Order Of Approximation Research Articles

About the Tikhonov Regularization Method for the Solution of Incorrect Problems

From time to time, papers are published containing gross errors when solving integral equations of the first kind. This paper is devoted to the analysis of these errors. The paper considers Tikhonov’s weak and operator regularization. To construct a solution to the integral equation, the local splines of the Lagrangian type of the second order of approximation, as well as the local splines of the Hermitian type of the fourth order of approximation of the first level, are used. The results of numerical experiments are presented.

Application of Splines of the Seventh Order Approximation to the Solution of the Fredholm Integral Equations with Weekly Singularity

We consider the construction of a numerical solution to the Fredholm integral equation of the second kind with weekly singularity using polynomial spline approximations of the seventh order of approximation. The support of the basis spline of the seventh order of approximation occupies seven grid intervals. In the beginning, in the middle, and at the end of the integration interval, we apply various modifications of the basis splines of the seventh order of approximation. We use the Gaussian-type quadrature formulas to calculate the integrals with a weakly singularity. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the required error of approximation. Numerical examples of the application of spline approximations of the seventh order to solve integral equations are given.

A kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator with higher approximation order

In this paper, a kind of bivariate Bernoulli-type multiquadric quasi-interpolation operator is studied by combining the known multiquadric quasi-interpolation operator with the generalized Taylor polynomial as the expansion in the bivariate Bernoulli polynomials. Some error bounds and convergence rates of the combined operators are studied. A selection of numerical examples is presented to compare the performances of the obtained scheme. Furthermore, our method can be applied to time-dependent differential equations. Its advantage is that the algorithm is very simple and easy to implement.

A Special Relativistic Exploitation of the Second Law of Thermodynamics and Its Non-Relativistic Limit.

A thermodynamic process is a solution of the balance equations fulfilling the second law of thermodynamics. This implies restrictions on the constitutive relations. The most general way to exploit these restrictions is the method introduced by Liu. This method is applied here, in contrast to most of the literature on relativistic thermodynamic constitutive theory, which goes back to a relativistic extension of the Thermodynamics of Irreversible Processes. In the present work, the balance equations and the entropy inequality are formulated in the special relativistic four-dimensional form for an observer with four-velocity parallel to the particle current. The restrictions on constitutive functions are exploited in the relativistic formulation. The domain of the constitutive functions, the state space, is chosen to include the particle number density, the internal energy density, the space derivatives of these quantities, and the space derivative of the material velocity for a chosen observer. The resulting restrictions on constitutive functions, as well as the resulting entropy production are investigated in the non-relativistic limit, and relativistic correction terms of the lowest order are derived. The restrictions on constitutive functions and the entropy production in the low energy limit are compared to the results of an exploitation of the non-relativistic balance equations and entropy inequality. In the next order of approximation our results are compared to the Thermodynamics of Irreversible Processes.

Covariance properties under natural image transformations for the generalised Gaussian derivative model for visual receptive fields.

The property of covariance, also referred to as equivariance, means that an image operator is well-behaved under image transformations, in the sense that the result of applying the image operator to a transformed input image gives essentially a similar result as applying the same image transformation to the output of applying the image operator to the original image. This paper presents a theory of geometric covariance properties in vision, developed for a generalised Gaussian derivative model of receptive fields in the primary visual cortex and the lateral geniculate nucleus, which, in turn, enable geometric invariance properties at higher levels in the visual hierarchy. It is shown how the studied generalised Gaussian derivative model for visual receptive fields obeys true covariance properties under spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations. These covariance properties imply that a vision system, based on image and video measurements in terms of the receptive fields according to the generalised Gaussian derivative model, can, to first order of approximation, handle the image and video deformations between multiple views of objects delimited by smooth surfaces, as well as between multiple views of spatio-temporal events, under varying relative motions between the objects and events in the world and the observer. We conclude by describing implications of the presented theory for biological vision, regarding connections between the variabilities of the shapes of biological visual receptive fields and the variabilities of spatial and spatio-temporal image structures under natural image transformations. Specifically, we formulate experimentally testable biological hypotheses as well as needs for measuring population statistics of receptive field characteristics, originating from predictions from the presented theory, concerning the extent to which the shapes of the biological receptive fields in the primary visual cortex span the variabilities of spatial and spatio-temporal image structures induced by natural image transformations, based on geometric covariance properties.

Yavno-neyavnye skhemy rascheta dinamiki uprugovyazkoplasticheskikh sred s malym vremenem relaksatsii

We consider the dynamic behavior of elastoviscoplastic media under the action of an external load. For the general case of a nonlinear viscosity function describing high-speed hardening, we construct an explicit–implicit calculation scheme of the second order of approximation that permits one to obtain a numerical solution of the original semilinear hyperbolic problem. A distinctive feature of this approach is that it does not use the method of splitting by physical processes. Despite this, an explicit computational algorithm was obtained that allows efficient implementation on modern computing systems.

