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We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production. • A sequence of sub-problems is constructed by linearizing the dynamic system and constraints. • Each sub-problem can be transformed into an equivalent nonlinear optimization problem with linear equality constraints through Chebyshev polynomials. • Any feasible solution of the approximate sub-problem will satisfy its dynamic system and constraints over the entire time horizon. • Global convergence of the proposed method is established. [ABSTRACT FROM AUTHOR]

In this work, the dynamic responses of an uncertain flexible shaft-disk-drum rotor under elastic support and connection are investigated, where uncertain parameters are considered as unknown but bounded interval variables. The energy method is used to derive deterministic analytical model for the flexible shaft-disk-drum rotor, which is validated by modal tests. A non-intrusive interval method based on Chebyshev polynomial approximation is introduced to evaluate the uncertain dynamic response of the flexible shaft-disk-drum rotor system. Comparative studies of Monte Carlo simulation, scanning method and the interval method are conducted to illustrate the validity and accuracy of the approach proposed here. [ABSTRACT FROM AUTHOR]

In many situations, exact solutions to complex problems may be challenging or impossible to obtain, making approximation techniques necessary for making informed decisions. Since function implementation on FPGAs can be difficult and resource-consuming, therefore it would be a better idea to approximate them. In this study, the Chebyshev approximation technique is thoroughly investigated, and an FPGA implementation is proposed and analyzed. This implementation will be compared to other implementations of approximation techniques with parameters such as accuracy, speed, and design size taken into consideration. Applications of this FPGA design are also discussed and shown such as approximating the sigmoid function for machine learning in an efficient manner. This study proved the adequacy of the Chebyshev approximation and its accelerated FPGA implementation for various applications including machine learning, filter design, signal processing, and many other practical applications in engineering and science. [ABSTRACT FROM AUTHOR]

Tensor robust principal component analysis (TRPCA) is a useful approach for obtaining low-rank data corrupted by noise or outliers. However, existing TRPCA methods face certain challenges when it comes to estimating the tensor rank and the sparsity accurately. The commonly used tensor nuclear norm (TNN) may lead to sub-optimal solutions due to the gap between TNN and the tensor rank. Additionally, the ℓ 1 -norm is not an ideal estimation of the ℓ 0 -norm, and solving TNN minimization can be computationally intensive because of the tensor singular value thresholding (t-SVT) scheme. To address these issues, a method called fast correction TNN (FC-TNN) is proposed for TRPCA. In contrast to existing methods, FC-TNN introduces a correction term to bridge the gap between TNN and the tensor rank. Furthermore, a new correction term is employed for the ℓ 1 -norm to achieve the desired solution. To improve computational efficiency, the Chebyshev polynomial approximation (CPA) technique is presented for computing t-SVT without requiring tensor singular value decomposition (t-SVD). The CPA technique is incorporated into the alternating direction method of multipliers (ADMM) algorithm to solve the proposed model effectively. Theoretical analysis demonstrates that FC-TNN offers a lower error bound compared to TNN under certain conditions. Extensive experiments conducted on various tensor-based datasets illustrate that the proposed method outperforms several state-of-the-art methods. • We consider the corrected tensor nuclear norm and l1 norm to solve the tensor recovery problem. • The proposed model is a more general framework, including several current methods as special cases. • A fast ADMM algorithm with chebyshev polynomial approximation technique is proposed. • Numerical experiments conducted on real-word datasets demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this study, we propose a moving least squares approximation with shifted Chebyshev polynomials to solve linear and nonlinear third-kind Volterra delay integral equations (VDIEs). The suggested approach does not use meshing and does not rely on the geometry of the domain; therefore, we may consider it as a meshless method. This method approximates the solution using the collocation method based on the moving least squares approximation. The formulation of the technique for the suggested equations is described, and its convergence is analysed. Numerical results are presented to demonstrate the high resolution of the proposed approach and confirm its capability to provide accurate and efficient computations for Volterra delay integral equations of the third kind. [ABSTRACT FROM AUTHOR]

In this study, we investigate and analyze an approximation of the Chebyshev polynomials for pantograph‐type fractional‐order differential equations. First, we construct the operational matrices of pantograph and Caputo fractional derivatives using Chebyshev interpolation. Then, the obtained matrices are utilized to approximate the fractional derivative. We also provide a detailed convergence analysis in terms of the weighted square norm. Finally, we describe and discuss the results of three numerical experiments conducted to confirm the applicability and accuracy of the computational scheme. [ABSTRACT FROM AUTHOR]

The best polynomial approximation and the Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as better than the Chebyshev approximation, because it is usually considered in the uniform norm. However, it is not always superior to the latter, as noticed by Trefethen (Math. Today 47:184–188, 2011) for the function f (x) = | x - 0.25 | . Recently, Wang (arxiv, 2106.03456, 2021) observed a similar phenomenon for a function with an algebraic singularity and proved it in theory. In this paper, we find that for functions with logarithmic endpoint singularities, the pointwise errors of the Chebyshev approximation are smaller than those of the best approximations of the same degree, except only at the narrow boundary layer. The pointwise error for the Chebyshev series truncated at the degree n is O (n - κ) ( κ = min { 2 γ + 1 , 2 δ + 1 } ), but increases as O (n - (κ - 1)) at the endpoint singularities. Theorems are given to explain this effect. [ABSTRACT FROM AUTHOR]

