Order Approximation Research Articles

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})} .

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.

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.

Elementary steps of catalytic reactions occurring on metallic alloy nanoparticles

Catalytic reactions occurring 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.

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.

Compositional Analysis of Ancient Glassware Based on CRITIC Weighting Method and Superior Order Approximation Model

Compositional Analysis of Ancient Glassware Based on CRITIC Weighting Method and Superior Order Approximation Model

Influence of initial correlations on evolution over time of an open quantum system

A novel approach to accounting for the influence of initial system–bath correlations on the dynamics of an open quantum system, based on the conventional projection operator technique, is suggested. To avoid the difficulties of treating the initial correlations, the conventional Nakajima–Zwanzig inhomogeneous generalized master equations (GMEs) for a system’s reduced statistical operator and correlation function are exactly converted into the homogeneous GMEs (HGMEs), which take into account the initial correlations in the kernel governing the evolution of these HGMEs. In the second order (Born) approximation in the system–bath interaction, the obtained HGMEs are local in time and valid at all timescales. They are further specialized for a realistic equilibrium Gibbs initial (at t=t0) system+bath state (for a system reduced statistical operator an external force at t>t0 is applied) and then for a bath of oscillators (Boson field). As an example, the evolution of a selected quantum oscillator (a localized mode) interacting with a Boson field (Fano-like model) is considered at different timescales. It is shown explicitly how the initial correlations influence the oscillator evolution process. In particular, it is shown that the equilibrium system’s correlation function acquires at the large timescale the additional constant phase factor conditioned by survived initial system–bath correlations.

Higher order approximation of option prices in Barndorff-Nielsen and Shephard models

We present an approximation method based on the mixing formula [Hull, J. and White, A., The pricing of options on assets with stochastic volatilities. J. Finance, 1987, 42, 281–300; Romano, M. and Touzi, N., Contingent claims and market completeness in a stochastic volatility model. Math. Finance, 1997, 7, 399–412] for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das and Langrené [Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility. Stochastics, 2022, 94, 745–788]. We obtain asymptotic results for the error of the N th order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein–Uhlenbeck process or a gamma Ornstein–Uhlenbeck process.

Reduced order model enhanced source iteration with synthetic acceleration for parametric radiative transfer equation

Applications such as uncertainty quantification, shape optimization, and optical tomography, require solving the radiative transfer equation (RTE) many times for various parameters. Efficient solvers for RTE are highly desired.Source Iteration with Synthetic Acceleration (SISA) is a popular and successful iterative solver for RTE. Synthetic Acceleration (SA) acts as a preconditioning step to accelerate the convergence of Source Iteration (SI). After each source iteration, classical SA strategies introduce a correction to the macroscopic particle density by solving a low order approximation to a kinetic correction equation. For example, Diffusion Synthetic Acceleration (DSA) uses the diffusion limit. However, these strategies may become less effective when the underlying low order approximations are not accurate enough. Furthermore, they do not exploit low rank structures concerning the parameters of parametric problems.To address these issues, we propose enhancing SISA with data-driven ROMs for the parametric problem and the corresponding kinetic correction equation. First, the ROM for the parametric problem can be utilized to obtain an improved initial guess. Second, the ROM for the kinetic correction equation can be utilized to design a low rank approximation to it. Unlike the diffusion limit, this ROM-based approximation builds on the kinetic description of the correction equation and leverages low rank structures concerning the parameters. We further introduce a novel SA strategy called ROMSAD. ROMSAD initially adopts our ROM-based approximation to exploit its greater efficiency in the early stage of SISA, and then automatically switches to DSA to leverage its robustness in the later stage. Additionally, we propose an approach to construct the ROM for the kinetic correction equation without directly solving it.Through a series of numerical tests, we compare the proposed methods with SI-DSA and DSA preconditioned Krylov solver. Particularly, for a multiscale parametric pin-cell problem, ROMSAD achieves approximately 10 times the acceleration compared to SI-DSA and 4 times acceleration compared to DSA preconditioned GMRES.

Improved modified estimator for estimation of median using auxiliary information under simple random sampling

This article aims to suggest an improved estimator for estimation of population median using auxiliary information under simple random sampling. The expression for the bias and mean square error are obtained up to first order approximation. We determine the MLE of the optimal values of the describing scalars. The proficiency of the suggested estimator is evaluated in comparison to the preliminary estimators using the MSE threshold. The suggested estimators are compared numerically to the ones that are currently studied in this study. The performance and novelty of the estimators was evaluated using real data sets and a simulation study. To check the efficiency of estimators empirical and theoretical study has been studied. Based on numerical result it is to be noted that our suggested estimator is more efficient as compared to existing estimators which is considered in this article in terms of least MSE and greater PRE.

Thermodynamic potential and drag force of the impurity-polariton many-body interaction

Investigating the many-body interaction among polaritons is a fundamental goal in comprehending nonlinear behaviors in microcavity semiconductor systems. Hence, the present study focuses on exploring the many-body interaction between excitons and impurities. It was observed that the energy eigenequation governing this system is of sixth-order, rendering it unsolvable. Consequently, a strategy involving the utilization of impurity weight was employed to partition the initial energy eigenequation into two fourth-order equations, a method deemed suitable for lower order approximations. Subsequently, Green’s function was applied to these two fourth-order equations. The analysis of the computed results, encompassing the energy spectrum, polariton density, and thermodynamic potential of the system, reveals that the many-body interaction between excitons and impurities induces the emergence of novel bound states of quasiparticles. Moreover, the inclusion of impurities leads to a decrease in both energy and density of polaritons. Furthermore, the investigation delves into the drag force exerted by impurities on excitons, demonstrating that the drag force escalates in a second-order fashion with impurity velocity and persists even as impurity velocity increases, surpassing the Landau’s equilibrium theory.

Magnetic corrections to the QCD coupling: Strong field approximation

We compute the one-loop vertex function of the QCD coupling in the presence of an ultraintense magnetic field. From the vertex function, we extract the effective coupling and show that it grows with increasing magnetic field. We consider the quark-gluon vertex and the three-gluon vertex, accounting for the propagators of charged particles within the loops using the lowest Landau level approximation in order to satisfy the condition where the magnetic field is the largest energy scale. Under this approximation, we find that the contribution from the three-gluon vertex vanishes. Therefore, this result arises from the competition between the color charge associated to gluons and to quarks as well, with the former being larger than the latter. The behavior of the QCD coupling as a function of the magnetic field strength is analogous to that exhibited by the light-quark condensate, indicating the magnetic catalysis occurs. This increasing behavior stems from the dominant contribution of color charge associated to gluons in the vertex function. Published by the American Physical Society 2024

Unsteady solute dispersion of electro-osmotic flow of micropolar fluid in a rectangular microchannel

This study scrutinizes the two-dimensional concentration distribution for a solute cloud containing a micropolar fluid in a rectangular microchannel under the influence of an applied electric field. The concentration distribution is obtained up to second order approximation using Mei's homogenization method. Analytical formulas are derived for dispersion coefficient, mean and two-dimensional concentration distributions. This study also includes the analytical expressions for electric potential, velocity, and microrotation profiles. This study discusses the impact of coupling number, couple stress parameter, electric double layer thickness, and Péclet number on solute concentration distribution. The results of fluid velocity and dispersion coefficient are validated with available works in the literature. The non-Newtonian parameter and electric double layer thickness are shown to have a significant impact on dispersion. Our study reveals that concentration distribution rises but spreading of solute reduces when the coupling number increases. This is also true when the Debye length decreases. It is also obtained that the solute spreads more in the Newtonian fluid case compared to the micropolar fluid case. Finally, coupling number and electric double layer thickness show a symmetric pattern to the indicator function for the transverse concentration variation rate. The findings of this work have broad implications in deoxyribonucleic acid analysis, chemical mixing, and separation.

Enhanced direct and synthetic estimators for domain mean with simulation and applications

This article considers the issue of domain mean estimation utilizing bivariate auxiliary information based enhanced direct and synthetic logarithmic type estimators under simple random sampling (SRS). The expressions of mean square error (MSE) of the proposed estimators are provided to the 1st order approximation. The efficiency criteria are derived to exhibit the dominance of the proposed estimators. To exemplify the theoretical results, a simulation study is conducted on a hypothetically drawn trivariate normal population from R programming language. Some applications of the suggested methods are also provided by analyzing the actual data from the municipalities of Sweden and acreage of paddy crop in the Mohanlal Ganj tehsil of the Indian state of Uttar Pradesh. The findings of the simulation and real data application exhibit that the proposed direct and synthetic logarithmic estimators dominate the conventional direct and synthetic mean, ratio, and logarithmic estimators in terms of least MSE and highest percent relative efficiency.

Алгоритм аппроксимации сигналов для классификации электроэнцефалограмм человека

The article examines the problem of classification of electroencephalograms (EEG), where noise in the signals, caused by various factors, prevents effective analysis and interpretation of the data. The main goal of the study is to analyze the effectiveness of a signal approximation algorithm using a wavelet technique with the next piece-wise approximation in order to effectively remove noise and subsequently solve the problem of signal classification using a convolutional neural network. The classification accuracy of the proposed algorithm with a low-pass filter is compared at different cutoff frequencies. One of the key findings of the study is a two-step approach to signal processing. In the first stage, the model is trained on the raw data, and then the trained convolution kernel with the highest variance is applied to the original signals. This solves the problem of choosing the mother function. This approach aims to enhance the informative components in the signal. The second processing step involves applying the proposed approximation algorithm after the convolution, which complements the first step to create a comprehensive method. This method not only effectively reduces noise in the data, but also has high potential to improve the overall accuracy in solving EEG signal classification problems. Thus, the results of the study provide important practical and theoretical implications, highlighting the prospects for applying the proposed method in the field of signal analysis.