Truncated Series Expansion Research Articles

An estimation approach for the influential–imitator diffusion

This paper presents a numerical estimation procedure for the influential–imitator diffusion, an extension to the Bass model in which a population is partitioned into two segments: influentials (who influence each other) and imitators (whose choices are affected by the ones of influentials). Focusing on the estimation of the model parameters, we propose a maximum likelihood approach and investigate its numerical solvability, building on an asymptotic approximation of the underlying differential equation. Specifically, we develop a truncated series expansion, exhibiting an increasing accuracy when the spontaneous innovation decreases. After uncovering the theoretical properties of the proposed methodology, we propose a specialized block coordinate descent method for the numerical maximization of the likelihood function. Empirical and computational tests are provided using the Michell and West dataset about the cannabis consumption of a cohort of students over their second, third and fourth year at a secondary school in Glasgow. The estimated imitation pattern confirms the well-known hypothesis on peer influences, where the choices of popular children represent the leading effects to determine the habits of others.

Improved convergence based on two-point Padé approximants: Simple shear, uniaxial elongation, and flow past a sphere

We investigate the performance of two-point Padé approximants on the truncated series solutions for the shear and elongational viscosities (obtained directly from the exact solutions or by asymptotic methods) for viscoelastic flows modeled with the Giesekus constitutive equation. Τhe major dimensionless groups are the Weissenberg number ( Wi), the Giesekus mobility parameter ( α), and the ratio of the polymer viscosity to the total viscosity of the fluid ( ηp) in terms of which the exact analytical solutions for the shear and elongational viscosities are found. The exact solutions are used to derive truncated series expansions in terms of the Weissenberg number at the Newtonian limit (zero Wi) and in terms of the inverse of the Weissenberg number at the purely elastic limit (infinite Wi). These series are then used to construct low- and high-order two-point diagonal Padé approximants. It is found that both the low- and high-order formulas predict the right trends over the entire range of Wi, from the Newtonian limit to the pure elastic limit. As expected, the high-order formula is much more accurate than the low-order formula. The results show that viscoelastic phenomena, such as shear thinning and extensional thickening, at any value of the Weissenberg number can be predicted analytically with acceptable accuracy using the proposed two-point Padé non-linear analysis. Finally, a fourth-order formula in terms of the Weissenberg number (or in terms of the Deborah number) for the drag force on a rigid sphere that translates with constant speed in a Giesekus fluid is also provided along with a discussion regarding the missing information from the literature, that is, required to perform the proposed non-linear analysis.

Solution of the population balance equation for wet granulation using second kind Chebyshev polynomials

Wet granulation is used in many industrial fields, such as pharmaceutical and nutraceutical, to improve compressibility properties of powders. The comprehension of the physical phenomena of the granulation process, such as simultaneous particle breakage, aggregation, and nucleation, is fundamental to optimize the operating conditions. In this work the population balance equation (PBE), used to mathematically describe these physical phenomena, has been numerical solved with a new finite element method with expansion coefficients in terms of a truncated series expansion with the orthogonal second kind Chebyshev basis polynomials. Such a numerical method is a straightforward and effective method, which has the advantage of concurrently giving the distribution and the different required moments. Moreover, the proposed model accounting for the breakage, aggregation, and nucleation phenomena was used in an ad-hoc optimization procedure to describe the particle size distribution of experimental results of a wet granulation process reaching satisfactorily RMS values.

On the effects of higher order stress terms in pure mode III loading of bi‐material notches

AbstractThe effects of higher order terms (HOTs) on the stress distribution at the vicinity of the sharp corner of two bonded dissimilar materials were investigated under pure out‐of‐plane (mode III) loading. To achieve this aim, finite element (FE) analysis in conjunction with the over‐deterministic technique were performed for two specimens, each containing four different notch opening angles. The coefficients of singular and non‐singular stress terms were then computed from the displacement field determined from the FE analysis. The elastic stresses obtained directly from the FE analysis were compared with those reconstructed from the truncated series expansions, in order to validate the accuracy of the calculated coefficients. The numerical results show that although the contribution of HOTs in the out‐of‐plane stress field is less than that of the in‐plane notch problems, ignoring their effects can lead to considerable errors in the stress field estimation. The magnitude of numerical error depends on the geometry and loading conditions of the bi‐material notch.

An Artificial Neural Network Approach for Solving Space Fractional Differential Equations

The linear algebraic system generated by the discretization of fractional differential equations has asymmetry, and the numerical solution of this kind of problems is more complex than that of symmetric problems due to the nonlocality of fractional order operators. In this paper, we propose the artificial neural network (ANN) algorithm to approximate the solutions of the fractional differential equations (FDEs). First, we apply truncated series expansion terms to replace unknown function in equations, then we use the neural network to get series coefficients, and the obtained series solution can make the norm value of loss function reach a satisfactory error. In the part of numerical experiments, the results verify that the proposed ANN algorithm can make the numerical results achieve high accuracy and good stability.

Outage Analysis for Tag Selection in Reciprocal Backscatter Communication Systems

Applying tag selection to backscatter networks can greatly improve the system outage performance. In this letter, the analytical expressions on outage probability for both ideal and outdated tag selection in reciprocal backscatter networks are derived with a truncated series expansion. Furthermore, the asymptotic analysis with large transmission power is conducted to give an insight into the effects of system parameters. From the asymptotic analysis, it is found that the diversity order of outage probability for ideal tag selection is equal to half of the number of tags. However, when applying outdated tag selection, the diversity order is equal to one, and thereby no diversity gain can be harvested, regardless of the temporal correlation and the number of tags.

Wavelet-based robust estimation and variable selection in nonparametric additive models

This article studies M-type estimators for fitting robust additive models in the presence of anomalous data. The components in the additive model are allowed to have different degrees of smoothness. We introduce a new class of wavelet-based robust M-type estimators for performing simultaneous additive component estimation and variable selection in such inhomogeneous additive models. Each additive component is approximated by a truncated series expansion of wavelet bases, making it feasible to apply the method to nonequispaced data and sample sizes that are not necessarily a power of 2. Sparsity of the additive components together with sparsity of the wavelet coefficients within each component (group), results into a bi-level group variable selection problem. In this framework, we discuss robust estimation and variable selection. A two-stage computational algorithm, consisting of a fast accelerated proximal gradient algorithm of coordinate descend type, and thresholding, is proposed. When using nonconvex redescending loss functions, and appropriate nonconvex penalty functions at the group level, we establish optimal convergence rates of the estimates. We prove variable selection consistency under a weak compatibility condition for sparse additive models. The theoretical results are complemented with some simulations and real data analysis, as well as a comparison to other existing methods.

Higher order stress terms in sharp notch problems under pure‐out‐of‐plane loading

AbstractCurrent paper studies the effects of higher order terms (HOTs) on the stress and displacement fields around sharp V‐notched components under pure antiplane shear (mode III) loading. For this aim, two V‐notched test samples each containing four different notch angles are simulated in a finite element (FE) commercial software. Subsequently, the over‐deterministic technique is used as an approach to calculate the coefficients of singular and nonsingular terms out of the nodal displacement values. To verify the accuracy of the values calculated as the coefficients, the stress field reconstructed from the truncated series expansions is compared with the direct nodal outputs of FE analysis. The results demonstrate that the use of nonsingular terms of the asymptotic series expansion is necessary only in the case that the distribution of stress at far distances from the notch tip is sought and the notch angle is small. To check the stability of the results, some of the parameters which affect accurate determination of unknown coefficients and convergence of the numerical results are studied.

Approximation of Probability Density Functions for PDEs with Random Parameters Using Truncated Series Expansions

The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two stochastic finite element methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the random PDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram-Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of PDEs with random parameters has not yet been explored.

Temporal large-eddy simulation based on direct deconvolution

In this work, we propose an approach for Temporal Large-Eddy Simulation (TLES) with direct deconvolution. In contrast to previous approaches such as the Temporal Approximate Deconvolution Model (TADM) by Pruett et al. [“A temporal approximate deconvolution model for large-eddy simulation,” Phys. Fluids 18, 028104 (2006)], the non-filtered velocity field is recovered using the differential form of the filter operation rather than from a truncated series expansion of the inverse filter operator. This direct deconvolution is used to obtain formal closure of an analytic evolution equation of the temporal residual-stress tensor. Thus, the Temporal Direct Deconvolution Model (TDDM) has advantages relative to the TADM in being both more accurate and requiring less computational effort. As for the TADM, a secondary regularization term based on selective frequency damping is employed. The TDDM was implemented in the spectral element code Nek5000 (Argonne National Laboratory, NEK5000 Version 17.0, 2019, https://nek5000.mcs.anl.gov) to simulate two canonical incompressible flows as three test cases: an a priori test case of Homogeneous Isotropic Turbulence (HIT) at Reλ = 50, a greatly coarsened a posteriori HIT case at Reλ = 190, and an a posteriori highly anisotropic turbulent channel flow at Reτ = 180. Analyses of the energy spectrum, the mean flow, the root-mean-square of the velocity fluctuations, and the Reynolds stresses are presented. The results demonstrate a significant improvement compared to no-model solutions regarding the mean flow in the turbulent channel and the energy spectrum in the HIT case, while the computational cost is reduced dramatically compared to the direct numerical simulation.

Improved Massive MIMO RZF Precoding Algorithm Based on Truncated Kapteyn Series Expansion

In order to reduce the computational complexity of the inverse matrix in the regularized zero-forcing (RZF) precoding algorithm, this paper expands and approximates the inverse matrix based on the truncated Kapteyn series expansion and the corresponding low-complexity RZF precoding algorithm is obtained. In addition, the expansion coefficients of the truncated Kapteyn series in our proposed algorithm are optimized, leading to further improvement of the convergence speed of the precoding algorithm under the premise of the same computational complexity as the traditional RZF precoding. Moreover, the computational complexity and the downlink channel performance in terms of the average achievable rate of the proposed RZF precoding algorithm and other RZF precoding algorithms with typical truncated series expansion approaches are analyzed, and further evaluated by numerical simulations in a large-scale single-cell multiple-input-multiple-output (MIMO) system. Simulation results show that the proposed improved RZF precoding algorithm based on the truncated Kapteyn series expansion performs better than other compared algorithms while keeping low computational complexity.

Boundary integral equation analysis for suspension of spheres in Stokes flow

We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the far-field, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magneto-hydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.

Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications

This work is concerned with the Fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions. For radially symmetric kernels, the nonlocal operators under consideration are diagonalizable in the Fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or Fourier symbols consisting of possibly singular and highly oscillatory integrals. For a large class of fractional power-like kernels, we propose a new approach based on reformulating the Fourier symbols both as coefficients of a series expansion and solutions of some simple ODE models. We then propose a hybrid algorithm that utilizes both truncated series expansions and high order Runge–Kutta ODE solvers to provide fast evaluation of Fourier symbols in both one and higher dimensional spaces. It is shown that this hybrid algorithm is robust, efficient and accurate. As applications, we combine this hybrid spectral discretization in the spatial variables and the fourth-order exponential time differencing Runge–Kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal Allen–Cahn equations, nonlocal Cahn–Hilliard equations, and nonlocal phase-field crystal models. Numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models.

Fast and Accurate Approximations for the Colebrook Equation

The standard expression for pipe friction calculations, the Colebrook equation, is in an implicit form. Here, we present two accurate approximate solutions, given by replacing the numerically unstable term in Keady's exact Lambert function solution with a truncated series expansion. The resulting expressions have a higher accuracy than most advanced approximations and a lower computational cost than basic engineering formulas. The simplest expression, given by retaining only three terms in the series expansion, has a maximum error of less than 0.153% for Re ≥ 4000. The slightly more involved expression, based on five terms, has a maximum error of 0.0061%.

Numerical modeling of soot formation in a turbulent C2H4/air diffusion flame

Soot formation in a lifted C 2 H 4 -Air turbulent diffusion flame is studied using two different paths for soot nucleation and oxidation; by a 2D axisymmetric RANS simulation using ANSYS FLUENT 15.0. The turbulence-chemistry interactions are modeled using two different approaches: steady laminar flamelet approach and flamelet-generated manifold. Chemical mechanism is represented by POLIMI to study the effect of species concentration on soot formation. P1 approximation is employed to approximate the radiative transfer equation into truncated series expansion in spherical harmonics while the weighted sum of gray gases is invoked to model the absorption coefficient while the soot model accounts for nucleation, coagulation, surface growth, and oxidation. The first route for nucleation considers acetylene concentration as a linear function of soot nucleation rate, whereas the second route considers two and three ring aromatic species as function of nucleation rate. Equilibrium-based and instantaneous approach has been used to estimate the OH concentration for soot oxidation. Lee and Fenimore-Jones soot oxidation models are studied to shed light on the effect of OH on soot oxidation. Moreover, the soot-radiation interactions are also included in terms of absorption coefficient of soot. Furthermore, the soot-turbulence interactions have been invoked using a temperature/mixture fraction-based single variable PDF. Both the turbulence-chemistry interaction models are able to accurately predict the flame liftoff height, and for accurate prediction of flame length, radiative heat loss should be accounted in an accurate way. The soot-turbulence interactions are found sensitive to the PDF used in present study.

An Abel transform for deriving line‐of‐sight wind profiles from LEO‐LEO infrared laser occultation measurements

AbstractWe have developed a formula for the retrieval of the line‐of‐sight (l.o.s.) wind speed from future low Earth orbit (LEO) satellite‐to‐satellite infrared laser occultation measurements. The formula involves an Abelian integral transform akin to the Abel transform widely used for deriving refractive index from bending angle in Global Navigation Satellite System radio occultation measurements. Besides the Abelian integral transform, the formula is derived from a truncated series expansion of the volume absorption coefficient as a function of frequency and includes a simple absorption‐line‐asymmetry correction term. A first‐order formulation (referred to as the standard formula) is complemented by higher‐order terms that can be used for high‐accuracy computations. Under the assumptions of spherical symmetry and perfect knowledge of spectroscopy, the residual l.o.s. wind error from using the standard formula rather than the high‐accuracy formula is assessed to be small compared to that anticipated from measurement errors in a real experiment. Applying the new formula just in standard form to future infrared laser transmission profiles would therefore enable the retrieval of l.o.s. stratospheric wind profiles with an accuracy limited mainly by measurement errors, residual spectroscopic errors, and deviations from spherical symmetry.

Closed-form representation of matrix functions in the formulation of nonlinear material models

The paper presents new approach to the evaluation of matrix functions operating over tensors that are essential part of formulation of complex nonlinear material models in mechanics of solids. A method is presented how to automatically derive numerically efficient closed-form representation of an arbitrary matrix function and its first and second derivatives for 3×3 matrices with real eigenvalues. The method offers an unique solution to the standard problem of ill-conditioning in the vicinity of multiple equal eigenvalues which is characteristic for all closed-form representations. A compiled library of subroutines with derived closed-form representation of most commonly used matrix functions along with their first and second derivatives has been created and is available for the use in general finite element environments. Consequently, the matrix functions can become as accurate, efficient and commonly available as their scalar counterparts, resulting in more common use of advanced strain and stress measures, such as Hencky strain measure which have so far been considered difficult for implementation. Accuracy and efficiency of the derived closed-form representations was compared with corresponding truncated series expansion and a speed up between 20 and 80 times has been observed depending on the matrix. The proposed methodology was tested on a set of selected nonlinear material models where matrix functions play an essential part in nonlinear finite element formulation.

Asymptotic properties of Lasso in high-dimensional partially linear models

We study the properties of the Lasso in the high-dimensional partially linear model where the number of variables in the linear part can be greater than the sample size. We use truncated series expansion based on polynomial splines to approximate the nonparametric component in this model. Under a sparsity assumption on the regression coefficients of the linear component and some regularity conditions, we derive the oracle inequalities for the prediction risk and the estimation error. We also provide sufficient conditions under which the Lasso estimator is selection consistent for the variables in the linear part of the model. In addition, we derive the rate of convergence of the estimator of the nonparametric function. We conduct simulation studies to evaluate the finite sample performance of variable selection and nonparametric function estimation.

Multivariate Canonical Polynomials in the Tau Method with Applications to Optimal Control Problems

The tau method is a highly accurate technique that approximates differential equations efficiently. It has three approaches: recursive, spectral and operational. Only the first two approaches concern this paper. In the recursive Tau method, the approximate solution of the differential equation is obtained in terms of a special polynomial basis called {\it canonical polynomials}. The present paper extends this concept to the {\it multivariate canonical polynomial vectors} and proposes a self starting algorithm to generate those vectors. In the spectral Tau method, the approximate solution is obtained as a truncated series expansions in terms of a set of orthogonal polynomials where the coefficients of the expansions are obtained by forcing the defect of the differential equation to vanish at the some selected points. In this paper we illustrated how the spectral tau can be used to solve a class of optimal control problem associated with a nonlinear system of differential equations. Some numerical examples that confirm our method are given.

The Barone-Adesi Whaley Formula to Price American Options Revisited

This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.