Structures with inhomogeneous materials, non-uniform cross-sections, non-uniform supports, and subject to non-uniform loads are increasingly common in aerospace applications. This paper presents a simple and unified numerical dynamics model for all beams with arbitrarily axially varying cross-sections, materials, foundations, loads, and general boundary conditions. These spatially varying properties are all approximated by high-order Chebyshev expansions, and discretized by Gauss–Lobatto sampling. The discrete governing equation of non-uniform axially functionally graded beams resting on variable Winkler–Pasternak foundations subjected to non-uniformly distributed loads is derived based on the Euler–Bernoulli beam theory. A projection matrix method is employed to simultaneously assemble spectral elements and impose general boundary conditions. Numerical experiments are performed to validate the proposed method, considering different inhomogeneous materials, boundary conditions, foundations, cross-sections, and loads. The results are compared with those reported in the literature and obtained by the finite element method, and excellent agreement is observed. The convergence, accuracy, and efficiency of the proposed method are demonstrated. [ABSTRACT FROM AUTHOR]

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The structure and dynamics of confined suspensions of particles of arbitrary shape are of interest in multiple disciplines from biology to engineering. Theoretical studies are often limited by the complexity of long-range particle–particle and particle–wall forces, including many-body fluctuating hydrodynamic interactions. Here, we report a computational study on the diffusion of spherical and cylindrical particles confined in a spherical cavity. We rely on an immersed-boundary general geometry Ewald-like method to capture lubrication and long-range hydrodynamics and include appropriate non-slip conditions at the confining walls. A Chebyshev polynomial approximation is used to satisfy the fluctuation–dissipation theorem for the Brownian suspension. We explore how lubrication, long-range hydrodynamics, particle volume fraction, and shape affect the equilibrium structure and the diffusion of the particles. It is found that once the particle volume fraction is greater than 10%, the particles start to form layered aggregates that greatly influence particle dynamics. Hydrodynamic interactions strongly influence the particle diffusion by inducing spatially dependent short-time diffusion coefficients, stronger wall effects on the particle diffusion toward the walls, and a sub-diffusive regime—caused by crowding—in the long-time particle mobility. The level of asymmetry of the cylindrical particles considered here is enough to induce an orientational order in the layered structure, decreasing the diffusion rate and facilitating a transition to the crowded mobility regime at low particle concentrations. Our results offer fundamental insights into the diffusion and distribution of globular and fibrillar proteins inside cells. [ABSTRACT FROM AUTHOR]

The method for constructing the Chebyshev approximation of multivariable functions by a logarithmic expression with an absolute error is proposed. It implies constructing an intermediate Chebyshev approximation of the values of the exponent of an approximated function by a polynomial with the relative error. The construction of the Chebyshev approximation by a polynomial is based on calculating the boundary mean-power approximation by an iterative scheme based on the least squares method with properly formed values of variable weight function. The presented results of test examples' solving confirm the fast convergence of the method in calculating the parameters of the Chebyshev approximation of the functions of one, two, and three variables by the logarithmic expression. [ABSTRACT FROM AUTHOR]

The tracking of magnetic field lines can be very expensive, in terms of computational burden, when the field sources are numerous and have complex geometries, especially when accuracy is a priority, because an evaluation of the field is required in many situations. In some important applications, the computational cost can be significantly reduced by using a suitable approximation of the field in the integrated regions. This paper shows how Chebyshev polynomials are well-suited for field interpolation in magnetic field-line tracking, then discusses the conditions in which they are most appropriate, and quantifies the effectiveness of parallel computing in the approximation procedures. [ABSTRACT FROM AUTHOR]

The method for calculating the Poiseuille number as a function of the rarefaction parameter, the ratio of radii of the cylinders that form a long annular channel, and the accommodation coefficients of the tangential impulse of gas molecules on channel walls is proposed. These quantities are obtained using the Chebyshev polynomial approximation in the intermediate flow regime on the basis of the Shakhov linearized kinetic equation with the Maxwell specular-diffuse boundary conditions. An analysis of the results obtained in the neighborhood of the hydrodynamic gas flow regime with slip is given and the results obtained are compared with similar results found within the framework of BGK model. [ABSTRACT FROM AUTHOR]

The nonlinear piezoelectric vibration energy harvesting system offers a solution to the power supply challenge for low-power sensors. However, In practical engineering, structural parameter uncertainties arise from manufacturing errors, environmental temperature fluctuations, and changes in material properties. Besides, due to inherent flaws in the structure and materials, asymmetry may occur in the potential well, which leads to the complexity of the dynamic behavior exhibited in energy harvesting system. In this paper, in order to investigate the dynamics of the asymmetric monostable piezoelectric energy harvesting system under an uncertain parameter, the orthogonal polynomial approximation method is applied to transform the stochastic system into an equivalent deterministic system. Then, the responses of the equivalent deterministic system are used to reveal the stochastic behavior of the uncertainty system. The results indicate that, when the uncertainty intensity of the cubic nonlinear coefficient is 0, an increase in the asymmetry level of the potential well leads to an increase in the output voltage. Increasing the other parameters, the output voltage is reduced. In addition, as the uncertainty intensity increases, the parameter intervals of chaotic response become wider and the energy harvesting efficiency decreases. Moreover, as the potential well asymmetry increases, the sensitivity of energy harvesting system to uncertain parameter increases. • The dynamic behaviors of stochastic asymmetric monostable energy harvester are studied. • The Chebyshev polynomial approximation is applied to transform the stochastic system into a high-dimensional equivalent deterministic system. • The influences of the uncertain parameter on system responses are revealed. • The results show that the uncertainty of parameter can make the output voltage unstable. [ABSTRACT FROM AUTHOR]

