Chebyshev Basis Research Articles

An Optimal Linear-combination-of-unitaries-based Quantum Linear System Solver

Solving systems of linear equations is one of the most important primitives in many different areas, including in optimization, simulation, and machine learning. Quantum algorithms for solving linear systems have the potential to provide a quantum advantage for these problems. In this work, we recall the Chebyshev iterative method and the corresponding optimal polynomial approximation of the inverse. We show that the Chebyshev iteration polynomial can be efficiently evaluated both using quantum singular value transformation (QSVT) as well as linear combination of unitaries (LCU). We achieve this by bounding the 1-norm of the coefficients of the polynomial expressed in the Chebyshev basis. This leads to a considerable constant-factor improvement in the runtime of quantum linear system solvers that are based on LCU or QSVT (or, conversely, a several orders of magnitude smaller error with the same runtime/circuit depth).

ANOVA approximation with mixed tensor product basis on scattered points

In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. Using this, we derive the ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results.

Solving Linear Volterra-Fredholm Integro-Differential Equations using Chebyshev-Galerkin with Error Estimation

The semi-analytic method based on the Galerkin technique with Chebyshev basis is presented in this paper to solve the linear integro-differential equations in the Volterra-Fredholm type. To facilitate, the use of the Chebyshev-Galerkin technique, new theorems and lemmas are established. Different numerical examples are introduced to demonstrate how the proposed method has efficacy and is easier to apply for this problem type. Some issues’ domains don’t match the Chebyshev’s domain. So, how to adequate the solution domain is presented. The error estimate is computed to prove the method's applicability and accuracy of the proposed method for any problem, especially in the case of the exact solution is not easy to determine.

Application of Trigonometric Polynomial Fitting Method in Simulating the Spatial Distribution of PM2.5 Concentration in South-Central China

Near-surface PM2.5 estimates remain a global scientific research challenge due to their effect on human fitness and atmospheric environmental quality. However, practical near-surface PM2.5 estimates are impeded by the incomplete monitoring data. In this study, we propose the trigonometric polynomial fitting (TPF) method to estimate near-surface PM2.5 concentrations in south-central China during 2015. We employ 10-fold cross-validation (CV) to assess the reliability of TPF in estimating practical PM2.5 values. When compared to alternative methods such as the orthogonal polynomial fitting (OBF) method based on Chebyshev basis functions, Kriging interpolation, and radial basis function (RBF) interpolation, our results show that utilizing TPF31, with a maximum order of 3 in the x direction and a maximum order of 1 in the y direction, leads to superior efficiency through error minimization. TPF31 reduces MAE and RMSE by 1.93%, 24%, 6.96% and 3.6%, 23.07%, 10.43%, respectively, compared to the other three methods. In addition, the TPF31 method effectively reconstructs the spatial distribution of PM2.5 concentrations in the unevenly distributed observation stations of Inner Mongolia and the marginal regions of the study area. The reconstructed spatial distribution is remarkably smooth. Despite the non-uniform distribution of observation stations and the presence of missing data, the TPF31 method demonstrates exceptional effectiveness in accurately capturing the inherent physical attributes of spatial distribution. The theoretical and experimental results emphasize that the TPF method holds significant potential for accurately reconstructing the spatial distribution of PM2.5 in China.

Expansions and Characterizations of Sieved Random Walk Polynomials

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x),$ $n\in\mathbb{N}$ with $(c_n)_{n\in\mathbb{N}}\subseteq(0,1)$. For every $k\in\mathbb{N}$, the $k$-sieved polynomials $(P_n(x;k))_{n\in\mathbb{N}_0}$ arise from the recurrence coefficients $c(n;k):=c_{n/k}$ if $k|n$ and $c(n;k):=1/2$ otherwise. A main objective of this paper is to study expansions in the Chebyshev basis $\{T_n(x)\colon n\in\mathbb{N}_0\}$. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version $\mathrm{D}_k$ of the Askey-Wilson operator $\mathcal{D}_q$. It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative and obtained from $\mathcal{D}_q$ by letting $q$ approach a $k$-th root of unity. However, for $k\geq2$ the new operator $\mathrm{D}_k$ on $\mathbb{R}[x]$ has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for $k$-sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator $\mathrm{A}_k$.

