Global Polynomials Research Articles

Non-linear integral equations on unbounded domain with global polynomials

Solving Hammerstein - Fredholm integral equations on unbounded domains is challenging due to its domain of integration. Therefore, this paper discusses a numerical solution to the proposed problem using the Galerkin method, with low computational complexity and high convergence rates. In order to reduce the computational complexity, Laguerre polynomials are used as basis functions and to analyze the better convergence rates Kumar-Sloan (KS) approach has been applied. Further, the multi-Galerkin method with KS approach is used to improve the convergence results of Galerkin method. In addition, we have shown that the derivatives of the approximation solution converge to the corresponding derivatives of the exact solution with the same convergence rates as we attained for approximate solutions. Finally, numerical examples lend credibility to the presented theoretical framework.

Superconvergence of Legendre spectral projection methods for [formula omitted]th order integro-differential equations with weakly singular kernels

In this article, we apply Legendre spectral Galerkin, Legendre spectral multi-projection methods and their iterated versions to find the approximate solution of mth order Fredholm integro-differential equations with weakly singular kernel. Motivated by Mandal et al. (2023), we use Cauchy repeated integral theorem to transform the integro-differential equation to an single integral equation and obtain superconvergence results by iterated Legendre spectral Galerkin method, in spite of the singularity in the kernel function and unbounded differential operator. We have further improved the convergence rate of the approximate solution by using iterated Legendre spectral multi-projection method. Global Legendre polynomials are used as a basis for the approximating space, to reduce the computational cost of our proposed methods. In this article, we have derived theoretical error bounds and obtained the global convergence rates for all the discussed methods in both L2 and infinity norms. Numerical methods are implemented on examples to justify the theoretical results.

Space−time spectral approximations of a parabolic optimal control problem with an L2‐norm control constraint

AbstractThis article deals with the spectral approximation of an optimal control problem governed by a parabolic partial differential equation (PDE) with an ‐norm control constraint. The investigations employ the space−time spectral method, which is, more precisely, a dual Petrov‐Galerkin spectral method in time and a spectral method in space to discrete the continuous system. As a global method, it uses the global polynomials as the trial functions for discretization of PDEs. After obtaining the optimality condition of the continuous system and that of its spectral discrete surrogate, we establish a priori and a posteriori error estimates for the spectral approximation in detail. Three numerical examples in different spatial dimensions then confirm the theoretical results and also show the efficiency as well as a good precision of the adopted space−time spectral method.

Projection methods for approximate solution of a class of nonlinear Fredholm integro-differential equations

The aim of this article is to find the approximate solution of the nonlinear Fredholm integro-differential equations of second kind with smooth kernels with less computational complexity and investigate the asymptotic behavior of convergence of the approximate solutions by using global polynomials based projection methods. We develop the theoretical framework for the nonlinear Fredholm integro-differential equations to obtain the superconvergence results by Legendre polynomial based projection methods and their iterated versions. Numerical examples are considered to demonstrate the theoretical results.

Transferable species distribution modelling: Comparative performance of Generalised Functional Response models

Predictive species distribution models (SDMs) are becoming increasingly important in ecology, in the light of rapid environmental change. However, the predictions of most current SDMs are specific to the habitat composition of the environments in which they were fitted. This may limit SDM predictive power because species may respond differently to a given habitat depending on the availability of all habitats in their environment, a phenomenon known as a functional response in resource selection. The Generalised Functional Response (GFR) framework captures this dependence by formulating the SDM coefficients as functions of habitat availability. The original GFR implementation used global polynomial functions of habitat availability to describe the functional responses. In this study, we develop several refinements of this approach and compare their predictive performance using two simulated and two real datasets. We first use local radial basis functions (RBF), a more flexible approach than global polynomials, to represent the habitat selection coefficients, and balance bias with precision via regularization to prevent overfitting. Second, we use the RBF-GFR and GFR models in combination with the classification and regression tree CART, which has more flexibility and better predictive powers for non-linear modelling. As further extensions, we use random forests (RFs) and extreme gradient boosting (XGBoost), ensemble approaches that consistently lead to variance reduction in generalization error. We find that the different methods are ranked consistently across the datasets for out-of-data prediction. The traditional stationary approach to SDMs and the GFR model consistently perform at the bottom of the ranking (simple SDMs underfit, and polynomial GFRs overfit the data). The best methods in our list provide non-negligible improvements in predictive performance, in some cases taking the out-of-sample R2 from 0.3 up to 0.7 across datasets. At times of rapid environmental change and spatial non-stationarity ignoring the effects of functional responses on SDMs, results in two different types of prediction bias (under-prediction or mis-positioning of distribution hotspots). However, not all functional response models perform equally well. The more volatile polynomial GFR models can generate biases through over-prediction. Our results indicate that there are consistently robust GFR approaches that achieve impressive gains in transferability across very different datasets.