The Fuzzy transform is applied mainly to 1-D signals and 2-D data organized as a regular grid (e.g., 2-D images), thus, limiting its potential application to arbitrary data in terms of dimensionality and structure. This article defines and analyzes the properties of the data-driven F-transform, with a focus on the construction of the class of data-driven membership functions, which are multiscale, local, linearly independent, intrinsic, and robust to data discretization. Data-driven membership functions are defined by applying a 1-D filter to the Laplace–Beltrami operator, which encodes the geometric and topological properties of the input data. Then, we address the efficient computation of the data-driven F-transform through a polynomial or a rational polynomial approximation of the input filter. In this way, the computation of the data-driven F-transform is independent of the evaluation of the membership functions at any point of the input domain and reduces to the solution of a small set of sparse and symmetric linear systems. Finally, the data-driven F-transform is efficiently evaluated on large and arbitrary data, in terms of dimensionality, structure, and size. [ABSTRACT FROM AUTHOR]

We study the problem of off-policy evaluation in the multi-armed bandit model with bounded rewards, and develop minimax rate-optimal procedures under three settings. First, when the behavior policy is known, we show that the Switch estimator, a method that alternates between the plug-in and importance sampling estimators, is minimax rate-optimal for all sample sizes. Second, when the behavior policy is unknown, we analyze performance in terms of the competitive ratio, thereby revealing a fundamental gap between the settings of known and unknown behavior policies. When the behavior policy is unknown, any estimator must have mean-squared error larger—relative to the oracle estimator equipped with the knowledge of the behavior policy— by a multiplicative factor proportional to the support size of the target policy. Moreover, we demonstrate that the plug-in approach achieves this worst-case competitive ratio up to a logarithmic factor. Third, we initiate the study of the partial knowledge setting in which it is assumed that the minimum probability taken by the behavior policy is known. We show that the plug-in estimator is optimal for relatively large values of the minimum probability, but is sub-optimal when the minimum probability is low. In order to remedy this gap, we propose a new estimator based on approximation by Chebyshev polynomials that provably achieves the optimal estimation error. Numerical experiments on both simulated and real data corroborate our theoretical findings. [ABSTRACT FROM AUTHOR]

Approximations based on Chebyshev polynomials have several astrodynamic applications. The performance of these approximations can be improved by parallel implementations exploiting parallel architectures, such as OpenMP and CUDA. In this paper, we introduce the parallel implementation to two astrodynamic applications. The first is the gravitational finite element model (FEM): a piecewise Chebyshev approximation that replaces high degree and order gravitational spherical harmonic models (SHMs). Thus, much lower degree, locally valid functions can efficiently model and compute local gravity perturbations in parallel structure for efficient performance. For this model, the total gravity acceleration is split into a reference and disturbance term. The reference includes two-body plus J 2 , which are relatively cheap to compute. The FEM approximates the higher-order gravity terms. It is developed from a 2D mesh grid covering a sphere of a specified radius, and a family of spherical shells is sampled using a cosine distribution in the radial direction. To reduce the required memory when seeking a specific accuracy, an adaptive version of the gravitational FEM is introduced. In addition, a parallel implementation of the FEM using OpenMP is preseneted. We show the runtime comparison for the 200 degree × 200 order EGM2008 SHM and the serial and parallel equivalent FEM algorithms. The other application is the Chebyshev-Picard method (CPM): a numerical integrator that solves an ordinary differential equation by approximating the integrand using a Chebyshev approximant and iterates over the trajectory via Picard iteration. A parallel CUDA implementation of the CPM method in conjunction with the EGM2008 SHM and the FEM is introduced. We present numerical examples for propagating four Earth-orbiting satellites considering both the 200 × 200 EGM2008 SHM and the equivalent FEM representation to test the algorithm's performance via parallel and serial computation (i.e., a single CPU thread). [ABSTRACT FROM AUTHOR]

Over the last decades, there has been an increasing interest in dedicated preclinical imaging modalities for research in biomedicine. Especially in the case of positron emission tomography (PET), reconstructed images provide useful information of the morphology and function of an internal organ. PET data, stored as sinograms, involve the Radon transform of the image under investigation. The analytical approach to PET image reconstruction incorporates the derivative of the Hilbert transform of the sinogram. In this direction, in the present work we present a novel numerical algorithm for the inversion of the Radon transform based on Chebyshev polynomials of the first kind. By employing these polynomials, the computation of the derivative of the Hilbert transform of the sinogram is significantly simplified. Extending the mathematical setting of previous research based on Chebyshev polynomials, we are able to efficiently apply our new Chebyshev inversion scheme for the case of analytic preclinical PET image reconstruction. We evaluated our reconstruction algorithm on projection data from a small-animal image quality (IQ) simulated phantom study, in accordance with the NEMA NU 4-2008 standards protocol. In particular, we quantified our reconstructions via the image quality metrics of percentage standard deviation, recovery coefficient, and spill-over ratio. The projection data employed were acquired for three different Poisson noise levels: 100% (NL1), 50% (NL2), and 20% (NL3) of the total counts, respectively. In the uniform region of the IQ phantom, Chebyshev reconstructions were consistently improved over filtered backprojection (FBP), in terms of percentage standard deviation (up to 29% lower, depending on the noise level). For all rods, we measured the contrast-to-noise-ratio, indicating an improvement of up to 68% depending on the noise level. In order to compare our reconstruction method with FBP, at equal noise levels, plots of recovery coefficient and spill-over ratio as functions of the percentage standard deviation were generated, after smoothing the NL3 reconstructions with three different Gaussian filters. When post-smoothing was applied, Chebyshev demonstrated recovery coefficient values up to 14% and 42% higher, for rods 1–3 mm and 4–5 mm, respectively, compared to FBP, depending on the smoothing sigma values. Our results indicate that our Chebyshev-based analytic reconstruction method may provide PET reconstructions that are comparable to FBP, thus yielding a good alternative to standard analytic preclinical PET reconstruction methods. [ABSTRACT FROM AUTHOR]