Enabling attractive-repulsive potentials in binary-collision-approximation monte-carlo codes for ion-surface interactions

Binary Collision Approximation (BCA) codes for ion-material interactions, such as SRIM, Tridyn, F-TRIDYN, and SDtrimSP, have historically been limited to screened Coulomb potentials even at low energies due to the difficulty in numerically solving the Distance of Closest Approach (DOCA) problem for attractive-repulsive potentials. Techniques such as direct n-body simulation or modifications to Newton’s method are either prohibitively costly or not guaranteed to work for all potentials. Advanced rootfinding techniques, such as companion matrix solvers, offer a solution. For many attractive-repulsive potentials, however, a companion matrix cannot be used directly, because there is no way to put the associated functions into a monomial basis form. A complementary technique is proxy rootfinding—by finding the best-fit polynomial approximant of a function, the zeros of the approximant can be guaranteed to be close to the zeros of the function. Using the Chebyshev basis and grid offers additional guarantees with regards to the quality of the approximation, the speed of convergence, and the avoidance of Runge’s phenomenon. By finding Chebyshev interpolants and using the Chebyshev-Frobenius companion matrix, the zeros of any real function on a bounded domain can be found. Here we show that using an Adaptive Chebyshev Proxy Rootfinder with Automatic Subdivision (ACPRAS) with appropriate scaling functions, numerical issues presented by attractive-repulsive potentials, including those of scale, can be handled. Using these techniques, we show that it is possible to include any physically reasonable interatomic potential in a BCA code, and to guarantee correctness of the resulting scattering angle calculations.

Parity of polynomial multiplier sequences for the Chebyshev basis

Parity of polynomial multiplier sequences for the Chebyshev basis

Spectral method for the gravitational perturbations of black holes: Schwarzschild background case

We develop a novel technique through spectral decompositions to study the gravitational perturbations of a black hole, without needing to decouple the linearized field equations into master equations and separate their radial and angular dependence. We first spectrally decompose the metric perturbation in a Legendre and Chebyshev basis for the angular and radial sectors respectively, using input from the asymptotic behavior of the perturbation at spatial infinity and at the black hole event horizon. This spectral decomposition allows us to then transform the linearized Einstein equations (a coupled set of partial differential equations) into a linear matrix equation. By solving the linear matrix equation for its generalized eigenvalues, we can estimate the complex quasinormal frequencies of the fundamental mode and various overtones of the gravitational perturbations simultaneously and to high accuracy. We apply this technique to perturbations of a nonspinning, Schwarzschild black hole in general relativity and find the complex quasinormal frequencies of two fundamental modes and their first two overtones. We demonstrate that the technique is robust and accurate, in the Schwarzschild case leading to relative fractional errors of $\leq 10^{-10} - 10^{-8}$ for the fundamental modes, $\leq 10^{-7} - 10^{-6}$ for their first overtones, $\leq 10^{-7} - 10^{-4}$ for their second overtones. This method can be applied to any black hole spacetime, irrespective of its Petrov type, making the numerical technique extremely powerful in the study of black hole ringdown in and outside general relativity.

High-Precision Isogeometric Static Bending Analysis of Functionally Graded Plates Using a New Quasi-3D Spectral Displacement Formulation

A new quasi-three-dimensional (3D) shear deformation theory, called the spectral displacement formulation (SDF), is proposed for high-precision static bending analyses of functionally graded plates. The main idea is to expand unknown displacement fields into Chebyshev series of a unique form in the thickness direction; the truncation numbers are set to be adjustable to meet various application requirements. Specifically, 3D elasticity solutions and traction-free boundary conditions can be approached by increasing the number of Chebyshev bases. The SDF is also an extension of the classical plate theory and naturally avoids the shear locking problem, making it versatile for functionally graded material (FGM) plates of arbitrary thicknesses. The C1 continuity requirement for the discretization of the generalized displacements is conveniently fulfilled by the nonuniform rational B-splines (NURBS)-based isogeometric method. Numerical examples demonstrate the excellent performance of the proposed method for the displacement and stress analyses of functionally graded plates. The high precision and versatility of the present method have manifested its great potential applications in strain-based or stress-based reliability analysis, optimization design, fatigue analysis, and fracture analysis of FGM plates, and other related fields.

