Order Approximation Research Articles

Higher order approximation of functions by modified Goodman-Sharma operators

Here we study the approximation properties of a modified Goodman-Sharma operator recently considered by Acu and Agrawal in [1]. This operator is linear but not positive. It has the advantage of a higher order of approximation of functions compared with the Goodman-Sharma operator. We prove direct and strong converse theorems in terms of a related K-functional.

Longitudinal Doppler for Observers in Uniform Acceleration

This study derives the exact frequency observed for electromagnetic waves emitted during uniform acceleration. In the classic case the frequency observed is typically calculated using the formula f = f0 (1+gH/c2) where ‘f0‘ is relevant to the source and g represents the constant acceleration. At least in Feynman’s Lectures, such formula is derived directly as a first order approximation from the relativistic Doppler effect. Relativistic relations are used to determine the exact light-time between the source and observer. It is found that, if g represents the proper acceleration, the frequency ratio remains the same as in the classic case (1+gH/c2). Being that value exact, no higher order relation exists where it can be derived from. The absence of higher order terms depends on a peculiar compensation of two relativistic effects.

Multi-patch Isogeometric convolution hierarchical deep-learning neural network

A seamless integration of neural networks with Isogeometric Analysis (IGA) was first introduced in [1] under the name of Hierarchical Deep-learning Neural Network (HiDeNN) and has systematically evolved into Isogeometric Convolution HiDeNN (in short, C-IGA) [2]. C-IGA achieves higher order approximations without increasing the degree of freedom. Due to the Kronecker delta property of C-IGA shape functions, one can refine the mesh in the physical domain like standard finite element method (FEM) while maintaining the exact geometrical mapping of IGA. In this article, C-IGA theory is generalized for multi-CAD-patch systems with a mathematical investigation of the compatibility conditions at patch interfaces and convergence of error estimates. Two compatibility conditions (nodal compatibility and G0 (i.e., global C0) compatibility) are presented and validated through numerical examples.

Molecular anatomy of the pressure anisotropy in the interface of one and two component fluids: Local thermodynamic description of the interfacial tension.

Through the decomposition of the pressure into the kinetic and the intermolecular contributions, we show that the pressure anisotropy in the fluid interface, which is the source of the interfacial tension, comes solely from the latter contribution. The pressure anisotropy due to the intermolecular force between the fluid particles in the same or the different fluid components is approximately proportional to the multiplication of the corresponding fluid density gradients, and from the molecular dynamics simulation of the liquid-vapor and liquid-liquid interfaces, we demonstrate that the density gradient theory by van der Waals gives the leading order approximation of the free energy density in inhomogeneous systems, neglecting the Tolman length.

Elementary Steps of Catalytic Reactions Occurring on Metallic Alloy Nanoparticles.

Catalytic reactions running in an adsorbed overlayer on metallic alloy nanoparticles are of high interest in the context of applications in the chemical industry. The understanding of the corresponding kinetics is, however, still limited. One of the reasons of this state of the art is the interplay between adsorption and adsorbate-influenced segregation of metal atoms inside alloy nanoparticles. I scrutinize this interplay by using a generic field model of segregation and the mean-field approximation in order to describe adsorption, desorption, and elementary catalytic reactions. Under steady-state conditions, the segregation is demonstrated to be manifested in the change of the dependence of the activation energies of desorption or elementary reactions on coverage, and the sign of this change is positive. The effect of this change on the apparent reaction orders is briefly discussed as well.

Efficient Imputation Methods for Estimating Population Proportion in Diagonal Systematic Sampling

Accurate estimation of proportions is essential for effective decision-making, resource management, and ensuring the reliability of statistical analyses. It underpins many critical processes across various domains, making it a foundational aspect of data analysis and interpretation. In this paper, we have proposed a regression-type and exponential type imputation methods which are free of unknown parameters for estimating missing values or non-responses while estimating population proportion under diagonal systematic sampling design. Imputation methods are crucial in data analysis and machine learning because they address the issue of missing data. Some key significance includes improved data quality, preservation of sample size, bias reduction and enhanced predictive power. The estimators of the proposed imputation methods were derived. The properties (biases and MSEs) of the class of estimators of the proposed imputation methods were derived up to first order approximation. Results of numerical illustration using simulated data revealed that the proposed estimators are more efficient and practicable than exiting estimators considered in the study.

Higher order approximations in quantum speed limit time of a qubit coupled to XXZ spin chain environment

Higher order approximations in quantum speed limit time of a qubit coupled to XXZ spin chain environment

Non-Equilibrium Quantum Brain Dynamics: Water Coupled with Phonons and Photons.