In this paper the effect of stratification on vertical variation of sediment concentration in the suspended-load layer in turbulent flows through open channels under unsteady and nonequilibrium condition is investigated. Unlike previous studies, the model includes the effect of stratification in terms of reduction of the eddy diffusivity. A new approach for solving the governing advection-diffusion equation using orthogonal shifted Chebyshev polynomials of fourth kind is presented. In this approach, the solution of the problem is obtained using a finite degree Chebyshev polynomial. The distribution of the sediment diffusivity is considered including the effect of stratification. Three different types (constant, linear and parabolic) of sediment diffusivity distribution is considered both in neutral and stratified flow condition. The final transport equation is solved under both neutral and stratified condition and appropriate bottom boundary condition using the conventional backward Euler scheme. The method is unconditionally convergent and converges more rapidly than previous methods as Laplace transformation and generalized integral transformation technique. The solutions are compared with previous methods and satisfactory results are obtained. The proposed solutions are also validated with experimental data under steady flow conditions. Results show that under stratified flow condition, sediment concentration also decreases when flow is still unsteady. [ABSTRACT FROM AUTHOR]

A rectangular-window sidelobe magnitude reduction method by widening the main lobe width was proposed by Saramäki. A technique for trading off main lobe width against sidelobe magnitude for any arbitrary window was reported by Lim et al.; a fast convergence algorithm for its implementation was proposed by the same authors in another article, where the derivatives of the window function are expressed in Chebyshev polynomials which have high arithmetic complexity. All the coefficients of a rectangular-window are equal; this special property is exploited, in this paper, for deriving the window function’s derivatives without the use of Chebyshev polynomials resulting in a great reduction in the arithmetic complexity. [ABSTRACT FROM AUTHOR]

In the stage of modelling, measuring, mechanical processing and manufacturing of the nonlinear energy harvesting system, deviations and errors of system parameters are inevitable. Even slight variation of key parameters may have a significant influence on the output voltages, especially for the multi-stable nonlinear case. Therefore, the investigation of dynamic behaviors for the tristable energy harvesting system with uncertain parameters is of important value both for research and application. In this paper, the uncertainty of a tristable piezoelectric vibration energy harvester with a random coefficient ahead of the nonlinear term is studied. By using the Chebyshev polynomial approximation, this tristable energy harvesting system is first reduced into an equivalent deterministic form, the ensemble mean responses of which are derived to exhibit the stochastic behaviors. The periodic and chaotic motions, bifurcations and crises under different conditions are analyzed. The results show that the output voltage is sensitive to the uncertainty of the nonlinear coefficient, which leads to unstable behavior around the bifurcation and crisis points particularly. Exploring the influence pattern of uncertain parameters on the output voltage and avoiding the unstable parameter intervals are essential for optimizing the structure. It can further improve the efficiency of the nonlinear energy harvesting system. [ABSTRACT FROM AUTHOR]

The generalized Chebyshev (pseudoelliptical) filter functions may not be optimal for certain applications due to physical constrains. With redundancy in filter polynomials, this article breaks down the “mental barrier” in conventional synthesis and directly synthesizes the “Chebyshev-like” functions to achieve desired filter performance under physical limitations. By exploiting the redundant restrictions on exact equal-ripple response, the reflection zeros can be moved to the complex plane, thus forming the “Chebyshev-like functions.” A class of inline filters with a block of second-order dangling resonators is synthesized, which can generate a pair of symmetric transmission zeros (Tzs) but is considered unrealizable using standard Chebyshev functions. With the proposed theories, an accurate and much improved filter prototype is directly derived from the “Chebyshev-like” polynomials, avoiding additional optimization in the design process. For verification, a group of examples with synthesis results are elucidated. Eight ten-pole filters with four Tzs are synthesized and fabricated. The simulated and measured results demonstrate the effectiveness and application of the proposed theories. [ABSTRACT FROM AUTHOR]

This article introduces a new method for discretizing and solving integral equation formulations of Maxwell’s equations, which achieves spectral accuracy for smooth surfaces. The approach is based on a hybrid Nyström-collocation method using Chebyshev polynomials to expand the unknown current densities over curvilinear quadrilateral surface patches. As an example, the proposed strategy is applied to the magnetic field integral equation (MFIE) and the N-Müller formulation for scattering from metallic and dielectric objects, respectively. The convergence is studied for several different geometries, including spheres, cubes, and complex NURBS geometries imported from CAD software, and the results are compared against a commercial Method-of-Moments solver using RWG basis functions. [ABSTRACT FROM AUTHOR]