Enforcement of interface laws for dissipative vibro-acoustic problems within high-order X-FEM

This work elaborates on the numerical solution to cope with the enforcement of a general robin-type weak/strong discontinuity interface in acoustic problems, where energy dissipative media with complex-valued properties are involved. We propose a discrete variational formulation based on the idea of Nitsche’s method in the eXtended Finite Element Method (X-FEM) framework. The formulation is stable in the sense that it is able to treat any interface parameters accurately within any mesh configuration and approximation order. We prove the stability of the method, fully considering the effects of energy dissipation, and provide a robust strategy involving basis conversion to obtain the stability parameter. Numerical examples demonstrate that the proposed discrete formulation gives an optimal solution in terms of accuracy and conditioning, compared to a penalty-like formulation. For all interface types, optimal rates of convergence are achieved on both primal and dual variables. Our method exhibits high robustness and stability regardless of discretization, geometries, materials, excitation frequency and interface parameters.

Unified plate finite elements for the large strain analysis of hyperelastic material structures

This paper proposes a high-order two-dimensional (2D) finite element model for the analysis of isotropic, nearly incompressible hyperelastic material structures based on a decoupled Neo–Hookean strain energy function. The model is based on the Carrera Unified Formulation (CUF) , which allows to automatically implement different kinematics by using an opportune recursive notation. The principle of virtual work and a finite element approximation are exploited to obtain the nonlinear governing equations. Considering the three-dimensional full Green–Lagrange strain components and given the material Jacobian tensor, the explicit forms of tangent stiffness matrices of unified plate elements are presented in terms of the fundamental nuclei, which are independent of the theory approximation order. Several problems of soft material plates under uniform pressure are investigated, including a silicone rubber clamped plate and a simply supported plate made of biological material. The proposed model is compared with literature results including those coming from experiments and numerical solutions. The numerical investigation demonstrated the validity and accuracy of the proposed methodology for the analysis of hyperelastic plates.

Efficient representation of head-related transfer functions in continuous space–frequency domains

Utilizing spherical harmonic (SH) domain has been established as the default method of obtaining continuity over space in head-related transfer functions (HRTFs). This paper concerns different variants of extending this solution by replacing SHs with four-dimensional (4D) continuous functional models in which frequency is imagined as another physical dimension. Recently developed hyperspherical harmonic (HSH) representation is compared with models defined in spherindrical coordinate system by merging SHs with one-dimensional basis functions. The efficiency of both approaches is evaluated based on the reproduction errors for individual HRTFs from HUTUBS database, including detailed analysis of its dependency on chosen orders of approximation in frequency and space. Employing continuous functional models defined in 4D coordinate systems allows HRTF magnitude spectra to be expressed as a small set of coefficients which can be decoded back into values at any direction and frequency. The best performance was noted for HSHs and SHs merged with reverse Fourier–Bessel series, with the former featuring better compression abilities, achieving slightly higher accuracy for low number of coefficients. The presented models can serve multiple purposes, such as interpolation, compression or parametrization for machine learning applications, and can be applied not only to HRTFs but also to other types of directivity functions, e.g. sound source directivity.

Shift-invariant spline subspaces of Sobolev spaces and their order of approximation

Closed shift-invariant subspaces of Hilbert space are the core idea of important classes of signal and image analysis methods such as wavelets, shearlets, radial basis functions and many more. In 1994 de Boor, DeVore and Ron [4] gave a full characterization of the order of approximation in such shift-invariant subspaces of L2(Rn), n∈N. Ron and Holtz extended this work in [10] to Sobolev norms, and gave specific criteria on the generating functions to guarantee certain orders of approximation. However, not for all generating functions these criteria are easy to verify. We here give new criteria which are designed for generating functions given explicitly in Fourier domain. We show by various examples on spline functions that in these cases our new criteria give better estimates on the order of approximation.

Quantum-gravitational corrections to the power spectrum for a closed universe

We study the quantum-gravitational corrections to the power spectrum of a gauge-invariant inflationary scalar perturbations in a closed model of a universe. We consider canonical quantum gravity as an approach to quantizing gravity. This leads to the Wheeler-DeWitt equation, which has been studied by applying a semiclassical Born–Oppenheimer type of approximation. At the corresponding orders of approximation, we recover both the uncorrected and quantum-gravitationally corrected Schrödinger equations for the perturbation modes from which we calculate the quantum-gravitational corrections to the power spectrum in the slow-roll regime. The results are compared to the power spectra for the flat model of the universe.