A numerical procedure for approximating time fractional nonlinear Burgers–Fisher models and its error analysis

Burger and Fisher diffusion transfer properties and reactions from the characteristics are studied using a non-linear equation. The nonlinear fractional Burgers–Fisher equation (NFB-FE) appears in realistic physical situations such as ultra-slow kinetics, Brownian motion of particles, anomalous diffusion, polymerases of ribonucleic acid and deoxyribonucleic acid, continuous random movement, and formation of wave patterns. The present study focuses on the collocation scheme based on the shifted Chebyshev basis (SCB) and the compact finite difference method to obtain the numerical scheme of the NFB-FE. The simulation model is created in the two steps: Initially, a semi-discrete is formed in a temporal sense, applying a linear approximation with an accuracy order of two. Next, we examine the unconditional stability and the convergence order. In the second stage, the collocation approach based on the SCB of the fourth type is used to discretize the spatial derivative parts and generate the full-discrete scheme.

On Mixed Steps-Collocation Schemes for Nonlinear Fractional Delay Differential Equations

This research deals with the numerical solution of fractional differential equations with delay using the method of steps and shifted Legendre (Chebyshev) collocation method. This article presents a new formula for the fractional derivatives (in the Caputo sense) of shifted Legendre polynomials. With the help of this tool and previous work of the authors, efficient numerical schemes for solving nonlinear continuous fractional delay differential equations are proposed. The proposed schemes transform the nonlinear fractional delay differential equations to a non-delay one by employing the method of steps. Then, the approximate solution is expanded in terms of Legendre (Chebyshev) basis functions. Furthermore, the convergence analysis of the proposed schemes is provided. Several practical model examples are considered to illustrate the efficiency and accuracy of the proposed schemes.

Computation of large-genus solutions of the Korteweg–de Vries equation

We consider the numerical computation of finite-genus solutions of the Korteweg–de Vries equation when the genus is large. Our method applies both to the initial-value problem when spectral data can be computed and to dressing scenarios when spectral data is specified arbitrarily. In order to compute large genus solutions, we employ a weighted Chebyshev basis to solve an associated singular integral equation. We also extend previous work to compute period matrices and the Abel map when the genus is large, maintaining numerical stability. We demonstrate our method on four different classes of solutions. Specifically, we demonstrate dispersive quantization for “box” initial data and demonstrate how a large genus limit can be taken to produce a new class of potentials.

On Kosloff Tal-Ezer least-squares quadrature formulas

In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on quasi-uniform grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes that give well-conditioned and quickly converging interpolatory quadrature rules at the same time. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the combination of the Kosloff Tal-Ezer map and least-squares approximation (KTL) for numerical quadrature: a careful selection of the mapping parameter ensures stability of the scheme, a high accuracy of the approximation and, at the same time, an asymptotically optimal ratio between the degree of the polynomial and the spacing of the grid. We will investigate the properties of this KTL quadrature and focus on the symmetry of the quadrature weights, the limit relations for the mapping parameter, as well as the computation of the quadrature weights in the standard monomial and in the Chebyshev bases with help of a cosine transform. Numerical tests on equispaced nodes show that a static choice of the map’s parameter improve the results of the composite trapezoidal rule, while a dynamic approach achieves larger stability and faster convergence, even when the sampling nodes are perturbed. From a computational point of view the proposed method is practical and can be implemented in a simple and efficient way.

Simulating and Pricing CAT Bonds Using the Spectral Method Based on Chebyshev Basis

Simulating and Pricing CAT Bonds Using the Spectral Method Based on Chebyshev Basis

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

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

GPUMD: A package for constructing accurate machine-learned potentials and performing highly efficient atomistic simulations.

We present our latest advancements of machine-learned potentials (MLPs) based on the neuroevolution potential (NEP) framework introduced in Fan et al. [Phys. Rev. B 104, 104309 (2021)] and their implementation in the open-source package gpumd. We increase the accuracy of NEP models both by improving the radial functions in the atomic-environment descriptor using a linear combination of Chebyshev basis functions and by extending the angular descriptor with some four-body and five-body contributions as in the atomic cluster expansion approach. We also detail our efficient implementation of the NEP approach in graphics processing units as well as our workflow for the construction of NEP models and demonstrate their application in large-scale atomistic simulations. By comparing to state-of-the-art MLPs, we show that the NEP approach not only achieves above-average accuracy but also is far more computationally efficient. These results demonstrate that the gpumd package is a promising tool for solving challenging problems requiring highly accurate, large-scale atomistic simulations. To enable the construction of MLPs using a minimal training set, we propose an active-learning scheme based on the latent space of a pre-trained NEP model. Finally, we introduce three separate Python packages, viz., gpyumd, calorine, and pynep, that enable the integration of gpumd into Python workflows.

Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases

When one solves differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the “Chebyshev difference basis” ςn(x) ≡ Tn+2(x)−Tn(x) with coefficients here denoted by bn and the “quadratic factor basis” ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundary, then the coefficients an decrease proportionally to \({\cal O}\left( {A\left( n \right)/{n^\kappa }} \right)\) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as \({\cal O}\left( {A\left( n \right)/{n^{\kappa - 2}}} \right)\). The error for the unconstrained Chebyshev series, truncated at degree n = N, is \({\cal O}\left( {\left| {A\left( N \right)} \right|/{N^\kappa }} \right)\) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms in interpolation, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x. Meanwhile, for Chebyshev polynomials, the values of their derivatives at the endpoints are \({\cal O}\left( {{N^2}} \right)\), but only \({\cal O}\left( N \right)\) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing; actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is under the same basis, and when involving different bases, it may not be established yet.

Optimal control of linear fractional-order delay systems with a piecewise constant order based on a generalized fractional Chebyshev basis

This research studies optimal control of a new subclass of variable-order fractional delay systems whose its order is a piecewise constant function. This category of systems has not been discussed in the literature yet. An effective methodology based on a generalization of the fractional-order Chebyshev functions is offered for providing a solution with high level of precision. A detailed consideration regarding the convergence of the new framework is furnished. Moreover, two important estimates connected to the best approximation of the mentioned fractional basis in the Sobolev space and Hilbert space are achieved. Because direct implementation of the Riemann-Liouville integral operator (RLIP) leads to probably some serious drawbacks, such as numerical challenges, instability and unexpected oscillatory behaviour of the system under examination, a key integral operator connected to the basis under consideration is attained. The capacity and capability of the suggested numerical scheme are illustrated and verified through our numerical findings.

Non-hyperbolicity at large scales of a high-dimensional chaotic system

The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively uniformly hyperbolic, which implies many felicitous statistical properties. We obtain direct and reliable numerical evidence, contrary to the chaotic hypothesis, of the existence of non-hyperbolic large-scale dynamical structures in a mean-field coupled system. To do this, we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev basis transfer operator discretization. This enables us to obtain a high-precision estimate of a homoclinic tangency, implying a failure of uniform hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be a priori assumed to hold in all systems, and a better understanding of the domain of its validity is required.

Analytical expressions for electrodynamic parameters of the shielded microstrip line

On the basis of an electrodynamic model of a screened microstrip line, built on the basis of the projection method using the Chebyshev basis, which explicitly takes into account the edge features of the field, a mathematical model of a microstrip line with a strip conductor was developed. The line width does not exceed the height of the substrate. In this case, the current density on the strip conductor is approximated by only one basis function. Analytical expressions are presented in the form of a sum of slowly and rapidly converging series to determine the main electrodynamic parameters of the line – wave resistance and deceleration coefficient. Due to logarithmic features, slowly converging series are summed up and transformed into rapidly converging power series. In addition, limit expressions in the form of improper integrals are given for the main electrodynamic parameters of an open microstrip line in the quasi-static approximation. Due to the logarithmic features, these integrals are also converted to rapidly converging power series. As a result, simple approximate formulas were obtained. They allow calculating the deceleration coefficient and wave impedance of the line with an error not exceeding 1%, when the width of the strip conductor is less than twice the thickness of the substrate. The results of calculating the electrodynamic parameters obtained on the basis of the developed mathematical model and on the basis of the projection method with an accuracy of up to 5 significant digits are presented. These results make it possible to establish the limits of applicability of the quasi-static approximation and to determine the error in calculating the deceleration coefficient and wave resistance using the obtained analytical expressions. The error does not exceed 0.1%, if the width of the strip conductor is less than twice the thickness of the substrate in a wide range of changes in the substrate dielectric constant and frequency.