We investigate Quantum Electrodynamics (QED) of water coupled with sound and light, namely Quantum Brain Dynamics (QBD) of water, phonons and photons. We provide phonon degrees of freedom as additional quanta in the framework of QBD in this paper. We begin with the Lagrangian density QED with non-relativistic charged bosons, photons and phonons, and derive time-evolution equations of coherent fields and Kadanoff-Baym (KB) equations for incoherent particles. We next show an acoustic super-radiance solution in our model. We also introduce a kinetic entropy current in KB equations in 1st order approximation in the gradient expansion and show the H-theorem for self-energy in Hartree-Fock approximation. We finally derive conserved number density of charged bosons and conserved energy density in spatially homogeneous system.

Algorithms for solving the inverse scattering problem for the Manakov model

The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton).

Investigation on the dominant mechanism of chatter in high-load robot milling process based on theoretical and experimental analysis

Chatter has always been a key problem restricting the improvement of robotic milling quality and efficiency. To avoid chatter, it is necessary to determine what is the dominant chatter mechanism (mode coupling or regenerative) of the robot milling system. Therefore, this paper focus on the dominant chatter mechanism in high-load (600kg) robot milling. The modal test results show that the dynamic flexibility of spindle-tool structure mode in high-load robot is significantly higher than that of the body structure mode, which is significantly different from the low-load robot in other studies. The mode coupling chatter stability prediction models are established based on eigenvalue method and zeroth order approximation, and the predicted stability boundaries are compared with the experimental results. The results show that only high-frequency chatter exists in the high speed region (1000-8000rpm), and no low frequency chatter occurs. The low-frequency chatter around the robot body mode is found in the low-speed region (400-1000rpm), but the mode coupling chatter theory could not explain the chatter varies periodically with the spindle speed. However, the stability boundary predicted by the regenerative chatter theory also changes periodically with the spindle speed. This indicates that the milling chatter dominant mechanism of high load robot is regenerative chatter. This study analyzes the milling chatter dominant mechanism of high-load robot through theoretical and experimental verification, which can provide theoretical support for high-load robot milling chatter control.

Simplified inverse distance weighting-immersed boundary method for simulation of fluid-structure interaction

When simulating fluid-structure interaction (FSI) problems involving moving objects, the implicit inverse distance weighting-immersed boundary method (IDW-IBM) developed by Du et al. [1] has to construct a large square correlation matrix and solve its inversion at each time step. In this work, a simplified inverse distance weighting-immersed boundary method (SIDW-IBM) is proposed to eliminate the intrinsic limitations in the original implicit IDW-IBM. Through error analysis using Taylor series expansion, a second order approximation can be derived, which allows us to approximate the large square correlation matrix into a diagonal matrix; thereby, we proposed the SIDW-IBM based on this second order approximation to explicitly evaluate the velocity corrections, where the needs to assemble the large correlation matrix and inverse it are circumvented. Owing to the fact that the inverse distance weighting interpolation removes the limitations in the Dirac delta function, the proposed SIDW-IBM has been successfully implemented on the non-uniform meshes to further improve the computational efficiency. The proposed SIDW-IBM is integrated with the reconstructed lattice Boltzmann flux solver (RLBFS) [2] to simulate some classic incompressible viscous flows, including flow past an in-line oscillating cylinder, flow past a heaving airfoil, and flow past a three-dimensional flexible plate. The good agreement between the present results and reference data demonstrates the capability and feasibility of the SIDW-IBM for simulating FSI problems with moving boundaries and large deformations.

Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation

Abstract In [L. Bourhrara, A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes, J. Comput. Phys. 397 2019, Article ID 108801], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed. One of its features is that a streamline weight is added to the test function to obtain the variational formulation. In the present paper, restricting our attention to the advective part of the Boltzmann equation, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space. The use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a Céa’s type lemma and standard interpolation estimates are sufficient for our analysis. For our numerical scheme, based on ℙ k {\mathbb{P}^{k}} discontinuous Galerkin finite elements (in space) on a mesh of size h and a spherical harmonics approximation of order N (in the angular variable), the convergence rate is of order 𝒪 ⁢ ( N - t + h k ) {\mathcal{O}(N^{-t}+h^{k})} for a smooth solution which admits partial derivatives of order k + 1 {k+1} and t with respect to the spatial and angular variables, respectively. For k = 0 {k=0} (piecewise constant finite elements) we also obtain a convergence result of order 𝒪 ⁢ ( N - t + h 1 2 ) {\mathcal{O}(N^{-t}+h^{\frac{1}{2}})} . Numerical experiments in one, two and three dimensions are provided, showing a better convergence behavior for the L 2 {L^{2}} -norm, typically of one more order, 𝒪 ⁢ ( N - t + h k + 1 ) {\mathcal{O}(N^{-t}+h^{k+1})} .