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra-Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example. [ABSTRACT FROM AUTHOR]

We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case. [ABSTRACT FROM AUTHOR]

A stochastic dynamic model of a Kaplan turbine is established in this paper during the transient process. When the Kaplan turbine operates with fluctuating load, the synergistic relationship between the guide vanes and blades experiences random fluctuation resulting from the mechanical, hydraulic and signal factors. To study the effect of stochastic fluctuations of the synergistic relationship, Chebyshev polynomial approximation method is adopted to analyze the stochastic dynamic characteristics of the Kaplan turbine during the transient process. Using Chebyshev polynomial approximation, the stochastic model of the Kaplan turbine is simplified to its equivalent deterministic model, and the stochastic dynamic characteristics of the model are investigated in the transient process. The effects of stochastic intensity on the dynamic behaviors of the Kaplan turbine are analyzed by means of numerical simulation. Moreover, the influences of PID parameters on the stochastic dynamic characteristics of the Kaplan turbine are studied through bifurcation diagrams. Analysis of stochastic characteristics and dynamic behaviors suggests that transient performance improvement can be obtained by controlling the synergistic stochastic intensity and PID parameters. [ABSTRACT FROM AUTHOR]

We study the numerical stability of polynomial based encoding methods, which has emerged to be a powerful class of techniques for providing straggler and fault tolerance in the area of coded computing. Our contributions are as follows: 1)We construct new codes for matrix multiplication that achieve the same fault/straggler tolerance as the previously constructed MatDot Codes and Polynomial Codes. 2)We show that the condition number of every m × m sub-matrix of an m × n, n ≥ m Chebyshev-Vandermonde matrix, evaluated on the n-point Chebyshev grid, grows as O(n2(n-m) for n > m. 3)By specializing our orthogonal polynomial based constructions to Chebyshev polynomials, and using our condition number bound for Chebyshev-Vandermonde matrices, we construct new numerically stable techniques for coded matrix multiplication. We empirically demonstrate that our constructions have significantly lower numerical errors compared to previous approaches which involve inversion of Vandermonde matrices. We generalize our constructions to explore the trade-off between computation/communication and fault-tolerance. 4)We propose a numerically stable specialization of Lagrange coded computing. Our approach involves the choice of evaluation points and a suitable decoding procedure. Our approach is demonstrated empirically to have lower numerical errors as compared to standard methods. [ABSTRACT FROM AUTHOR]

A balanced substrate-integrated waveguide (SIW) bandpass filter (BPF) is designed and fabricated based on through-quartz-via (TQV) technologies in this article. Sixth-order filtering response centered at 41.4 GHz with equal source and load termination is achieved by the transformation of the Chebyshev polynomial. By exploiting cross-coupled resonators, the 3-dB fractional bandwidth (FBW) achieves 12.1%, and the out-of-band rejection is better than 30 dB with frequencies up to $2f_{0}$. High CM suppression can be achieved up to $2.2f_{0}$ with the help of two slots on the bottom metal layer. Based on the theory of the proposed SIW BPF, asymmetric diplexer with different resonance orders for two channels is studied with high isolation and wide stopband. The mixed dual-mode (TE101 and TE301) and single-mode (TE101) junction resonators provide a transition between the input port and two channels. Diplexer with lower and upper channels centered at 20.2 and 37.5 GHz, respectively, is fabricated and measured to verify the proposed structure, which exhibits high isolation better than 40 dB over the whole band and a wide stopband up to $3f_{0}$ of the lower channel. Measured results are in good agreement with the simulations. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to develop an efficient numerical method for mid-frequency analysis of built-up structures with large convex uncertainties. Design/methodology/approach: Based on the Chebyshev polynomial approximation technique, a Chebyshev convex method (CCM) combined with the hybrid finite element/statistical energy analysis (FE-SEA) framework is proposed to fulfil the purpose. In CCM, the Chebyshev polynomials for approximating the response functions of built-up structures are constructed over the uncertain domain by using the marginal intervals of convex parameters; the bounds of the response functions are calculated by applying the convex Monte–Carlo simulation to the approximate functions. A relative improvement method is introduced to evaluate the truncated order of CCM. Findings: CCM has an advantage in accuracy over CPM when the considered order is the same. Furthermore, it is readily to consider the CCM with the higher order terms of the Chebyshev polynomials for handling the larger convex parametric uncertainty, and the truncated order can be effectively evaluated by the relative improvement method. Originality/value: The proposed CCM combined with FE-SEA is the first endeavor to efficiently handling large convex uncertainty in mid-frequency vibro-acoustic analysis of built-up structures. It also has the potential to serve as a powerful tool for other kinds of system analysis when large convex uncertainty is involved. [ABSTRACT FROM AUTHOR]

For robust design optimization (RDO) of electrical machines, cases with random uncertainty and interval uncertainty are generally investigated separately. Accordingly, the performance fluctuation analysis under uncertainty is based on random method and interval method respectively. However, the problem with hybrid uncertainties can also be met yet rarely researched. Under this circumstance, the uncertainty analysis methods for a single type of uncertainty may no longer be applicable, which challenges the robust optimization conduction. For effective RDO of electrical machines with hybrid uncertainties, this article presents a robust optimizer based on evolutionary algorithms and the polynomial chaos Chebyshev interval (PCCI) method. The PCCI method is utilized for effectively modeling the fluctuations caused by the hybrid uncertainty with a small number of samples. As additional enhancements, the filtering strategy for the algorithm with deterministic constraints is proposed to reduce the solutions that require robustness analysis in each iteration and accelerate the optimization further while not affecting the global convergence ability. A design example of a brushless DC motor considering hybrid uncertainties is analyzed and optimized. The results confirm the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

Design space exploration of approximate computing is deemed tough. While restricting the optimization to reduce approximate errors at the individual-function level might simplify the problem to be tackled, the increased hardware cost should be well justified by the improvement of the quality of service (QoS) at the algorithm level. In light of the loose correlation between atomic errors and algorithmic QoS, it is imperative but computationally expensive to explore approximate design space incorporating a variety of alternatives for evaluation. Despite being addressed extensively in the literature, we consider the transcendental sigmoid ($\sigma$) and hyperbolic tangent (tanh) functions as typical examples manifesting the dilemma of approximate computing. In this work, we leverage Pareto-front optimization at three hierarchies, from the parameter layer up to the structure and algorithm layers, to effectively tailor the approximate design space of the $\sigma $ and tanh functions for hardware-efficient as well as algorithm-feasible implementations. Our investigations are performed based on a comprehensive design library consisting of representative approximate schemes in radically different hardware structures featuring both linear and nonlinear approximations. As tested on MNIST with 99% accuracy and Wisconsin Breast Cancer data set with 96.6% accuracy, we identify a novel and compact shift-based approximation that directly applies to the two’s complement numbers achieving a $1.5\times $ area reduction and a $3.4\times $ energy reduction compared with the prior art. Provided the flexibility in approximate functions is of concern, we also present a uniform yet concise structure for implementing the Chebyshev-polynomials-based approximation adaptive in silicon to arbitrary nonlinear functions and error constraints. [ABSTRACT FROM AUTHOR]

Existing light field depth map estimation approaches only utilize partial angular views in occlusion areas and local spatial dependencies in the optimization. This paper proposes a novel two-stage light field depth estimation method via graph spectral analysis to exploit the complete correlations and dependencies within angular patches and spatial images. The initial depth map estimation leverages the undirected graph to jointly consider occluded and unoccluded views within each angular patch. The estimated depth minimizes the structural incoherence of its corresponding angular patch with the focused one by evaluating the highest graph frequency component. Subsequently, depth map refinement optimizes the initial depth map with the color consistency and smoothness formulated by weighted adjacency matrix. The structural constraints are efficiently employed using low-pass graph filtering with Chebyshev polynomial approximation. Experimental results demonstrate that the proposed method improves the depth map estimation, especially in the edge regions. [ABSTRACT FROM AUTHOR]

A public-key cryptosystem using Chebyshev polynomials defined on a finite set has recently been developed, which is a kind of chaos-based cryptography. The security of this cryptosystem relies on the difficulty of finding the degree of Chebyshev polynomials from given parameters. In this brief, we propose polynomial time algorithms to identify the degree of Chebyshev polynomials modulo a prime power. We demonstrate that the cryptosystem based on Chebyshev polynomials modulo a prime power is not secure. This result also means that there are no commutative polynomials for constructing public-key cryptosystems modulo a prime power. [ABSTRACT FROM AUTHOR]

The Savitzky–Golay (SG) filtering is a widely used denoising method employed in different applications. The SG filter has two design parameters: the window length and the filter order. As the window length increases, the estimation variance is reduced, but at the same time, the bias error is increased. In this article, we obtain the optimal window length of an SG filter with an arbitrary order, based on minimizing the mean square error (mse), a well-known performance measure considering both the estimation bias and variance. To achieve the optimal window length, we propose an algorithm the performance of which is much better than the existing methods. In this article, we follow the viewpoint proposed by Persson and Strang and design the filter on the basis of the Chebyshev orthogonal polynomials. [ABSTRACT FROM AUTHOR]

Vehicular ad hoc networks (VANETs) are wireless self-organizing networks, whose communication is conducted through open wireless channels, thus making it vulnerable to various attacks. It is therefore necessary to employ encryption technology to ensure secure communication. In view of the foregoing, key agreement techniques are introduced to VANET communications. The group key agreement (GKA) protocol allows a group of participants to establish a public session key for a secure communication channel over an insecure network. The traditional GKA protocol is inefficient, however, and consumes considerable amounts of resources. With the aim of resolving problems related to information security and privacy preservation in VANETs, this paper proposes a full session key agreement scheme based on chaos mapping. In this paper, the first phase describes the key agreement between the fog server and group administrator; the second phase describes the group key agreement for vehicle nodes in the group where the vehicle can only communicate with vehicles having the same group key without the aid of a trusted key center or centralized key distributor. The proposed scheme is based on Chebyshev chaotic mapping algorithm, in which a chaotic map is utilized to provide a one-way hash; Chebyshev polynomial is also employed to establish a public multi-party key; it avoids the modular multiplication index or scalar multiplication on the elliptic curve. The proposed scheme does not only satisfy the security requirements in VANETs but also performs detailed analysis. [ABSTRACT FROM AUTHOR]

