Low-order Polynomial Research Articles

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.

An Image Compensation-Based Range–Doppler Model for SAR High-Precision Positioning

The range–Doppler (R–D) model is extensively employed for the geometric processing of synthetic aperture radar (SAR) images. Refining the sensor motion state and imaging parameters is the most common method for achieving high-precision geometric processing using the R–D model, comprising a process that involves numerous parameters and complex computations. In order to reduce the specialization and complexity of parameter optimization in the classic R–D model, we introduced a novel approach called ICRD (image compensation-based range–Doppler) to improve the positioning accuracy of the R–D model, implementing a low-order polynomial to compensate for the original imaging errors without altering the initial positioning parameters. We also designed low-order polynomial compensation models with different parameters. The models were evaluated on various SAR images from different platforms and bands, including spaceborne TerraSAR-X and Gaofen3-C images, manned airborne SAR-X images, and unmanned aerial vehicle-mounted miniSAR-Ku images. Furthermore, image positioning experiments involving the use of different polynomial compensation models and various numbers and distributions of ground control points (GCPs) were conducted. The experimental results demonstrate that geometric processing accuracy comparable to that of the classical rigorous positioning method can be achieved, even when applying only an affine transformation model to the images. Compared to classical refinement models, however, the proposed image-compensated R–D model is much simpler and easy to implement. Thus, this study provides a convenient, robust, and widely applicable method for the geometric-positioning processing of SAR images, offering a potential approach for the joint-positioning processing of multi-source SAR images.

Combining approach of collocation and finite difference methods for fractional parabolic PDEs

This research aims to estimate the solutions of fractional-order partial differential equations of spacial fractional and both time-space fractional order. For this, we use finite differences for time derivatives and the well-known collocation method for space derivatives with lower-order Bernstein polynomials as basis functions. We explain the mathematical formulations in detail. Convergence and stability analysis of the space–time fractional diffusion equation with the source term is reported subsequently. Three numerical examples are considered for demonstrating the accuracy and reliability of the proposed method.

A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks

Kolmogorov–Arnold Networks (KANs) were recently introduced as an alternative representation model to MLP. Herein, we employ KANs to construct physics-informed machine learning models (PIKANs) and deep operator models (DeepOKANs) for solving differential equations for forward and inverse problems. In particular, we compare them with physics-informed neural networks (PINNs) and deep operator networks (DeepONets), which are based on the standard MLP representation. We find that although the original KANs based on the B-splines parameterization lack accuracy and efficiency, modified versions based on low-order orthogonal polynomials have comparable performance to PINNs and DeepONet, although they still lack robustness as they may diverge for different random seeds or higher order orthogonal polynomials. We visualize their corresponding loss landscapes and analyze their learning dynamics using information bottleneck theory. Our study follows the FAIR principles so that other researchers can use our benchmarks to further advance this emerging topic.

Robust adaptive control of delayed systems with multiple uncertainties and polynomial nonlinearities

This work discusses the issue of robust adaptive control for delayed systems with multiple uncertain factors. The system uncertainties cover uncertain dynamics, unknown system parameters, and external disturbances. Besides, the growing conditions have both low-order and high-order polynomial nonlinearities, which bring in numerous challenges for control design. The suitable transformations are first introduced to obtain a new system with a dynamic gain. Then, the control design of an auxiliary system is considered. By introducing the Lyapunov–Krasovskii (L–K) functional method, and providing an important result for an auxiliary system, the adaptive homogeneous control strategy is proposed to achieve global asymptotic stability (GAS) of the entire system. Besides, by considering the external disturbances, the method is also extended to design robust adaptive tracking controller. Numerical and practical examples are provided to verify the theory.

MODELING OF A PEM FUEL CELL POLARIZATION CURVE BY LOW-ORDER POLYNOMIALS FOR THE OUTPUT POWER CALCULATION ALGORITHMS

The control and energy algorithms that govern the operation of a PEM fuel cell within an energy system must account for the nonlinear phenomena in the cell; it is often captured by the polarization curve. However, the modeling approaches used for simulation studies could be more suitable for the abovementioned algorithms due to their higher computational burden. That leads to the development of simplified approaches. Our study focuses on the algorithms used for the fuel cell output power calculation. Many of them are directed towards the maximum power point operation of the cell. We propose that the fuel cell representation for the algorithms can be used in the low-order polynomial form, thus decreasing the computational load compared to the other approaches. For the maximum power point prediction, we propose using the truncated form of polarization curve input data (ignoring the activation loss region). Our study has demonstrated that based on the same input data and MATLAB curve fitting commands, the 3rd-order polynomial provides the comparable RMSE 3.5…3.9 for the power curve approximation vs. 3.9 for the 5th-order polynomial representation. The value of the maximum PowerPoint is obtained with a 1.3 % relative error with the 2nd-order polynomial using the truncated input data compared to 1.2 % for the 5th-order polynomial representation.