On Theoretical Aspect of Photons Emission From Quark-Gluon Interaction Upon Hard Collision

Abstract This work applies a quantum chromodynamics theory QCD to photon emission from heavy quark-gluon scattering in the strong interaction quark-gluon to study and calculate photon flow in hard interaction processes. The photon flow from the dense quark-gluon interaction was calculated using critical temperature, quantum chromodynamic coupling and fugacity at chemical potential μq (500 MeV) for the anti-top with gluon interaction in t ¯ g → s ¯ γ system. The photon flow spectrum in hard collisions is described under the first leading order approximation for QCD theory at finite chemical potential and critical temperature using various photon energy values dependent on the fugacity of quark and gluon. Photon flux calculation provided valuable insights into QGP conditions. The coupling strength αc (T) calculation is a strongly decreasing function of critical temperature ranging 107MeV ≤ Tc ≤ 145MeV and also find appreciable increasing as a result of increasing temperature of system ranging 180MeV ≤ T ≤ 360MeV. The photon flow produced in the t ¯ g → s ¯ γ system was observed in the low range of photon energy with two critical temperatures. The total photon emission strongly decreases marginally as a function of the increase in photon energy. The photon flow emission was found to have appreciable enhancement at the anti-top quark interaction with gluon and it increases highly as a result of the increased temperature of the system and reaches to maximum near 360 MeV at minimum strength coupling in the energy of the photon 1GeV≤E≤2 GeV.

Grazing-sliding bifurcation in a dry-friction oscillator on a moving belt under periodic excitation.

In this paper, we consider the grazing-sliding bifurcations in a dry-friction oscillator on a moving belt under periodic excitation. The system is a nonlinear piecewise smooth system defined in two zones whose analytical expressions of the solutions are not available. Thus, we obtain conditions of the existence of grazing-sliding orbits numerically by the shooting method. Then, we compute the lower and higher order approximations of the stroboscopic Poincaré map, respectively, near the grazing-sliding bifurcation point by the method of local zero-time discontinuity mapping. The results of computing the bifurcation diagrams obtained by the lower and higher order maps, respectively, are compared with those from direct simulations of the original system. We find that there are big differences between the lower order map and the original system, while the higher order map can effectively reduce such disagreements. By using the higher order map and numerical simulations, we find that the system undergoes very complicated dynamical behaviors near the grazing-sliding bifurcation point, such as period-adding cascades and chaos.

Найкраще наближення в інтегральній метриці лінійної комбінації періодичних функцій різної парності

In many areas of mathematics, extremal problems often arise related to approximation characteristics of both the approximating functions and the properties of the elements being approximated. For example, in the case of polynomials, the problem can involve the number of points where the values of the function coincide with the values of the polynomial used to replace the function on the studied interval. In practice, the problem of approximating a function from a given set R is reduced to replacing it, according to a defined algorithm, with a function (polynomial) of a fixed degree that is close to it in a certain sense. In uniform and integral metrics, the problem of finding the exact values of the best approximations for classes of r-times differentiable functions, where r is a natural number, has been explored in the works of J. Favard [1], N. I. Akhiezer, M. G. Krein [2], B. Nadi, S. M. Nikolsky [3], V. K. Dzyadyk [4], S. B. Stechkin, Sun Yun-shen, and others. The final results concerning the solution of the best approximation problem on Weyl-Nagy classes for arbitrary values of the parameters defining these classes belong to the Ukrainian mathematician V. K. Dzyadyk [4]. In his works on function classes generated by the well-known Bernoulli kernels, it was established that the number of coincidence points between the kernel and the approximating polynomial of degree n − 1, including their multiplicities, does not exceed 2n, which allowed for obtaining final results. The work [5] presents cases of such linear combinations of even or odd kernels for which the number of uniformly distributed interpolation points equals 2n + 2 for a polynomial of degree n − 1 that deviates the least in the metric of the L – space from the studied linear combination. The idea of studying composite kernels expressed as a linear combination of component terms belongs to O. I. Stepanyets [6], and it was implemented in problems of joint approximation of functions and their derivatives. In the 1980s and 1990s, O. I. Stepanets developed a new approach to classifying periodic functions, which allowed for a fine classification of extremely broad sets of periodic functions. The results obtained for these classes are, on one hand, of a general nature, and on the other, they provide a whole series of new, previously unknown results that could not be achieved with previously known classes. Following the approaches to function classification, we can consider a linear combination of function classes as a certain single class of a more complex nature. Then, the problem of finding the exact values of the upper bounds of the best approximations reduces to the problem of the best approximation for this composite class, which corresponds to convolutions with the generating composite kernel. In this work, we investigate linear combinations of three continuous 2π-periodic kernels of different parity, and it is established that there exist such composite kernels that their trigonometric polynomial of the best approximation of order n − 1 in the integral metric interpolates the kernel at 2n + 2 uniformly distributed points