A simulation study: An enhanced generalized class of estimators for estimation of population proportion using twofold auxiliary attribute

The article introduces a novel class of estimators designed for estimating finite population proportions. These estimators utilize dual auxiliary attributes and are applicable under simple random sampling. The proposed class of estimators includes various members with distinct characteristics. The article provides numerical terminologies for the bias and MSE of the estimators, acquire up to first order of approximation. Four actual data sets are used. Additionally, a simulation study is accompanied to perceive the presentations of estimators. The MSE criterion is used to assess how well the proposed estimator performed as likened to the preliminary estimators. The simulation analysis revealed that, in contrast to other examined estimators, the suggested class of estimators provided better results. The empirical investigation offers evidence to substantiate the findings of the argument. Theoretical research also displays that the suggested class of estimators outperforms its competitors.

Spherical Harmonics for the 1D Radiative Transfer Equation. I. Reflected Light

A significant challenge in radiative transfer theory for atmospheres of exoplanets and brown dwarfs is the derivation of computationally efficient methods that have adequate fidelity to more precise, numerically demanding solutions. In this work, we extend the capability of the first open-source radiative transfer model for computing the reflected light of exoplanets at any phase geometry, PICASO (Planetary Intensity Code for Atmospheric Spectroscopy Observations). Until now, PICASO has implemented two-stream approaches to the solving the radiative transfer equation for reflected light, in particular following the derivations of Toon et al. In order to improve the model accuracy, we have considered higher-order approximations of the phase functions; namely, we have increased the order of approximation from two to four, using spherical harmonics. The spherical harmonics approximation decouples spatial and directional dependencies by expanding the intensity and phase function into a series of spherical harmonics, or Legendre polynomials, allowing for analytical solutions for low-order approximations to optimize computational efficiency. We rigorously derive the spherical harmonics method for reflected light and benchmark the four-term method (SH4) against Toon et al. and two independent and higher-fidelity methods (CDISORT and doubling method). On average, the SH4 method provides an order-of-magnitude increase in accuracy, compared to Toon et al. Finally, we implement SH4 within PICASO and observe only a modest increase in computational time, compared to two-stream methods (20% increase).

Numerical and series solutions for Von-Kármán flow of viscoelastic fluid inspired by viscous dissipation and Joule heating effects

Present article aims at developing similarity solutions for heat transportation in swirling flow of viscoelastic fluid (obeying Jeffrey model) generated by an infinite porous rotating surface. The resulting thermal energy equation comprising frictional heating and Ohmic dissipation terms is mainly focused. Relevant equations simplified under boundary layer approximations have been solved analytically by a powerful homotopy based analytical scheme, while numerical solutions are constructed by a reliable MATLAB solver bvp4c. For homotopy based series solutions, total squared residual of the system is evaluated, and it is found to decline for increasing order of approximations. Both applied methods are shown to be in compliance with each other for wide range of viscoelastic fluid parameters. Computed solutions are utilized to understand the combined influence of viscoelasticity and viscous dissipation on heat transport phenomena associated with the Von-Kármán configuration. Subtle fluid dynamics entities such as resisting torque and Nusselt number are computed and elaborated. Results predict that the torque required by the disk drastically reduces when viscoelastic fluid assumption is applied. However, cooling rate of the disk in Newtonian fluid is better than that in viscoelastic fluid. Present results are found in agreement with that of the previous studies in a limiting situation.

Finite temperature considerations in the structure of quadratic GUP-modified white dwarfs

In quantum gravity phenomenology, the effect of the generalized uncertainty principle (GUP) on white dwarf structure has been given much attention in recent literature. However, these studies assume a zero temperature equation of state (EoS), excluding young white dwarfs whose initial temperatures are substantially high. To that cause, this paper calculates the Chandrasekhar EoS and resulting mass-radius relations of finite temperature white dwarfs modified by the quadratic GUP, an approach that extends Heisenberg’s uncertainty principle by a quadratic term in momenta. The EoS was first approximated by treating the quadratic GUP parameter as perturbative, causing the EoS to exhibit expected thermal deviations at low pressures, and conflicting behaviors at high pressures, depending on the order of approximation. We then proceeded with a full numerical simulation of the modified EoS, and showed that in general, finite temperatures cause the EoS at low pressures to soften, while the quadratic GUP stiffens the EoS at high pressures. This modified EoS was then applied to the Tolman–Oppenheimer–Volkoff equations and its classical approximation to obtain the modified mass-radius relations for general relativistic and Newtonian white dwarfs. The relations for both cases were found to exhibit the expected thermal deviations at small masses, where low-mass white dwarfs are shifted to the high-mass regime at large radii, while high-mass white dwarfs acquire larger masses, beyond the Chandrasekhar limit. Additionally, we find that for sufficiently large values of the GUP parameter and temperature, we obtain mass-radius relations that are completely removed from the ideal case, as high-mass deviations due to GUP and low-mass deviations due to temperature are no longer mutually exclusive.