Multiplier-less Broadband and Linear Phase Digital Hilbert Transformers

The Hilbert transformation for generating the analytic signal or signal envelope is widely used in modern communication receivers, in radar and sonar systems. It introduces a 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^{\\circ }$$\\end{document} phase shift of the input signal. Since the impulse response of the ideal Hilbert transformer is non-causal, it must be approximated by an FIR or IIR filter. This paper shows results of novel algorithms for designing broadband digital IIR Hilbert transformers and its implementation. The designs employ Galerkin or collocation techniques. The transfer function of the Hilbert transformer is a rational polynomial of low order and exhibits approximately linear phase. The filters match the 90∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$90^{\\circ }$$\\end{document} phase shift requirement of Hilbert transformers almost perfectly and exhibit approximately constant group delay in the passband. The achieved image rejection ratio is typically larger than 50 dB. The quantization of the filter coefficients is realized by a Canonical Signed Digit (CSD) representation, reducing the hardware resources compared with two’s complement. The resulting filters are multiplier-less, which is crucial for high-speed signal processing and low power consumption. The design techniques and the CSD representation are realized in a MATLAB toolbox. The filters were moreover implemented in VHDL and SystemC. Additionally, a MATLAB tool for automatically generating a VHDL package containing the filter parameters has been implemented.

Efficient Convolution-Based Representation of Smooth Reference Trajectories for Urban Air Mobility

Accurate trajectory-following control using dynamic inversion controllers requires suitable parameterizations of reference trajectories, such as energy-minimal flight paths generated with optimal control methods. These trajectory representations must enable efficient onboard storage and real-time evaluation while ensuring sufficient accuracy and smoothness to provide the required feedforward commands. We explore trajectory parameterizations based on the convolution of a piecewise low-order polynomial with an infinitely smooth kernel. These combine efficient storage with unlimited differentiability and allow for simple and efficient evaluation using numerical quadrature. To construct such parameterizations from given trajectories, we formulate quadratic programs that account for additional, mission-specific constraints. A numerical example demonstrates the parameterization of an approach and vertical landing trajectory based on optimal control results; we show that the convolution-based approach yields competitive results compared to a piecewise polynomial parameterization, with significant storage savings.

Coupling bit and modular arithmetic for efficient general-purpose fully homomorphic encryption

Fully Homomorphic Encryption (FHE) enables computation directly on encrypted data. This property is desirable for outsourced computation of sensitive data as it relies solely on the underlying security of the cryptosystem and not in access control policies. Even though FHE is still significantly slower than unencrypted computation, practical times are possible for applications easily representable as low-order polynomials, since most FHE schemes support modular addition and multiplication over ciphertexts. If, however, an application cannot be expressed with low-order polynomials, then Boolean logic must be emulated. This bit-level arithmetic enables any computation to be performed homomorphically. Nevertheless, as it runs on top of the natively supported modular arithmetic, it has poor performance, which hinders its use in the majority of scenarios. In this work, we propose Bridging, a technique that allows conversion from bit-level to modular arithmetic and vice-versa. This enables the use of the comprehensive computation provided by bit-level arithmetic and the performance of modular arithmetic within the same application. Experimental results show that Bridging can lead to 1-2 orders of magnitude performance improvement for tested benchmarks and two real-world applications: URL denylisting and genotype imputation. Bridging performance comes from two factors: reduced number of operations and smaller multiplicative depth.

