillations are identified. The theoretical analysis reveals nonlinear effects, such as the excess time in the prolate shape and frequency change for varying initial deformation amplitude. The influences of elasticity, measured by the stress relaxation and deformation retardation time scales, are quantified, and the effects are compared to the Newtonian case in the moderate-amplitude regime.

A Family of Generalized Cardinal Polishing Splines.

Spline functions have received widespread attention in the fields of image sampling and reconstruction. To enhance the performance of splines in reconstruction and reduce the computational burden of solving large linear equations, we propose a family of generalized cardinal polishing splines (GCP-splines) and provide a system of linear equations to obtain the expressions of GCP-splines. First, we propose a cardinal polishing spline basis function with high-precision. Then, we propose a class of GCP-splines and give a general theory of GCP-splines. To calculate the expressions of GCP-splines, we adopt a system of linear equations to obtain the time shifts operator and the convolutional coefficients based on the search spacing and number of terms. Finally, we propose continuous and discrete interpolation models based on GCP-splines, and demonstrate several valuable properties, such as order of approximation and the Riesz basis. To evaluate the performance of GCP-splines, we conduct several experiments on test images from different modalities. The experimental results demonstrate that the GCP-splines for image interpolation and image denoising have better performance and outperform other methods.

Монотонизация модифицированной схемы с эрмитовой интерполяцией для численного решения неоднородного уравнения переноса с поглощением

A hybrid scheme for the numerical solving of a transport equation is implemented. The high order scheme is a modified CIP (Cubic Interpolation Polynomial) scheme with Hermitian interpolation. The third order of approximation of the CIP scheme is achieved by including in the list of unknowns not only the nodal values of the function, but also the nodal values of its derivatives. In the modification under consideration the Euler–Maclaurin formula is used to calculate derivatives. The low order scheme is the characteristic scheme of the first order of approximation. Local, layerwise and global monotonization are considered, the hybridization is performed, respectively, after a cell, a time layer or the whole grid processing. It is shown that the best results are obtained by a scheme with local monotonization. The convergence orders of the hybrid scheme on tests of different smoothness of exact solution do not differ significantly from the convergence orders of the CIP scheme. It is proposed to calculate an integral along a characteristic using the Simpson formula for the Stieltjes integral in the case of the large optical thickness; it provides a significant reduction in the numerical solution errors.

On exploring the generalized mixture estimators under simple random sampling and application in health and finance sector

There are numerous studies where data on population units’ auxiliary variables and attributes are simultaneously available. Therefore, due to cost-effectiveness and ease of recording, the study variable and several linearly related auxiliary variables are recorded. These auxiliary variables are commonly observed as quantitative and qualitative (attributes) variables and are jointly used to estimate the study variable’s population mean using a mixture estimator. In order to achieve this, a family of generalized mixture estimators was proposed under simple random sampling with the aim of improving performance under symmetrical and asymmetrical distributions. In addition, the estimator’s behavior for various sample sizes was examined with regard to its convergence to the normal distribution. The suggested generalized mixture estimator’s mean square error is deduced up to the first order of approximation. It is discovered that for the normal, uniform, Weibull, and gamma distributions, the suggested estimator estimates the population mean of the study variable with greater accuracy than the competing estimators. It is also revealed that when the proposed estimator converges to normality, the sample size is at least taken as 110, 1000, and 120. Furthermore, the implementation of three real-life datasets related to the finance and health sectors is also presented to support the proposed estimator’s significance.

Approximation by half Bernstein polynomials

This paper defines and studies the half Bernstein polynomials to approximate functions in C[0,1]. The pointwise convergence in ordinary approximation is given for these polynomials. Also, the order of approximation, the Voronovskaya-type asymptotic theorem and the error estimate are introduced. Finally, to show the ability convergence of these polynomials we provide some examples to approximate these polynomials to test functions and compare numerical results with the results corresponding to classical Bernstein polynomials.

APPROXIMATION OF FUNCTIONS BY THE MATRIX MEANS IN WEIGHTED ORLICZ SPACES

Abstract. In this work, the approximation properties of the matrix submethods in weighted Orlicz spaces L T M , are investigated. We obtain some results related to trigonometric approximation using matrix submethods of partial sums of Fourier series of functions in weighted Orlicz spaces.The degree of trigonometric approximations by the matrix methods to the functions have been investigated in weighted Orlicz spaces. The error of estimations in this work is obtained in more general terms.

Spherical Harmonics for the 1D Radiative Transfer Equation. II. Thermal Emission

Approximate methods for radiative transfer equations that are fast, reliable, and accurate are essential for the understanding of atmospheres of exoplanets and brown dwarfs. The simplest and most popular choice is the “two-stream method,” which is often used to produce simple yet effective models for radiative transfer in scattering and absorbing media. Toon et al. (hereafter, Toon89) outlined a two-stream method for computing reflected light and thermal spectra that was later implemented in the open-source radiative transfer model PICASO. In Part I of this series, we developed an analytical spherical harmonics method for solving the radiative transfer equation for reflected solar radiation that was implemented in PICASO to increase the accuracy of the code by offering a higher-order approximation. This work is an extension of this spherical harmonics derivation, to study thermal emission spectroscopy. We highlight the model differences in the approach for thermal emission and benchmark the four-term method (SH4) against Toon89 and a high-stream discrete-ordinates method, CDISORT. By comparing the spectra produced by each model, we demonstrate that the SH4 method provides a significant increase in accuracy, compared to Toon89, which can be attributed to the increased order of approximation and to the choice of phase function. We also explore the trade-off between computational time and model accuracy. We find that our four-term method is twice as slow as our two-term method, but is up to five times more accurate, when compared with CDISORT. Therefore, SH4 provides excellent improvement in model accuracy with minimal sacrifice in numerical expense.

Approximation of functions in a certain Banach space by some generalized singular integrals

<abstract><p>We introduced the $ q $-Picard, the $ q $-Picard-Cauchy, the $ q $-Gauss-Weierstrass, and the $ q $-truncated Picard singular integrals. Using the last three mentioned integrals, the orders of approximation for functions from a generalized Hölder space were determined, both in the $ L^{p} $-norm and in the generalized Hölder-norm.</p></abstract>

Better Approximation Properties by New Modified Baskakov Operators

This paper introduces a new idea to obtain a better order of approximation for the Baskakov operator. We conclude two new operators from orders one and two of the Baskakov type. Also, we prove some directed results concerning the rate of convergence of these operators. We apply the Maple software to demonstrate how the operators converge to a specific function.

Высокоточная схема для уравнения переноса в задаче нейтронной защиты

The paper considers the problems of calculating the stationary transport of high-energy neutrons from an external source in a hollow metal structure. The spatial grid for calculation consists of tetrahedra, which allows to model complicated geometric shapes of objects. The calculated area is highly heterogeneous, since it consists of metal parts with strong absorption and scattering and practically non-absorbing internal parts. A scheme based on a minimal stencil, within a single tetrahedron for the calculation. The scheme has a high approximation order and is based on characteristic approaches. For a smooth solution and a constant absorption coefficient in each cell, the scheme has a third order of convergence. The cells are traversed using the graph topological sorting algorithm. A new version of the parallelization of the calculation for the OpenMP standard has been proposed. For discretization by an angular variable, a cubature formula of the direct product type over two angles with an algebraic order of accuracy of nine is used. The angular grid is constructed in such a way as to correctly calculate the moments for narrowly directed beams of external radiation. The standard 26-group approximation of the BNAB library is used for energy sampling. A decrease in the neutron flux inside the structure is shown for groups with an energy above 1 MeV. Graphs of the angular distribution inside the structure are given.

Bivariate Bernstein Chlodovsky Operators Preserving Exponential Functions and Their Convergence Properties

This paper is devoted to an extension of the bivariate generalized Bernstein-Chlodovsky operators preserving the exponential function exp ( 2 , 2 ) where exp ( α , β ) = e − α x − β y , α , β ∈ R 0 + and x , y ≥ 0 . For these operators, we first examine the weighted approximation properties for continuous functions in the weighted space, and in the latter case, we also obtain the convergence rate for these operators using a weighted modulus of continuity. Then, we investigate the order of approximation regarding local approximation results via Peetre’s K-functional. We introduce the GBS (Generalized Boolean Sum) operators of generalized Bernstein-Chlodovsky operators, and we estimate the degree of approximation in terms of the Lipschitz class of Bögel continuous functions and the mixed modulus of smoothness. Finally, we provide some numerical and graphical examples with different values to demonstrate the rate of convergence of the constructed operators.