The Application of Splines of the Seventh Order Approximation to the Solution of Integral Fredholm Equations

There are various numerical methods for solving integral equations. Among the new numerical methods, methods based on splines and spline wavelets should be noted. Local interpolation splines of a low order of approximation have proved themselves well in solving differential and integral equations. In this paper, we consider the construction of a numerical solution to the Fredholm integral equation of the second kind using spline approximations of the seventh order of approximation. The support of the basis spline of the seventh order of approximation occupies seven grid intervals. We apply various modifications of the basis splines of the seventh order of approximation at the beginning, the middle, and at the end of the integration interval. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the required error of approximation. Numerical examples of the application of spline approximations of the seventh order for solving integral equations are given.

An Extended Range of Stable Flux Reconstruction Schemes on Quadrilaterals for Various Polynomial Bases

An extended range of energy stable flux reconstruction schemes, developed using a summation-by-parts approach, is presented on quadrilateral elements for various sets of polynomial bases. For the maximal order bases, a new set of correction functions which result in stable schemes is found. However, for a range of orders it is shown that only a single correction function can be cast as a tensor-product. Subsequently, correction functions are identified using a generalised analytic framework that results in stable schemes for total order and approximate Euclidean order polynomial bases on quadrilaterals—which have not previously been explored in the context of flux reconstruction. It is shown that the approximate Euclidean order basis can provide similar numerical accuracy as the maximal order basis but with fewer points per element, and thus lower cost.

Conformal renormalization of scalar-tensor theories

We study a conformally coupled scalar-tensor theory with a quartic potential possessing local conformal symmetry up to a boundary term. We show that requiring the restoration of the full local conformal symmetry fixes the counterterms that render the on-shell action finite. The building block of the resulting action is a conformally covariant tensor which is constructed out of the metric and the scalar field and it has the same conformal weight as the Weyl tensor. This allows us to obtain the counterterms for the scalar-tensor sector in a closed form. The finiteness of the conformally complete version of the action is suggestive on the validity of the conformal renormalization prescription. We extend this theory by adding the conformal gravity action and also the Einstein-AdS action written in MacDowell-Mansouri form. Even though the latter breaks the conformal symmetry, we find that the action is still renormalized provided a suitable falloff of the scalar field when considering asymptotically locally anti-de Sitter solutions. Black hole solutions in these theories are studied, for which the Hawking temperature and the partition function to first order in the saddle-point approximation are calculated, providing a concrete example of this renormalization scheme.

An efficient high-order asymptotic approach for the elastodynamic homogenization of periodic composites

An efficient asymptotic approach, referred to as the Effective Non-Local Asymptotic Model (ENLAM), is developed for the elastodynamic homogenization of periodic composites. In comparison with the relevant two-scale asymptotic method reported in the literature, the effective motion equation obtained by ENLAM has a very compact form which holds for any order of asymptotic approximation. This compact form makes direct use of the effective displacement vector and of its different gradients whose highest order depends on the accuracy degree required. For every wavenumber belonging to the first Brillouin zone, the Floquet-Bloch wave expansion is employed, and compact recursive formulae are derived for determining the effective material parameters of periodic composites. For simplicity, all the results are presented only for a shear wave propagating in a periodic composite. Numerical examples for discussing the effective material parameters, the dispersion relation and the asymptotic approximation error are provided when a periodic composite with inclusions is concerned.

Stochastic bifurcations induced by Lévy noise in a fractional trirhythmic van der Pol system

Trirhythmical nature has aroused considerable interest in describing dynamic behaviors of a fractional self-sustained system. In this paper, the fractional trirhythmic system is considered and a bifurcation analysis in a stochastic fractional trirhythmic self sustained system subjected to Lévy noise perturbation is presented. The fractional electronic circuit has been used to model the system and the oscillations are described by a nonlinear fractional differential equation to show a new bifurcation parameter. Then, based on the minimum mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of the damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Using the predictor–corrector simulation method, it is shown a good agreement between the fractional and the equivalent integer order system. The detailed parameter space study reveals that the multi-rhythmic properties of the oscillation of this system can be efficiently controlled by fractional order parameter and Lévy noise parameters. Finally, the analytical solutions are validated by numerical results of the Monte Carlo simulation of the original fractional modified van der Pol oscillator. The trirhythmic region can be detected in the parameter space (α,β,γ,δ) by choosing an approximate fractional order. The stability index and the noise intensity can in any case govern the dynamics of the self sustained system, but regulating the skewness parameter cannot bring about transitions between unimodal, bimodal and trimodal in this multi-rhythmic system. More bifurcations appear by adjusting the fractional order in this system. These results may be conducive to further exploring bifurcations in the real-world applications. This study sheds new insight lights on the understanding nontrivial effects of fractional derivative on a trirhythmic self sustained system.