Artificial neural network-based sequential approximate optimization of metal sheet architecture and forming process

Abstract This paper introduces a sequential approximate optimization method that combines the finite element method (FEM), dynamic differential evolution (DDE), and artificial neural network (ANN) surrogate models. The developed method is applied to address two optimization problems. The first involves metamaterial design optimization for metal sheet architecture with binary design variables. The second pertains to optimizing process parameters in multi-stage metal forming, where the discrete nature arises owing to changing tool geometries across stages. This process is highly nonlinear, accumulating contact, geometric, and material nonlinear effects discretely through forming stages. The efficacy of the proposed optimization method, utilizing ANN surrogate models, is compared with traditionally used polynomial response surface (PRS) surrogate models, primarily based on low-order polynomials. Efficient learning of ANN surrogate models is facilitated through the FEM and Python integration framework. Initial data for surrogate model training is collected via Latin hypercube sampling and FEM simulations. DDE is employed for sequential approximate optimization, optimizing ANN or PRS surrogate models to determine optimal design variables. PRS surrogate models encounter challenges in dealing with nonlinear changes in sequential approximate optimization concerning discrete characteristics such as binary design variables and discrete nonlinear behavior found in multi-stage metal forming processes. Owing to the discrete nature, PRS surrogate models require more data and iterations for optimal design variables. In contrast, ANN surrogate models adeptly predict nonlinear behavior through the activation function's characteristics. In the optimization problem of Metal Sheet Architecture for design target C, the ANN surrogate model required an average of 4.6 times fewer iterations to satisfy stopping criteria compared to the PRS surrogate model. Furthermore, in the optimization of multi-stage deep drawing processes, the ANN surrogate model required an average of 6.1 times fewer iterations to satisfy stopping criteria compared to the PRS surrogate model. As a result, the sequential global optimization method utilizing ANN surrogate models achieves optimal design variables with fewer iterations than PRS surrogate models. Further confirmation of the method's efficiency is provided by comparing Pearson correlation coefficients and locus plots.

Indentation of a thin incompressible layer with finite friction

If a thin layer of an incompressible elastic material is pressed between two plane surfaces, the effective stiffness is very sensitive to the presence of frictional slip. This effect is investigated using a low-order polynomial representation of the through-thickness displacement profile. Results show good agreement with finite element studies and also show that the stiffness is significantly affected by that part of the layer [if any] outside the loaded region. The same result is then used in convolution to approximate the load–displacement response for a convex indenter.

Motion state factor driven for doubly-curved shallow shell deformation reconstruction

The conventional shape sensing method, based on C0 continuity, is recognized for its computational efficiency. However, it is prone to shear locking due to low-order polynomial interpolation in shell analysis, introducing two primary challenges. Firstly, it lacks consideration for C1 continuity. Secondly, existing high-order polynomial interpolation methods significantly increase the number of degrees of freedom (DOF), reducing computational efficiency. Therefore, a brand-new real-time structural health monitoring (SHM) method driven by motion state factor instead of nodal factors is proposed. This method accommodates all deformation modes of shallow shells with C1 continuity and improves the shape function's order without additional DOF. It begins with deriving motion states from compatibility equations, leading to a polynomial shape function expression. Subsequently, conversion equations for real constants and DOF are developed, establishing a least-squares model for theoretical strains and measured values. Following that, the higher-order element formula, optimized for minimal DOF, is then inferred from varied displacement data. Finally, simulations and experiments demonstrate this method's effectiveness, accuracy, and reliability in analyzing shallow shell elements.

Learning physics-based reduced-order models from data using nonlinear manifolds.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Choosing the Best Functions for the Rayleigh-Ritz Vibration Analysis of Beams

The Rayleigh-Ritz Method is used extensively for the vibration analysis of structures. The accuracy depends on the assumed functions. In this work several different groups of functions are examined and compared for the accuracy of the resulting natural frequencies, and for the overall mode shape error norms calculated with respect to the known exact solutions. It is concluded that a set that combines low order polynomials, odd cosine and odd sine functions, or, even cosine and even sine functions, is more likely to yield the best accuracy and convergence of both frequency and mode shapes for a general beam structure.

Analytical method for stable background reduction for Raman spectra of carbon‐containing meteorite and terrestrial samples suffering from intense fluorescence

AbstractChemical states of carbon in terrestrial (meta) sediments and carbonaceous chondrites gather attention as a geothermometer. As a nondestructive analytical method, Raman spectroscopy has been widely used to study their electronic properties, crystallinity, and structural defects through so‐called D and G bands. For the analysis of Raman spectra, a common problem is coexistence of a fluorescence background, which should be subtracted prior to the peak‐fitting analysis. However, we recently faced a problem that the band shape noticeably changed depending on the background function assumed although the background seemed to be well subtracted at a first glance regardless of the choice of the background function. For the application of the Raman spectroscopy as a geothermometer, a standard background subtraction method must be established to suppress the arbitrariness. In the present study, Raman spectra of seven carbon‐containing natural samples, whose background intensities were significantly different, were measured, and their background shape was evaluated by first‐, second‐, and third‐order polynomials. The results indicated that the third‐order polynomial was necessary and sufficient as a standard background function. Importantly, although lower order polynomials seem to successfully fit the background at a first glance, they falsely caused dispersion of the shoulder band shape.

A Spectral/hp-Based Stabilized Solver with Emphasis on the Euler Equations

The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/hp stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/hp framework. We introduce revised stabilization parameters, τsupg, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ: the first is described with entropy variables, and the other is the YZβ parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L1 and L2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries.

Degeneration of kernel regression with Matern kernels into low-order polynomial regression in high dimension.

Kernel methods such as kernel ridge regression and Gaussian process regression with Matern-type kernels have been increasingly used, in particular, to fit potential energy surfaces (PES) and density functionals, and for materials informatics. When the dimensionality of the feature space is high, these methods are used with necessarily sparse data. In this regime, the optimal length parameter of a Matern-type kernel may become so large that the method effectively degenerates into a low-order polynomial regression and, therefore, loses any advantage over such regression. This is demonstrated theoretically as well as numerically in the examples of six- and fifteen-dimensional molecular PES using squared exponential and simple exponential kernels. The results shed additional light on the success of polynomial approximations such as PIP for medium-size molecules and on the importance of orders-of-coupling-based models for preserving the advantages of kernel methods with Matern-type kernels of on the use of physically motivated (reproducing) kernels.

УСУНЕННЯ ЗАВАД У ДИНАМІЧНИХ СПЕКТРАХ ЗА НАЯВНОСТІ ПОТУЖНОГО СИГНАЛУ. Частина 2. ВУЗЬКОСМУГОВІ ЗАВАДИ УМОВНО-СТАЦІОНАРНОГО ХАРАКТЕРУ

Subject and Purpose. A new method for detecting narrow band interferences is discussed, with the use of an example record of Jovi- an Io-C decametric radio storm as obtained with the UTR-2 array on April 10, 2020. We aimed at developing an efficient and simple algorithm based on a detailed analysis of the effect the radio interference environment and the frequency response of the telescope may have on the efficiency of the interference mitigation procedure. Methods and Methodology. The ‘orthogonal detection’ method proposed for identifying linear interference patterns in dynamic spectra of powerful radio sources has been adapted for application to narrow band interferences of various spectral widths and variable brightness, which often happen to be located close to (or intersect with) the signal of interest. In order to minimize the impact of the telescope’s frequency response, a discretized frequency-scanning technique is used, which permits a steady approximation of the averaged spectrum by low order polynomials, as well as removal of the frequency trend which hinders distinguishing between the low-level interference signals and those coming from the source. Results. An efficient approach to the problem of detecting long lasting, narrowband interferences in dynamical spectra (con- ditionally stationary interferences) is proposed. The algorithm has been tested on the example of a powerful storm of Jovian radio emission. The proposed technique can be especially useful in situations where the signal of interest overlaps with the interference in the time-frequency domain. Conclusions. The new approach to the problem of interference mitigation, based on the development of combined algorithms of signal separation in the time-frequency domain, has been shown to offer an effective method of signal processing. In contrast to the previously used methods based on calculation of statistical moments for amplitudes, the new approach allows avoiding use of the same statistical indicators for both interference detection and subsequent steps of data analysis intended for building physical models and interpreting the observational data.

Simultaneous Model Change Detection in Multivariate Linear Regression With Application to Indonesian Economic Growth Data

In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three‐variate low‐order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID‐19 pandemic in 2020.

The hybrid method for finding the roots of a polynomial over finite fields based on affine expansion

This paper proposes a hybrid method to find the roots of a polynomial in a finite field based on combining the polynomial decomposition into the sum of multiples of the affine polynomial with the analytical method to find the roots of low-order polynomials. The proposed method allows a significant reduction in complexity and is applicable in the design of low-latency BCH and Reed-Solomon decoders.