Solving linear elasticity benchmark problems via the overset improved element-free Galerkin-finite element method

A novel approach for the solution of linear elasticity problems is introduced in this communication, which uses a hybrid chimera-type technique based on both finite element and improved element-free Galerkin methods. The proposed overset improved element-free Galerkin-finite element method (Ov-IEFG-FEM) for linear elasticity uses the finite element method (FEM) throughout the entire problem geometry, whereas a fine distribution of overlapping nodes is used to perform higher order approximations via the improved element-free Galerkin (IEFG) technique in regions demanding more computational accuracy. The method relies on keeping the FEM-based results in those regions where low order of approximation is enough to provide the required accuracy, i.e. outside the region where the solution will be enriched via the IEFG technique. The overlapping domains perform an iterative transfer of kinematics information through well-defined immersed boundaries, and a detailed explanation on this regard is also presented in this communication. The Ov-IEFG-FEM is used in a set of increasingly complex linear elasticity problems, and the outcomes demonstrate the suitability and reliability of this technique to solve such problems in an accurate and remarkably simple manner.

Circular phase shift migration based photoacoustic computational tomography

Abstract In the recent several decades, photoacoustic technique has proved to be capable of noninvasive biomedical imaging with scalable spatial resolution and high penetration depth. To achieve faithful image reconstruction, various method has been proposed, such as back projection, phase shift migration, model-based method, etc. These methods have simple implementations in homogeneous media, whereas in the cases of heterogeneous media (e.g., layered media), complicated modifications with respect to wave propagation at layer interface are always necessary. Here, we propose a frequency domain algorithm extension of phase shift migration in polar coordinates, which is obtained by making second order approximation to the optoacoustic point spread function in cylindrical coordinates and thereafter step by step wavenumber-frequency domain wave field extrapolation along the axial direction. Theoretical simulation and phantom experimental results demonstrate that the proposed circular phase shift migration method can well solve the layered heterogeneous reconstructive problem without modifying the wave propagation model. The acoustic diffraction limited acoustic resolution (up to 225 μm) and high imaging SNR (at least 16 dB) are obtained.

Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

We present a deterministic algorithm for the efficient evaluation of imaginary-time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary-time Green’s functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary-time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong-coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an Mth-order diagram at inverse temperature β and spectral width ωmax from O((βωmax)2M−1) for a direct quadrature to O(M(log(βωmax))M+1), with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multiband impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of Ca2RuO4, demonstrating the promise of the method for modeling realistic strongly correlated multiband materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust blackbox evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams. Published by the American Physical Society 2024

The pressure-wired Stokes element: a mesh-robust version of the Scott–Vogelius element

The Scott–Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order k and a discontinuous pressure approximation of order k-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k-1$$\\end{document}. It employs a “singular distance” (measured by some geometric mesh quantity Θz≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Theta \\left( \ extbf{z}\\right) \\ge 0$$\\end{document} for triangle vertices z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{z}$$\\end{document}) and imposes a local side condition on the pressure space associated to vertices z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{z}$$\\end{document} with Θz=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta \\left( \ extbf{z}\\right) =0$$\\end{document}. The method is inf-sup stable for any fixed regular triangulation and k≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 4$$\\end{document}. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices 0<Θz≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\Theta \\left( \ extbf{z}\\right) \\ll 1$$\\end{document}. In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the “singular distance”. We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence.

High-Order Approximation to Caputo Derivative on Graded Mesh and Time-Fractional Diffusion Equation for Nonsmooth Solutions

Abstract In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations (TFDEs) involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial interpolation to approximate the Caputo derivative on the nonuniform mesh. The truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as min{4−α,rα} and (4−α)/α, respectively, where α∈(0,1) is the order of fractional derivative and r≥1 is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on the graded temporal mesh is formulated. The scheme proves to be uniquely solvable for general r. Then, we derive the unconditional stability of the scheme on uniform mesh. The convergence of the scheme, in particular for r = 1, is analyzed for nonsmooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples.