Polynomial Approximation Research Articles

Rotor angle stability of a microgrid generator through polynomial approximation based on RFID data collection and deep learning.

The article proposes a novel approach to assess rotor angle stability in microgrids by enhancing the Modified Galerkin Method (MGM), which is based on the Polynomial Approximation, using real-time RFID data acquisition. Due to their reliance on assumptions, traditional rotor angle stability methodologies frequently fail in online transient stability testing. MGM successfully captures the dynamic behavior of microgrids by approximating state variables using a sequence of polynomials and coefficients. Redundant data, like as vibrations or noise signals, can cause delays in defect diagnosis and decrease diagnostic accuracy. This problem is addressed by integrating RFID technology. RFID technology could potentially be used with a hybrid CNN-LSTM model to develop a sophisticated fault diagnostic system. This entails identifying fault characteristics through the use of signal processing techniques and feature extraction methods, such as the Fourier transform and time-domain statistical features. In addition, we use Total Harmonic Distortion (THD) to reduce superfluous data. The suggested techniques significantly increase fault detection efficiency and precision, outperforming existing techniques with a 0.94 classification accuracy. An extensive case study on an IEEE 3-machine 9-bus system is used to illustrate its efficacy, showing observable improvements in fault detection speed and accuracy that make microgrid operations safer and more dependable.

Analysis of Spatial Effects of New Energy Vehicles Based on Semi-parametric Spatial Durbin Model

The global scientific and technological revolution and industrial change are developing rapidly, the integration of automotive technology with energy, transportation and information and communication fields is accelerating, and electrification, internet connectivity and intelligence have become the main trends. Therefore, studying the impact of digital development on the growth of new energy vehicle industry and consumer purchase intention is of great theoretical and practical significance for the digital transformation of automobile companies. In this paper, based on the provincial panel data of new energy vehicles in China from 2016 to 2023, we constructed the index system of digital economy and economic development, used the entropy method to calculate the level of development, and conducted spatial autocorrelation tests using global and local Moran indices. Considering the non-linear relationship between variables, a semi-parametric spatial Durbin model is established using local polynomial estimation to explore the impact of digital economy on new energy automobile industry. The results of the study show that the development of digital economy promotes the development of new energy automobile industry to a certain extent, but its development level may have a negative impact if it is too low or too high.

Fast and Efficient Lunar Finite Element Gravity Model

In this paper, the finite element method (FEM) is integrated with orthogonal polynomial approximation in high-dimensional spaces to innovatively model the Moon’s surface gravity anomaly. The aim is to approximate solutions to Laplace’s classical differential equations of gravity, employing classical Chebyshev polynomials as basis functions. Using classical Chebyshev polynomials as basis functions, the least-squares approximation was used to approximate discrete samples of the approximation function. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. The first application of this project is to substitute the globally valid classical spherical harmonic series of approximations with locally valid series of orthogonal polynomial approximations (i.e., using the FEM approach). With an error tolerance set at 10−9ms−2, this method is used to adapt the gravity model radially upwards from the lunar surface. The results showcase a need for a higher degree of approximation on and near the lunar surface, with the necessity decreasing as the radius increases. Notably, this method achieves a computational speedup of five orders of magnitude when applying the method to radial adaptation. More intrinsically, the second application involves using the methodology as an effective tool in solving boundary value problems. Specifically, this approach is implemented to solve classical differential equations involved with high-precision, long-term orbit propagation. This application provides a four-order-of-magnitude speedup in computational time while maintaining an error within the 10−10ms−2 error range for various orbit propagation tests. Alongside the advancements in orthogonal approximation theory, the FEM enables revolutionary speedups in orbit propagation without compromising accuracy.

Effects of carrier structure parameters on diesel particulate filter capture performance based on grey relational analysis

To reduce the pressure drop of diesel particulate filter (DPF) as well as improve its collection efficiency. A mathematical model of DPF pressure drop and capture efficiency was established. This research systematically investigates the effects of channels per square inch (CPSI), wall thickness, and carrier ratio on DPF performance through grey relational analysis (GRA) and polynomial approximation algorithm (PAA). The findings reveal that DPF exhibits lower pressure drop and higher collection efficiency in the case where the carrier ratio, wall thickness, and CPSI are within the ranges of 1.40 to 1.53, 9.8 to 10.5 mil, and 200 to 300, respectively. Particularly, at a carrier ratio of 1.52 and a wall thickness of 10.38 mil, the pressure drop reaches a minimum value of 4.23 kPa, with a collection efficiency of 90.28%. Meanwhile, at a carrier ratio of 1.52 and a CPSI of 200, the pressure drop reaches a minimum value of 4.17 kPa, with a collection efficiency of 90.21%. The model expressions for pressure drop and collection efficiency derived from the PAA are: y 1 = 51.19 + 3.815 x 1 − 8.726 x 2 + 3.794 x 1 2 − 1.735 x 1 x 2 + 0.524 x 2 2 , and y 2 = 0.383 + 0.105 x 1 + 0.085 x 2 + 0.032 x 1 2 − 0.016 x 1 x 2 − 0.0032 x 2 2 . The foregoing models exhibit excellent predictive accuracy and goodness of fit for the DPF performance. Under the interactive influence of carrier ratio and wall thickness, the root mean square errors (RMSE) for pressure drop and collection efficiency are 0.0110 and 0.0002, respectively, with corresponding goodness of fit values of 0.9996 and 0.9985. As revealed by the GRA results, wall thickness exerts the most significant impact on DPF performance, presenting a GRG of 0.84 for filtration efficiency and 0.79 for pressure drop. The overall influence on both collection efficiency and pressure drop follows the descending order: wall thickness > carrier ratio > CPSI. This work has important engineering significance for optimizing the DPF capture performance and extending its working time.

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Polynomial approximation of discounted moments

AbstractWe introduce an approximation strategy for the discounted moments of a stochastic process that can approximate the true moments for a large class of problems. These moments appear in pricing formulas of financial products such as bonds and credit derivatives. The approximation relies on a high-order power series expansion of the infinitesimal generator and draws parallels with the theory of polynomial processes. We demonstrate applications to bond pricing and credit derivatives. In the special cases that allow an analytical solution, the approximation error decreases to around 10 to 100 times machine precision for higher orders. When no analytical solution exists, we numerically compare the approximation with existing numerical techniques.

On the Lanczos Method for Computing Some Matrix Functions

The study of matrix functions is highly significant and has important applications in control theory, quantum mechanics, signal processing, and machine learning. Previous work has mainly focused on how to use the Krylov-type method to efficiently calculate matrix functions f(A)β and βTf(A)β when A is symmetric. In this paper, we mainly illustrate the convergence using the polynomial approximation theory for the case where A is symmetric positive definite. Numerical results illustrate the effectiveness of our theoretical results.

Optimal one-sided approximants of circular arc

The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants interpolating boundary points is assumed. However, the study of approximants, which additionally interpolate corresponding tangent directions and curvatures at the boundary of an arc, is also considered. Several low-degree polynomial approximants are studied in detail. When several solutions fulfilling the interpolation conditions exist, the optimal one is characterized, and a numerical algorithm for its construction is suggested. Theoretical results are demonstrated with several numerical examples and a comparison with general (i.e. non-one-sided) approximants are provided.

Uncertainty analysis of hypersonic flight dynamics under elastic effects

Abstract In addressing the aerodynamic-elastic coupling issues of hypersonic vehicles, this study delves into the analysis of the elastic deformation of the aircraft based on mechanistic understanding. It employs the Interval Analysis Method based on Polynomial Approximations to conduct uncertainty analysis of the aerodynamic model under elasticity, delineating the range of uncertainty in the aerodynamic model influenced by elastic deformation. The aerodynamic force of the aircraft under elasticity is connected with the rigid body, and then the aerodynamic layout of the aircraft fuselage and control surface after elastic deformation is calculated to study the influence of the elastic effect on the aerodynamic force. The results show that the change caused by uncertainty has a relatively small effect on lift but a relatively large effect on drag and pitching moment.

Bernstein Polynomials for Solving Fractional Differential Equations with Two Parameters

This work presents a general framework for solving generalized fractional differential equations based on operational matrices of the generalized Bernstein polynomials. This method effectively obtains approximate analytical solutions of many fractional differential equations. The generalized fractional derivative of the Caputo type with two parameters and its properties are studied. Using orthonormal Bernstein polynomials has led to the development of fractional polynomials, which offer an approximate solution for ordinary fractional differential equations. The approach employs the generalized orthogonal Bernstein polynomials (FOBPs) and constructs their operational matrices for fractional integration and derivative in the generalized Caputo sense to achieve this objective. Operational matrices convert ordinary differential equations into a system of algebraic equations, which can be solved using Newton’s method. The convergence analysis and error estimate associated with the proposed problem have been investigated using the approximation of generalized orthogonal Bernstein polynomials (FOBPs). The effect of the new parameters of the fractional derivative is presented in several examples. Finally, several examples are included to clarify the proposed technique’s validity, efficiency, and applicability via generalized orthogonal Bernstein polynomials (FOBPs) approximation.

A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order α(Dαf)(y)=1Γ(m−α)∫0y(y−x)m−α−1f(m)(x)dx,y>0, with m−1<α≤m,m∈N. The numerical procedure is based on approximating f(m) by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order α. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure.

A type II radio burst associated with solar filament–filament interaction

Aims. Solar radio type II bursts are often associated with coronal shocks driven by solar eruptions. In this study, we report a type II burst associated with filament–filament interaction. Methods. Combining the high-quality multiwavelength observations from CHASE, SDO, STEREO, and CALLISTO, we conducted a detailed study of the type II burst associated with filament–filament interaction. Results. On 2023 September 11, an erupting filament (F1) likely disturbed a nearby long filament (F2), causing F2 to subsequently erupt. As a result of possible magnetic reconnection between ejective materials from the two filaments, loop-like structures formed perpendicular to them. Subsequently, the expansion of these loop-like structures triggered a strong coronal mass ejection (CME). Interestingly, a type II burst appeared on the solar spectrum around the time when the loop-like structures formed and the CME appeared above the occulting disk of STEREO/COR1. By converting the frequency of the type II burst to the coronal height using polarization brightness data recorded by the COR1 coronagraph and the spherically symmetric polynomial approximation technique, we determined the formation height of the type II burst to be around 1.45 R⊙, with a speed of approximately 440 km s−1. This is comparable to the observed height of the CME (∼1.43 R⊙), although slightly lower in speed (540 km s−1). Conclusions. All these results indicate that the type II burst was closely associated with filament–filament interactions and was possibly excited by the accompanying CME at the flank. We suggest that the filament–filament interactions played an important role in producing the type II burst by acting as a piston to trigger a strong CME.

A Chebyshev Generated Block Method for Directly Solving Nonlinear and Ill-posed Fourth-Order ODEs

This study presents a block method induced by Chebyshev polynomials of first kind for directly solving initial value problems of fourth-order ordinary differential equations without reducing the problems to a system of first-order differential equations. The method was developed by applying interpolation and collocation procedures to a Chebyshev approximate polynomial. The unknown parameters were obtained using the Gaussian elimination method, then substituted into the approximate solution to get the continuous scheme and evaluated at the selected point to give the discrete scheme. The method was zero stable, consistent, and convergent, p-stable as shown by the region of absolute stability and has an order of seven. The accuracy and usability of the developed method were tested by applying it to solve six numerical examples. The method was found to be efficient as it gives minimal error. The numerical results of the method were compared with other works cited in the literature and found to be better as it gives minor errors.

(q, γ)-Bernstein Basis Functions on the Interval [a ; b

In this paper, we present a novel set of (q, γ)-Bernstein basis functions that are parameterized by q and and defined over an interval. We give detailed proofs for several key properties of these functions, including the partition of unity, recurrence relations, degree elevation, and the Marsden identity. Additionally, we introduce and validate the (q, γ)-De Casteljau algorithm, providing comprehensive examples to illustrate its implementation. These results are analyzed to highlight the theoretical and practical implications of these (q, γ)-Bernstein basis functions in various fields, such as polynomial approximation, numerical methods, and Computer Aided Geometric Design (CAGD). Furthermore, we discuss potential extensions and applications of these functions, considering their impact on future research and developments in the domain. By exploring these aspects, we aim to offer a robust framework for understanding and utilizing (q, γ)-Bernstein basis functions.

New control variates for pricing basket options

Abstract Accepted by: Giorgio Consigli Accurate pricing of basket options, which are financial derivatives on multiple underlying assets, is a challenging and practically important task for financial institutions. We propose several new control variates for accurate, fast and efficient pricing of basket options. The first approach to deriving new control variates is the use of Hermite polynomial approximation of appropriate function of the underlying asset prices, which leads to a Black–Scholes-like analytic solution. This approach is new in the option pricing context and opens up new possibilities in derivative pricing. Further control variates are analytically derived using Jensen’s inequality in one case, and distributional properties of multivariate Wiener processes in other cases. All the newly proposed control variates are shown to lead to excellent variance reduction in numerical experiments based on realistic data. The proposed methods are novel, computationally simple and have a strong potential to replace more conventional methods, such as the geometric lower bound in simulation-based pricing of basket options and similar products used in financial risk management.

The Hermite-Birkhoff Problem and Local Spline Approximation

This paper discusses the use of local spline approximations to solve the Hermite-Birkhoff problem. The solution to a specific problem using polynomial and non-polynomial local splines of the third order of approximation is considered. Here we discuss the case when the values of the function 𝑢(𝑥) and its derivative 𝑢’(𝑥) are given at the nodes of the grid in an alternative way: … , 𝑢(𝑥𝑗), 𝑢′(𝑥𝑗+1), 𝑢(𝑥𝑗+2), … . Note that when using polynomial and non-polynomial spline approximations, it is possible to obtain acceptable solutions in several interesting cases that are impossible when we use the classical approach. In the case of using local basis splines, many previously unsolvable problems turn out to be solvable. The results of the numerical experiments are presented.

Efficiency high-order discontinuous Galerkin method for low speeds flows with time-derivative preconditioning

A local preconditioning high-order discontinuous Galerkin (DG) method is introduced to enhance the efficiency and accuracy of the density-based approach for low-speed flows. The method is based on the time-derivative preconditioning technique typically employed in finite volume methods. Traditional approximation methods that only precondition low-order derivatives lead to incompatibility of the equations. To extend the preconditioning technique to the DG method, we employ an analytic preconditioning mass matrix derived from the DG formulation of the preconditioned Navier–Stokes equations. Modified numerical flux functions and a self-adaptive local velocity truncation parameter are employed for DG systems with preconditioning. We demonstrate the efficiency and accuracy of the preconditioned DG method using explicit and implicit schemes for steady flows and a dual-time scheme for unsteady flows. The method has been implemented and used to compute a variety of flow problems, including inviscid and laminar flows, utilizing various degrees of polynomial approximations. Computations with and without preconditioning are performed to analyze the influence of spatial discretization on the accuracy and convergence of the DG solutions at low Mach numbers. Numerical results underscore the enhanced accuracy and convergence properties of the proposed method for low-speed flows, indicating significant performance improvements over traditional methods.

Advancing instantaneous detonation prediction of condensed energetic materials based on high-order compressible multiphase fluid dynamics

The instantaneous detonation model (IDM) is widely used in simulating underwater explosions due to its efficiency and ability to ignore the detonation reaction process. In this study, we propose a new IDM to predict the fluid structure in the detonation zone of an RS211 explosive charge. This model is based on high-order solutions provided by the detonation shock dynamics model, where the spatial term is discretized using fifth-order WENO reconstruction in characteristic space and Lax–Friedrichs’s splitting and the temporal terms are discretized using a third-order TVD Runge–Kutta scheme. The interface motion is captured using the level-set method combined with MGFM, and a programmed burn model is provided to describe the generation and propagation of the detonation wave. The self-similarity of detonation wave propagation is validated, and the quantitative calculation formula of the instantaneous detonation model is obtained by averaging or curve fitting the dimensionless results. Consequently, the IDM of the RS211 charge is established using high-order polynomial approximations of the Taylor rarefaction zone and a constant static zone for 1D planar, cylindrical, and spherical RS211 charges. The application of the IDM involves direct mapping from the radial direction to the spatial structured grid for 1D planar, 2D cylindrical, and 3D spherical charges. Numerical results demonstrate that the IDM proposed in this paper shows good accuracy and high computational efficiency.

Local quadratic spectral and covariance matrix estimation

The problem of estimating the spectral density matrix of a multi‐variate time series is revisited with special focus on the frequencies and . Recognizing that the entries of the spectral density matrix at these two boundary points are real‐valued, we propose a new estimator constructed from a local polynomial regression of the real portion of the multi‐variate periodogram. The case is of particular importance, since is associated with the large‐sample covariance matrix of the sample mean; hence, estimating is crucial to conduct any sort of statistical inference on the mean. We explore the properties of the local polynomial estimator through theory and simulations, and discuss an application to inflation and unemployment.

Best one sided approximation of unbounded functions trigonometrical polynomials for one dimensional weighted-space

The aim of studying this research is to find the best one sided trigonometrical approximation of unbounded function of one variable in weighted space. Moreover we estimated the degree of the best one-sided trigonometrical approximation by averaged modulus of smoothness in same space.