High-degree Polynomial Research Articles

Polynomial Approximation over Arbitrary Shape Domains

In spectral/finite element methods, a robust and stable high-order polynomial approximation method for the solution can significantly reduce the required number of degrees of freedom (DOFs) to achieve a certain level of accuracy. In this work, a closed-form relation is proposed to approximate the Fekete points (AFPs) on arbitrary shape domains based on the singular value decomposition (SVD) of the Vandermonde matrix. In addition, a novel method is derived to compute the moments on highly complex domains, which may include discontinuities. Then, AFPs are used to generate compatible basis functions using SVD. Equations are derived and presented to determine orthogonal/orthonormal modal basis functions, as well as the Lagrange basis. Furthermore, theorems are proved to show the convergence and accuracy of the proposed method, together with an explicit form of the Weierstrass theorem for polynomial approximation. The method was implemented and some classical cases were analyzed. The results show the superior performance of the proposed method in terms of convergence and accuracy using many fewer DOFs and, thus, a much lower computational cost. It was shown that the orthogonal modal basis is the best choice to decrease the DOFs while maintaining a small Lebesgue constant when very high degree of polynomial is employed.

Statistical modelling of growth curve for longitudinal data on a feeding trial in goat breeds

The growth curve parameters were estimated extending a linear regression to higher degree polynomial with attempts to use also the biologically appealing Gombertz function. The data contained monthly weight records from weaning to the age of 14 months of kids in two Omani goat breeds Sahrawi and Jabbali used in a feeding trial assessing the effect of concentrate supplement (14% crude protein). The growth curve parameters were estimated within the fixed effects breed and feed concentrate level (2 or 3% of live body weight) with an extension to mixed models using animals as random effects. The parameters and model contrasting were performed with ML or REML as appropriate with relative comparison relying on AIC and significance testing of estimated parameters. The mixed model analyses were performed with publicly available R software package programs. The fixed effect cubic regression with linear random effect model fitted into the data. The higher level of concentrates improved the immediate post-weaning growth in the same way in the breeds while the subsequent growth curve differed with more pronounced weight increments in the Jabbali breed.

A new hybrid trigonometric WENO scheme for hyperbolic conservation laws and highly oscillatory problems

In the conventional hybrid WENO scheme, the computation of extremum points of high-order polynomials is required, which poses challenges when extending this methodology to the hybrid trigonometric WENO (TWENO) scheme due to the difficulties in determining the extremum points of high-order trigonometric polynomials. In this paper, we propose a novel hybrid strategy that circumvents the necessity of finding extremum points of high-degree polynomials, requiring only the extremum points of three lower-order polynomials. Based on this hybrid strategy, two hybrid TWENO schemes are designed, both of which significantly improve the numerical simulation performance of the TWENO scheme while saving approximately 80% of the computation time.

Hierarchical Stability Conditions for Two Types of Time-Varying Delay Generalized Neural Networks.

In this article, the stability analysis for generalized neural networks (GNNs) with a time-varying delay is investigated. About the delay, the differential has only an upper boundary or cannot be obtained. For the both two types of delayed GNNs, up to now, the second-order integral inequalities have been the highest-order integral inequalities utilized to derive the stability conditions. To establish the stability conditions on the basis of the high-order integral inequalities, two challenging issues are required to be resolved. One is the formulation of the Lyapunov-Krasovskii functional (LKF), the other is the high-degree polynomial negative conditions (NCs). By transforming the integrals in N-order generalized free-matrix-based integral inequalities (GFIIs) into the multiple integrals, the hierarchical LKFs are constructed by adopting these multiple integrals. Then, the novel modified matrix polynomial NCs are presented for the 2N-1 degree delay polynomials in the LKF differentials. Thus, the hierarchical linear matrix inequalities (LMIs) are set up and the nonlinear problems caused by the GFIIs are solved at the same time. Eventually, the superiority of the provided hierarchical stability criteria is demonstrated by several numeric examples.

Computing high-degree polynomial gradients in memory

Specialized function gradient computing hardware could greatly improve the performance of state-of-the-art optimization algorithms. Prior work on such hardware, performed in the context of Ising Machines and related concepts, is limited to quadratic polynomials and not scalable to commonly used higher-order functions. Here, we propose an approach for massively parallel gradient calculations of high-degree polynomials, which is conducive to efficient mixed-signal in-memory computing circuit implementations and whose area scales proportionally with the product of the number of variables and terms in the function and, most importantly, independent of its degree. Two flavors of such an approach are proposed. The first is limited to binary-variable polynomials typical in combinatorial optimization problems, while the second type is broader at the cost of a more complex periphery. To validate the former approach, we experimentally demonstrated solving a small-scale third-order Boolean satisfiability problem based on integrated metal-oxide memristor crossbar circuits, with competitive heuristics algorithm. Simulation results for larger-scale, more practical problems show orders of magnitude improvements in area, speed and energy efficiency compared to the state-of-the-art. We discuss how our work could enable even higher-performance systems after co-designing algorithms to exploit massively parallel gradient computation.

HIGH ORDER ANTI EVEN LEAST SQUARE FOR APPROXIMATING PRIME NUMBERS BELOW 1000

The study of prime numbers, pivotal in mathematics for centuries, holds significant importance in number theory and diverse applications like cryptography and computer science. This article introduces a novel approach, "High Order Anti Even Least Square," for approximating prime numbers below 1000. Integrating Least Square with specialized techniques tailored to complex prime number distributions, this method aims to enhance accuracy compared to traditional approaches. The research methodology involves polynomial approximation using both traditional and anti-even least squares methods, error calculation, visualization, and analysis. Results indicate that while traditional least squares generally performs better, the anti-even least squares method shows promise, particularly at higher polynomial degrees. This study contributes to advancing number theory and its practical applications by presenting a novel method for prime number approximation.

Mass lumping the dual cell method to arbitrary polynomial degree for acoustic and electromagnetic waves

We present a fundamental improvement of a high polynomial degree time domain cell method recently introduced by the last three authors. The published work introduced a method featuring block-diagonal system matrices where the block size and conditioning scaled poorly with respect to polynomial degree. The issue is herein bypassed by the construction of new basis functions exploiting quadrature rule based mass lumping techniques for arbitrary polynomial degrees in two dimensions for the Maxwell equations and the acoustic wave equation in the first order velocity pressure formulation. We characterize the degrees of freedom of all new discrete approximation spaces we employ for differential forms and show that the resulting block diagonal (inverse) mass matrices have block sizes independent of the polynomial degree. We demonstrate on an extensive number of examples how the new technique is applicable and efficient for large scale computations.

Biomass Higher Heating Value Estimation: A Comparative Analysis of Machine Learning Models

The research conducted focused on the capabilities of various non-linear and machine learning (ML) models in estimating the higher heating value (HHV) of biomass using proximate analysis data as inputs. The research was carried out to identify the most appropriate model for the estimation of HHV, which was determined by a statistical analysis of the modeling error. In this sense, artificial neural networks (ANNs), support vector machine (SVM), random forest regression (RFR), and higher-degree polynomial models were compared. After statistical analysis of the modeling error, the ANN model was found to be the most suitable for estimating the HHV biomass and showed the highest specific regression coefficient, with an R2 of 0.92. SVM (R2 = 0.81), RFR, and polynomial models (R2 = 0.84), on the other hand, also exhibit a high degree of estimation, albeit with somewhat larger modelling errors. The study conducted suggests that ANN models are best suited for the non-linear modeling of HHV of biomass, as they can generalize and search for links between input and output data that are more robust but also more complex in structure.

Fuzzy Predator-Prey Systems by Extended Runge-Kutta Method with Polynomial Interpolation Technique

The growing significance of uncertainty quantification in mathematical modeling of physical phenomena is evident, with fuzzy sets emerging as an alternative approach. This paper focuses on exploring a numerical method for addressing fuzzy differential equations in two-dimensional problems like fuzzy predator-prey systems. The numerical solution employed the extended Runge-Kutta fourth-order method. To overcome challenges related to polynomial fuzzy differential equations, a combination of the polynomial interpolation technique and the extended Runge-Kutta fourth-order method was applied, mitigating the issues associated with high-degree polynomials. The proposed approach for fuzzy predator-prey systems is briefly described, accompanied by a numerical example. The obtained results underscore the efficacy of the extended Runge-Kutta fourth-order method with polynomial interpolation, showcasing its remarkable accuracy and potential as an alternative method for addressing various uncertainty-related problems.

‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme

A new interpretation and applications of the ‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of 0 , 1 , … , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the ‘Diophantine’ and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm ‘eigenvectors’ of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

Method for considering real gas equations of state in through-flow programs

Numerical methods are significant in the turbomachine design process and off-design analysis. One of these methods is the through-flow method in the meridional plane, which is utilized in an early design phase. It provides a robust and quick numerical turbomachinery analysis by only giving a few characteristic geometric parameters.This work shows an extension of the through-flow method by considering real gas equations of state (EOSs) in throughflow programs. This is implemented in an in-house through-flow program called tFlow. This enables the method to support a more reliable calculation regarding, for instance, carbon dioxide (CO2) in a high-pressure region, where the gas does not behave like an ideal gas. The incorporation of real gas EOSs is validated with two-dimensional (2D) calculations of nozzle flows and with quasi-three-dimensional (3D) calculations of supercritical CO2 (sCO2) centrifugal compressors. A straightforward method for modifying the Riemann inlet boundary conditions for real gases is presented enabling a quick extension of through-flow codes to real gases and providing a stable solution process. Look-up tables (LUTs) are used instead of common high-degree polynomials to solve the real gas thermodynamic properties and reduce the computation time by a factor of about 20. Additionally, the calculation results verify the application of the Jameson–Schmidt–Turkel (JST) scheme in terms of real gas EOSs. Finally, the predictions are compared against 1D solutions for 2D cases and against the experimental data of the sCO2-HeRo compressor for 3D cases.

A new type of modified MR-WENO schemes with new troubled cell indicators for solving hyperbolic conservation laws in multi-dimensions

In this paper, a new type of increasingly high-order modified multi-resolution weighted essentially non-oscillatory (MMR-WENO) schemes with new troubled cell indicators is designed in the finite difference framework for solving hyperbolic conservation laws in one, two, and three dimensions. It is a first time to design new troubled cell indicators that based on two high-degree reconstruction polynomials, which are defined on the three-point, five-point, seven-point, and nine-point spatial stencils, respectively. Such new troubled cell indicators can automatically identify the discontinuous solutions without manually adjusting the parameters related to the problems. Subsequently, a series of MMR-WENO schemes are designed by using these new troubled cell indicators, which use the simple linear upwind schemes in smooth areas and the sophisticated MR-WENO schemes in discontinuous areas, thus achieving the goal of inheriting the excellent characteristics of original MR-WENO schemes while reducing computational costs. The new modified methodology is divided into two parts: if all extreme points of two reconstruction polynomials defined on the big spatial stencil are outward the smallest interval [xi−1/2,xi+1/2], the numerical flux is straightforwardly approximated by a high-degree reconstruction polynomial and the approximation has a high-order accuracy. Otherwise, the high-order MR-WENO spatial reconstruction procedures are adopted. The main benefits of new MMR-WENO schemes are their robustness and efficiency in comparison to original MR-WENO schemes, since the MMR-WENO schemes could save at most 63%-79% CPU time than the same-order MR-WENO schemes do for some numerical examples.

Algorithm 1041: HiPPIS—A High-order Positivity-preserving Mapping Software for Structured Meshes

Polynomial interpolation is an important component of many computational problems. In several of these computational problems, failure to preserve positivity when using polynomials to approximate or map data values between meshes can lead to negative unphysical quantities. Currently, most polynomial-based methods for enforcing positivity are based on splines and polynomial rescaling. The spline-based approaches build interpolants that are positive over the intervals in which they are defined and may require solving a minimization problem and/or system of equations. The linear polynomial rescaling methods allow for high-degree polynomials but enforce positivity only at limited locations (e.g., quadrature nodes). This work introduces open-source software (HiPPIS) for high-order data-bounded interpolation (DBI) and positivity-preserving interpolation (PPI) that addresses the limitations of both the spline and polynomial rescaling methods. HiPPIS is suitable for approximating and mapping physical quantities such as mass, density, and concentration between meshes while preserving positivity. This work provides Fortran and Matlab implementations of the DBI and PPI methods, presents an analysis of the mapping error in the context of PDEs, and uses several 1D and 2D numerical examples to demonstrate the benefits and limitations of HiPPIS.

A New Approach to Multiroot Vectorial Problems: Highly Efficient Parallel Computing Schemes

In this article, we construct an efficient family of simultaneous methods for finding all the distinct as well as multiple roots of polynomial equations. Convergence analysis proves that the order of convergence of newly constructed family of simultaneous methods is seventeen. Fractal-based simultaneous iterative algorithms are thoroughly examined. Using self-similar features, fractal-based simultaneous schemes can converge to solutions faster, saving computational time and resources necessary for solving nonlinear equations. Fractals analysis illustrates the newly developed method’s global convergence behavior when compared to single root-finding procedures for solving fractional order polynomials that arise in complex engineering applications. Some real problems from various branches of engineering along with some higher degree polynomials are considered as test examples to show the global convergence property of simultaneous methods, performance and efficiency of the proposed family of methods. Further computational efficiencies, CPU time and residual graphs are also drawn to validate the efficiency, robustness of the newly introduced family of methods as compared to the existing methods in the literature.

Multipolynomial Monte Carlo for trace estimation in lattice QCD

Estimating the trace of the inverse of a large matrix is an important problem in lattice quantum chromodynamics. A multilevel Monte Carlo method is proposed for this problem that uses different degree polynomials for the levels. The polynomials are developed from the GMRES algorithm for solving linear equations. To reduce orthogonalization expense, the highest degree polynomial is a composite or double polynomial found with a polynomial preconditioned GMRES iteration. Added to some of the Monte Carlo pieces is deflation of eigenvalues that reduces the variance. Deflation is also used for finding a reduced degree deflated polynomial. The new Multipolynomial Monte Carlo method can significantly improve the trace computation for matrices that have a difficult spectrum due to small eigenvalues.