Inertial navigation computation is to acquire the attitude, velocity, and position information of a moving body by integrating inertial measurements from gyroscopes and accelerometers. Over half a century has witnessed great efforts in coping with the motion noncommutativity errors to accurately compute the navigation information as far as possible, so as not to compromise the quality measurements of inertial sensors. Highly dynamic applications and the forthcoming cold-atom precision inertial navigation systems demand for even more accurate inertial navigation computation. The paper gives birth to an inertial navigation algorithm to fulfill that demand, named the iNavFIter, which is based on a brand-new framework of functional iterative integration and Chebyshev polynomials. Remarkably, the proposed iNavFIter reduces the noncommutativity errors to almost machine precision, namely, the coning/sculling/scrolling errors that have perplexed the navigation community for long. Numerical results are provided to demonstrate its accuracy superiority over the state-of-the-art inertial navigation algorithms at affordable computation cost. [ABSTRACT FROM AUTHOR]

Colloidal or nanoparticle mobility under confinement is of central importance for a wide range of physical and biological processes. Here, we introduce a minimal model of particles in a hydrodynamic continuum to examine how particle shape and concentration affect the transport of particles in spherical confinement. Specifically, an immersed boundary-general geometry Ewald-like approach is adopted to simulate the dynamics of spheres and cylinders under the influence of short- and long-range fluctuating hydrodynamic interactions with appropriate non-slip conditions at the confining walls. An efficient O(N) parallel finite element algorithm is used, thereby allowing simulations at high concentrations, while a Chebyshev polynomial approximation is implemented in order to satisfy the fluctuation–dissipation theorem. A concentration-dependent anomalous diffusion is observed for suspended particles. It is found that introducing cylinders in a background of spheres, i.e., particles with a simple degree of anisotropy, has a pronounced influence on the structure and dynamics of the particles. First, increasing the fraction of cylinders induces a particle segregation effect, where spheres are pushed toward the wall and cylinders remain near the center of the cavity. This segregation leads to a lower mobility for the spheres relative to that encountered in a system of pure spheres at the same volume fraction. Second, the diffusive-to-anomalous transition and the degree of anomaly quantified by the power law exponent in the mean square displacement vs time relation both increase as the fraction of cylinders becomes larger. These findings are of relevance for studies of diffusion in the cytoplasm, where proteins exhibit a distribution of size and shapes that could lead to some of the effects identified in the simulations reported here. [ABSTRACT FROM AUTHOR]

The main contribution of this brief is the introduction and the characterization of a novel Chebyshev-like rational map over finite fields. The referred map is identified as $t$ -Chebyshev map and depends on the definition of a finite field tangent function, which is also proposed. Among other new and interesting results, we demonstrate that the semigroup property holds for $t$ -Chebyshev maps and give a necessary and sufficient condition under which one of such maps induces a permutation on the set of elements of a field $\mathbb {F}_{q}$. This makes these maps suitable for practical use in both public- and secret-key cryptography. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to implement an approximate method for obtaining the solution of a physical model called the blood ethanol concentration system. This model can be expressed by a system of fractional differential equations (FDEs). Here, we will consider two forms of the fractional derivative namely, Caputo (with singular kernel) and Atangana–Baleanu–Caputo (ABC) (with nonsingular kernel). In this work, we use the spectral collocation method based on Chebyshev approximations of the third-kind. This procedure converts the given model to a system of algebraic equations. The implementation of the proposed method to solve fractional models in ABC-sense is the first time. We satisfy the efficiency and the accuracy of the given procedure by evaluating the relative errors. The results show that the implemented technique is an easy and efficient tool to simulate the solution of such models. [ABSTRACT FROM AUTHOR]

This paper presents a computationally fast method for solving gravitational accelerations near irregularly-shaped asteroids. This method is based on analytical three-dimensional Chebyshev polynomial approximation of the polyhedral gravity. For the purpose of improving the approximation accuracy, space partitioning schemes based on practical flight zones is used to avoid interpolation the whole space around the target asteroid. Specifically, a minimum ellipsoid close to the asteroid surface is defined to select the space for surrounding trajectories with safe distance and a cone connected to the surface is defined to select the space for descent trajectories. Moreover, interpolation points are sampled in a cosine sampling fashion according to the Chebyshev-Gauss-Lobatto nodes and a radial adaption technique. The performance of different space partitioning schemes is analyzed. The effectiveness of the proposed method is validated through simulations of solving gravitational accelerations at the test points near different shaped asteroids 1996 HW1, 433 Eros, 25143 Itokawa and 101955 Bennu. [ABSTRACT FROM AUTHOR]

In this paper, a high gain cascaded single stage distributed amplifier design method is proposed. This new design method is based on the Chebyshev polynomial approximation of the amplifier transducer gain. By this approximation an improvement of 12 dB in gain, is obtained, compared with the conventional cascaded single stage distributed amplifier with a simplified unilateral single field effect transistor model. This 12 dB are obtained by eliminating two resistors which consume the power; but, by doing this, a mismatch is provoked in the circuit, so a loss of gain is obtained at high frequencies. By the use of the proposed method a stable gain is obtained from dc to the cut-off frequency of the circuit. In addition, a more compact circuit is obtained, by eliminating four components, compared with the cascaded single stage distributed amplifier. [ABSTRACT FROM AUTHOR]

RodFIter is a promising method of attitude reconstruction from inertial measurements based on the functional iterative integration of Rodrigues vector. The Rodrigues vector is used to encode the attitude in place of the popular rotation vector because it has a polynomial-like rate equation and could be cast into theoretically sound and exact integration. This paper further applies the approach of RodFIter to the unity-norm quaternion for attitude reconstruction, named QuatFIter, and shows that it is identical to the previous Picard-type quaternion method. The Chebyshev polynomial approximation and truncation techniques from the RodFIter are exploited to speed up its implementation. Numerical results demonstrate that the QuatFIter is comparable in accuracy to the RodFIter, although its convergence rate is relatively slower with respect to the number of iterations. Notably, the QuatFIter has about two times better computational efficiency, thanks to the linear quaternion kinematic equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville fractional integrals/derivatives for optimal error estimates of Chebyshev polynomial approximations to functions with limited regularity. It naturally arises from exact representations of Chebyshev expansion coefficients. Here, the essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterise in existing literature. Then we can derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve available results in usual Sobolev spaces with integer regularity exponentials in several senses. As a byproduct, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi polynomial approximations. [ABSTRACT FROM AUTHOR]

A detailed comparison of the time-dependent wave packet method using the split operator propagator and recently introduced Chebyshev real wave packet approach for calculating reactive scattering processes is reported. As examples, the state-to-state differential cross sections of the H+H′D(v0=0,j0=1)→H′D+H/H′H+D reaction, the state-to-state reaction probabilities of the 16O+35O2 (v0=0,j0=0)→17O+16O18O/18O+16O17O reaction, the H+O2→O+HO reaction, and the F+HD→HF+D reaction are calculated, using an efficient reactant-coordinate-based method on an L-shape grid which allows the extraction of the state-to-state information of the two product channels simultaneously. These four reactions have quite different dynamic characteristics and thus provide a comprehensive picture of the relative advantages of these two propagation methods for describing reactive scattering dynamics. The results indicate that the Chebyshev real wave packet method is typically more accurate, particularly for reactions dominated by long-lived resonances. However, the split operator approach is often more cost effective, making it a method of choice for fast reactions. In addition, our results demonstrate accuracy of the reactant-coordinate-based method for extracting state-to-state information. [ABSTRACT FROM AUTHOR]

As a measure of vertex importance according to the graph structure, PageRank has been widely applied in various fields. While many PageRank algorithms have been proposed in the past decades, few of them take into account whether the graph under investigation is directed or not. Thus, some important properties of undirected graph—symmetry on edges, for example—is ignored. In this paper, we propose a parallel PageRank algorithm specifically designed for undirected graphs that can fully leverage their symmetry. Formally, our algorithm extends the Chebyshev Polynomial approximation from the field of real function to the field of matrix function. Essentially, it reflects the symmetry on edges of undirected graph and the density of diagonalizable matrix. Theoretical analysis indicates that our algorithm has a higher convergence rate and requires less computation than the Power method, with a 50% higher convergence rate when the damping factor c = 0.85. Experiments on six datasets illustrate that our algorithm with 38 parallelism can be up to 43 times faster than the single-thread Power method. • We propose an approach of computing PageRank via Chebyshev Polynomial approximation. • Based on this approach, we propose a parallel PageRank algorithm and provide an accurate convergence rate. • We conduct numerical experiments to validate our theoretical findings, and the results align with our expectations. [ABSTRACT FROM AUTHOR]

The concept of H -sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and H -sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, H -sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of H -sets to Linear Programming is studied. But on the downside, it is explained why H -sets have a very limited range of applicability in the times of large-scale computing. [ABSTRACT FROM AUTHOR]

Spectral polynomial approximation of smooth functions allows real-time manipulation of and computation with them, as in the Chebfun system. Extension of the technique to two-dimensional and three-dimensional functions on hyperrectangles has mainly focused on low-rank approximation. While this method is very effective for some functions, it is highly anisotropic and unacceptably slow for many functions of potential interest. A method based on automatic recursive domain splitting, with a partition of unity to define the global approximation, is easy to construct and manipulate. Experiments show it to be as fast as existing software for many low-rank functions, and much faster on other examples, even in serial computation. It is also much less sensitive to alignment with coordinate axes. Some steps are also taken toward approximation of functions on nonrectangular domains, by using least-squares polynomial approximations in a manner similar to Fourier extension methods, with promising results. [ABSTRACT FROM AUTHOR]

This paper focuses on the powered-descent guidance (PDG) problem involving uncertain-but-bounded parameters. An efficient numerical algorithm based on the high approximation accuracy of the Chebyshev series expansion is presented to solve the nonlinear optimal control problem with interval uncertain parameters. First, an infinite number of uncertain optimal control problems is translated into a series of deterministic optimal control problems by exploiting the Chebyshev interval inclusion. Subsequently, the deterministic optimal control problem is solved by a convex optimization method based on the lossless convexification of the PDG problem. Afterwards, these techniques are used to generate the enclosure of the PDG trajectory with uncertainty and the upper and lower bounds of the minimum fuel consumption during the pinpoint landing mission. Finally, the effectiveness and superiority of the proposed approach are validated by illustrative numerical simulations. • The proposed model accounts for uncertain mission parameters. • Only the upper and lower bounds of the uncertainty parameters are needed. • The enclosures of the trajectory and fuel consumption are generated. [ABSTRACT FROM AUTHOR]