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.

A Lower Bound for Splines on Tetrahedral Vertex Stars

A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly difficult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra---we call these closed vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of $C^r$ splines on closed vertex stars of degree at least $3r+2$. We show that this formula is a lower bound on the dimension of $C^r$ splines of degree at least $(3r+2)/2$. Our proof uses apolarity and the so-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. We furthermore observe that arguments of Alfeld, Schumaker, and Whiteley imply that the only splines of degree at most $(3r+1)/2$ on a generic closed vertex star are global polynomials.

Energy management optimal strategy of FCHEV based on the Radau Pseudospectral method

Energy management of the fuel cell hybrid electric vehicles (FCHEVs) is vital with respect to improving FCHEV's performance and durability. Radau Pseudospectral method (RPM)-based optimal control of energy management of FCHEV is proposed to optimise the fuel cell's lifetime by means of reducing its performance degradation. To utilise the RPM, both state and control variables are approximated by global interpolation polynomials, and differential equations of state variables are approximated by the derivatives of interpolation polynomials. Accordingly, the optimal control problem is transformed into nonlinear problem to be solved. The fuel cell's performance degradation is selected as objective function. The optimal results in NEDC show that battery with larger capacity is more beneficial than smaller one for reducing the fuel cell's performance degradation, with the total time of large load change of the fuel cell reducing. The RPM is also an effective way to optimise other objectives to energy management.

Energy management optimal strategy of FCHEV based on the Radau Pseudospectral method

Energy management of the fuel cell hybrid electric vehicles (FCHEVs) is vital with respect to improving FCHEV's performance and durability. Radau Pseudospectral method (RPM)-based optimal control of energy management of FCHEV is proposed to optimise the fuel cell's lifetime by means of reducing its performance degradation. To utilise the RPM, both state and control variables are approximated by global interpolation polynomials, and differential equations of state variables are approximated by the derivatives of interpolation polynomials. Accordingly, the optimal control problem is transformed into nonlinear problem to be solved. The fuel cell's performance degradation is selected as objective function. The optimal results in NEDC show that battery with larger capacity is more beneficial than smaller one for reducing the fuel cell's performance degradation, with the total time of large load change of the fuel cell reducing. The RPM is also an effective way to optimise other objectives to energy management.

Convergence rate for a Radau hp collocation method applied to constrained optimal control

For unconstrained control problems, a local convergence rate is established for an $hp$-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the $hp$-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.

Why High-Order Polynomials Should Not Be Used in Regression Discontinuity Designs

It is common in regression discontinuity analysis to control for third, fourth, or higher-degree polynomials of the forcing variable. There appears to be a perception that such methods are theoretically justified, even though they can lead to evidently nonsensical results. We argue that controlling for global high-order polynomials in regression discontinuity analysis is a flawed approach with three major problems: it leads to noisy estimates, sensitivity to the degree of the polynomial, and poor coverage of confidence intervals. We recommend researchers instead use estimators based on local linear or quadratic polynomials or other smooth functions.

Frequency scaling of slant‐path atmospheric attenuation in the absence of rain for millimeter‐wave links

AbstractBroadband satellite communications systems, either used for broadcast or fixed satellite services, have grown continuously in recent years. This has led to the use of higher frequency bands, from the Ku (14/11 GHz) to the Ka band (30/20 GHz) in the last decade, and with the expectation of using the Q/V band (50/40 GHz) and even the W band (75–110 GHz) in the future. As frequency increases, radio wave propagation effects in the slant‐path within the troposphere are becoming more and more relevant. The objective of this research is the proposal of frequency scaling approximations for the total attenuation in the absence of rain, a condition that occurs during the highest percentages of time, usually more than 95% in temperate climates. There is a strong relationship between total attenuation at different frequencies, as it arises from the same physical phenomena, namely, the presence of oxygen, water vapor, and clouds in the slant path. This strong relationship allows frequency scaling estimations to be proposed. In particular, polynomials for instantaneous frequency scaling of total attenuation under these conditions have been calculated for a set of frequencies in the range 10–100 GHz, based on atmospheric profiles of 60 sites from all over the world and physical models of attenuation. Global polynomials are provided for the 72 combinations of nine significant frequencies, which can be used to estimate attenuation at a frequency band from its known value at a different one. Refined expressions have also been calculated for different climatic zones, providing more precise estimations.

Convergence Rate for a Gauss Collocation Method Applied to Unconstrained Optimal Control

A local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as the number of collocation points increases, the discrete solution convergences exponentially fast in the sup-norm to the continuous solution. This is the first convergence rate result for an orthogonal collocation method based on global polynomials applied to an optimal control problem.

Weakly periodic boundary conditions for the homogenization of flow in porous media

Abstract Background Seepage in porous media is modeled as a Stokes flow in an open pore system contained in a rigid, impermeable and spatially periodic matrix. By homogenization, the problem is turned into a two-scale problem consisting of a Darcy type problem on the macroscale and a Stokes flow on the subscale. Methods The pertinent equations are derived by minimization of a potential and in order to satisfy the Variationally Consistent Macrohomogeneity Condition, Lagrange multipliers are used to impose periodicity on the subscale RVE. Special attention is given to the bounds produced by confining the solutions spaces of the subscale problem. Results In the numerical section, we choose to discretize the Lagrange multipliers as global polynomials along the boundary of the computational domain and investigate how the order of the polynomial influence the permeability of the RVE. Furthermore, we investigate how the size of the RVE affect its permeability for two types of domains. Conclusions The permeability of the RVE depends highly on the discretization of the Lagrange multipliers. However, the flow quickly converges towards strong periodicity as the multipliers are refined.

Symmetric global partition polynomials for reproducing kernel elements

The reproducing kernel element method is a numerical technique that combines finite element and meshless methods to construct shape functions of arbitrary order and continuity, yet retains the Kronecker- $$\delta $$ ? property. Central to constructing these shape functions is the construction of global partition polynomials on an element. This paper shows that asymmetric interpolations may arise due to such things as changes in the local to global node numbering and that may adversely affect the interpolation capability of the method. This issue arises due to the use of incomplete polynomials that are subsequently non-affine invariant. This paper lays out the new framework for generating general, symmetric, truly minimal and complete affine invariant global partition polynomials for triangular and tetrahedral elements. Optimal convergence rates were observed in the solution of Kirchhoff plate problems with rectangular domains.

Global Piecewise Fitting Method for Modal Parameter Identification by Orthogonal Polynomials Based on Dynamic Systems

Global estimation of modal parameters for dynamic Systems using fraction orthogonal polynomials possesses higher precision than the general rational fraction orthogonal polynomials algorithm. The object of this paper is to introduce a new technique to derive the global piecewise fitting method for modal parameter identification of dynamic Systems. Based on the global fraction orthogonal polynomials algorithm, a global piecewise fitting method for eliminating affections of modes outside of fitting bands is proposed. Both lower and higher modes outside of the fitting band are analyzed and processed. The frequency response data are revised by means of modes in two frequency bands close to the fitting band, and a curve fitting model is derived. The proposed method is compared with that of the traditional estimation without regard to the affections of modes outside of the fitting band. For this comparative study, simulated data are used. Simulation research indicates that the processed approach is effective.

Operational Modal Test of an Arch Bridge

A modal test was carried out on a 120m span concrete filled steel tubular arch bridge under operational conditions. The moving traffic load is considered as uncorrelated white noises at various points of the road surface and the test is of the type of multi-input multi-output (MIMO). The modal parameters are identified by Global Rational Fraction Polynomials (GRFP) method, a frequency domain method. In order to taking the mobile nature of the traffic load into consideration, a multi-input NExT method is utilized to retrieve the frequency response function needed by GRFP. A proof of the NExT method in the case of multi-input case is also provided in this paper.

Operational Modal Test of an Arch Bridge

A modal test was carried out on a 120m span concrete filled steel tubular arch bridge under operational conditions. The moving traffic load is considered as uncorrelated white noises at various points of the road surface and the test is of the type of multi-input multi-output (MIMO). The modal parameters are identified by Global Rational Fraction Polynomials (GRFP) method, a frequency domain method. In order to taking the mobile nature of the traffic load into consideration, a multi-input NExT method is utilized to retrieve the frequency response function needed by GRFP. A proof of the NExT method in the case of multi-input case is also provided in this paper.

Generalized global conjugate gradient squared algorithm

In this paper, a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right-hand sides is presented. The method can be derived by using products of two nearby global BiCG polynomials and formal orthogonal polynomials, of which global CGS and global BiCGSTAB are just particular cases. We also show to apply the method for solving the Sylvester matrix equation. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.