Improvements in digital meteor spectra reduction

This study addresses the complexity and importance of developing a method of calibrating digital observations of meteor spectra with all-sky cameras. It aims to present novel approaches to spectral sensitivity, atmospheric extinction and flat-field corrections. Images of a known line emission spectrum were captured at various positions within the field of view using a camera with a fish-eye lens and plastic holographic grating. The flat-field correction was separated into a wavelength-independent and wavelength-dependent component, both dependent on the position of the spectral line in the field of view (FoV). Total profile intensities of spectra obtained from the images were compared throughout the spectral range at different positions in the FoV. The flat-field was constructed by fitting those dependencies with high-degree polynomial functions. Using a simplified atmospheric model, a novel approach was constructed to determine the atmospheric extinction curve throughout the spectral range, allowing it to be separately considered from the spectral sensitivity which was previously not the case. A comparison of the newly developed and previously used methodology was tested on several meteor spectra of the same meteor captured from different stations of the European Fireball Network. It revealed a significantly improved correspondence of the analysed spectra in the part of the spectral range unaffected by the limitations imposed by the newly developed methodology. Failing to follow the correct calibration methodology precisely may introduce varying degrees of uncertainty in computations of elemental abundances and other physical properties, depending on the equipment’s specific effect magnitude.

High-degree polynomial noise subtraction for disconnected loops

In lattice QCD, the calculation of physical quantities from disconnected quark loop calculations have large variance due to the use of Monte Carlo methods for the estimation of the trace of the inverse lattice Dirac operator. In this work, we build upon our POLY and HFPOLY variance reduction methods by using high-degree polynomials. Previously, the GMRES polynomials used were only stable for low-degree polynomials, but through application of a new, stable form of the GMRES polynomial, we have achieved higher polynomial degrees than previously used. While the variance is not dependent on the trace correction term within the methods, the evaluation of this term will be necessary for forming the vacuum expectation value estimates. This requires computing the trace of high-degree polynomials, which can be evaluated stochastically through our new Multipolynomial Monte Carlo method. With these new high-degree noise subtraction polynomials, we obtained a variance reduction for the scalar operator of nearly an order of magnitude over that of no subtraction on a 243×32 quenched lattice at β=6.0 and κ=0.1570≈κcrit. Additionally, we observe that for sufficiently high polynomial degrees, POLY and HFPOLY approach the same level of effectiveness. We also explore the viability of using double polynomials for variance reduction as a means of reducing the required orthogonalization and memory costs associated with forming high-degree GMRES polynomials.

Stability Analysis of Discrete-Time Neural Networks With a Time-Varying Delay: Extended Free-Weighting Matrices Zero Equation Approach.

This research investigates the stability of discrete-time neural networks (DNNs) with a time-varying delay by using the Lyapunov-Krasovskii functional (LKF) method. Recent researches acquired some less conservatism stability criteria for time-varying delayed systems via some augmented LKFs. However, the forward difference of such LKFs resulted in high-degree time-varying delay-dependent polynomials. This research aims to develop some augmented state-related vectors and the corresponding extended free-weighting matrices zero equations to avoid the appearance of such high-degree polynomials and help to provide more freedom for the estimation results. Besides, an augmented delay-product-type LKF is also established for ameliorating the stability conditions of the time-varying delayed DNNs. Then, based on the above methods and Jensen's summation inequality, the auxiliary-function-based summation inequality, and the reciprocally convex matrix inequality, some less conservatism stability criteria for time-varying delayed DNNs are formulated. The validity of the proposed time-varying delay-dependent stability criteria is illustrated by two numerical examples.

An efficient forward semi-Lagrangian model

An efficient forward trajectory model is proposed, in which the property and position of the fluids advected from the Euler coordinates to the Lagrangian coordinates can be accurately evaluated. After sorting and aligning those fluid elements on the irregular Lagrangian curves, we apply the cubic or other high-degree polynomials to interpolate the properties of the elements from the irregular curves to the regular grids. There is no need to solve the cubic equations and the associated coefficients as proposed previously. The model is quite simple, accurate, and much more efficient than the previous models. It also allows higher-order polynomials to be employed in the interpolations. It is suitable for simulating the multi-dimensional fast-moving flows with large Courant Numbers, the transport of pollutants in the atmosphere and ocean, and movement of raindrops in atmospheric models.

Inexpensive polynomial-degree-robust equilibrated flux a posteriori estimates for isogeometric analysis

In this paper, we consider isogeometric discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e. vector-valued mapped piecewise polynomials lying in the [Formula: see text] space which appropriately approximate the desired divergence constraint. Our estimates are constant-free in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical B-splines, though not with respect to the smoothness and support overlaps. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments.