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This paper presents a novel approach to microwave lowpass filter design that addresses several challenges in modern microwave systems, including size, performance, and cost. With that in mind, we propose using monic Chebyshev polynomials as the characteristic function for the filter to achieve a low passband insertion loss and maximum cutoff slope. The half-power cutoff frequency characterizes these filters, and the ripple factor is not considered a design parameter. To demonstrate the effectiveness of the monic Chebyshev approximation, a comparison is made with chained function (CF) filters designed for microwave applications. It has been shown that the monic Chebyshev filter outperforms CF filters in all applications, including microwaves. [ABSTRACT FROM AUTHOR]

We consider a discrete best approximation problem formulated in the framework of tropical algebra, which deals with the theory and applications of algebraic systems with idempotent operations. Given a set of samples of input and output of an unknown function, the problem is to construct a generalized tropical Puiseux polynomial that best approximates the function in the sense of a tropical distance function. The construction of an approximate polynomial involves the evaluation of both unknown coefficient and exponent of each monomial in the polynomial. To solve the approximation problem, we first reduce the problem to an equation in unknown vector of coefficients, which is given by a matrix with entries parameterized by unknown exponents. We derive a best approximate solution of the equation, which yields both vector of coefficients and approximation error parameterized by the exponents. Optimal values of exponents are found by minimization of the approximation error, which is transformed into minimization of a function of exponents over all partitions of a finite set. We solve this minimization problem in terms of max-plus algebra (where addition is defined as maximum and multiplication as arithmetic addition) by using a computational procedure based on the agglomerative clustering technique. This solution is extended to the minimization problem of finding optimal exponents in the polynomial in terms of max-algebra (where addition is defined as maximum). The results obtained are applied to develop new solutions for conventional problems of discrete best Chebyshev approximation of real functions by piecewise linear functions and piecewise Puiseux polynomials. We discuss computational complexity of the proposed solution and estimate upper bounds on the computational time. We demonstrate examples of approximation problems solved in terms of max-plus and max-algebra, and give graphical illustrations. [ABSTRACT FROM AUTHOR]

The main goal of this study is to solve a class of linear and nonlinear Volterra integral equations (VIEs) of the third kind. The proposed technique estimates the solution of these equations using a technique based on MLS approximation and shifted Chebyshev polynomials. The approach is independent of the geometry of the domain and does not require any background meshes; therefore, it can be considered a meshless approach. The new method is effective and more flexible for many types of VIEs of the third kind, and its algorithm can be simply implemented on computers. The construction of a new technique for the proposed equations has been introduced. The convergence analysis is also studied. The convergence precision of the new approach is checked on classes of VIEs of the third kind, and the results validate the theoretical error estimates. Finally, numerical tests are provided to illustrate the accuracy and reliability of this approach. [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]

We present a hybrid quantum-classical framework for simulating generic matrix functions more amenable to early fault-tolerant quantum hardware than standard quantum singular-value transformations. The method is based on randomization over the Chebyshev approximation of the target function while keeping the matrix oracle quantum, and is assisted by a variant of the Hadamard test that removes the need for post-selection. The resulting statistical overhead is similar to the fully quantum case and does not incur any circuit depth degradation. On the contrary, the average circuit depth is shown to get smaller, yielding equivalent reductions in noise sensitivity, as explicitly shown for depolarizing noise and coherent errors. We apply our technique to partition-function estimation, linear system solvers, and ground-state energy estimation. For these cases, we prove advantages on average depths, including quadratic speed-ups on costly parameters and even the removal of the approximation-error dependence. [ABSTRACT FROM AUTHOR]

This paper explores geometric circulant matrices whose entries are Chebyshev polynomials of the first or second kind. Motivated by our previous research on r − circulant matrices and Chebyshev polynomials, we focus on calculating the Frobenius norm and deriving estimates for the spectral norm bounds of these matrices. Our analysis reveals that this approach yields notably improved results compared to previous methods. To validate the practical significance of our research, we apply it to existing studies on geometric circulant matrices involving the generalized k − Horadam numbers. The obtained results confirm the effectiveness and utility of our proposed approach. [ABSTRACT FROM AUTHOR]

In this study, we designed multi-section UWB Wilkinson power dividers for both equal and unequal power divider. The key innovation is successfully matching these power dividers using the Chebyshev method. The proposed method achieves very low insertion losses and high transmission coefficients across the ports. For the three-section equal power divider, Chebyshev polynomials were used to model the reflection coefficient, and the characteristic impedance for each section was calculated symmetrically. The design achieved a fractional bandwidth of 106% for a 20 dB return loss, with an insertion loss of 0.3 dB. For unequal multi-section dividers, power division ratios of 1.5:1 and 2:1 were chosen. Theoretical microwave calculations were used to determine the input and output transmission line impedances. Chebyshev approximation was then applied to design the three-section unequal power dividers. The return losses for these unequal dividers were measured at 17 dB and 15 dB, with fractional bandwidths of 120% and 126%. Both circuits had insertion losses of 0.25 dB. The consistent results from analytical calculations, simulations, and measurements confirm the effectiveness of the Chebyshev method for both equal and unequal power division. [ABSTRACT FROM AUTHOR]

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]

An ultra‐wideband low‐noise amplifier (LNA) design was developed based on distributed network synthesis to achieve impedance matching of an active device by parasitic absorption. The equivalent device models of a low‐noise device were generated using negative‐image device modeling and traveling‐wave matching, and matching circuits were produced using the Chebyshev approximation. The minimum insertion loss and ripple of the matching networks were adjusted correctly to allow the parasitic absorption to match the device models. This paper describes the design methodology for distributed matching circuits and its application to the design of an ultra‐wideband LNA. [ABSTRACT FROM AUTHOR]

Given samples of a real or complex-valued function on a set of distinct nodes, the traditional linear Chebyshev approximation is to compute the minimax approximation on a prescribed linear functional space. Lawson’s iteration is a classical and well-known method for the task. However, Lawson’s iteration converges only linearly and in many cases, the convergence is very slow. In this paper, relying upon the Lagrange duality, we establish an L q -weighted dual programming for the discrete linear Chebyshev approximation. In this framework of dual problem, we revisit the convergence of Lawson’s iteration and provide a new and self-contained proof for the well-known Alternation Theorem in the real case; moreover, we propose a Newton type iteration, the interior-point method, to solve the L 2 -weighted dual programming. Numerical experiments are reported to demonstrate its fast convergence and its capability in finding the reference points that characterize the unique minimax approximation. [ABSTRACT FROM AUTHOR]

We study the problem of the nonparametric estimation for the density π$$ \pi $$ of the stationary distribution of a d$$ d $$‐dimensional stochastic differential equation (Xt)t∈[0,T]$$ {\left({X}_t\right)}_{t\in \left[0,T\right]} $$. From the continuous observation of the sampling path on [0,T]$$ \left[0,T\right] $$, we study the estimation of π(x)$$ \pi (x) $$ as T$$ T $$ goes to infinity. For d≥2$$ d\ge 2 $$, we characterize the minimax rate for the L2$$ {\mathbf{L}}^2 $$‐risk in pointwise estimation over a class of anisotropic Hölder functions π$$ \pi $$ with regularity β=(β1,…,βd)$$ \beta =\left({\beta}_1,\dots, {\beta}_d\right) $$. For d≥3$$ d\ge 3 $$, our finding is that, having ordered the smoothness such that β1≤⋯≤βd$$ {\beta}_1\le \cdots \le {\beta}_d $$, the minimax rate depends on whether β2<β3$$ {\beta}_2<{\beta}_3 $$ or β2=β3$$ {\beta}_2={\beta}_3 $$. In the first case, this rate is logTTγ$$ {\left(\frac{\log T}{T}\right)}^{\gamma } $$, and in the second case, it is (1T)γ$$ {\left(\frac{1}{T}\right)}^{\gamma } $$, where γ$$ \gamma $$ is an explicit exponent dependent on the dimension and β‾3$$ {\overline{\beta}}_3 $$, the harmonic mean of smoothness over the d$$ d $$ directions after excluding β1$$ {\beta}_1 $$ and β2$$ {\beta}_2 $$, the smallest ones. We also demonstrate that kernel‐based estimators achieve the optimal minimax rate. Furthermore, we propose an adaptive procedure for both L2$$ {L}^2 $$ integrated and pointwise risk. In the two‐dimensional case, we show that kernel density estimators achieve the rate logTT$$ \frac{\log T}{T} $$, which is optimal in the minimax sense. Finally we illustrate the validity of our theoretical findings by proposing numerical results. [ABSTRACT FROM AUTHOR]

Statistical inverse learning aims at recovering an unknown function f from randomly scattered and possibly noisy point evaluations of another function g, connected to f via an illposed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization strategy of applying finite-dimensional projections. Our key finding is that coupling the number of random point evaluations with the choice of projection dimension, one can derive probabilistic convergence rates for the reconstruction error of the maximum likelihood (ML) estimator. Convergence rates in expectation are derived with a ML estimator complemented with a norm-based cutoff operation. Moreover, we prove that the obtained rates are minimax optimal. [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]

The paper proposes the method for constructing Chebyshev approximation of multivariable functions with a relative error using a power expression. It involves building an intermediate Chebyshev approximation of values rooted to the corresponding degree of the approximated function by a polynomial with a relative error. Parameters for the polynomial approximation are computed as the boundary mean-power approximation through an iterative scheme using the least squares method with a variable weight function. The authors provide test examples, confirming the fast convergence of the method for constructing the Chebyshev approximation for the functions of one, two, and three variables using power expression. [ABSTRACT FROM AUTHOR]

In recent years, the problem of traffic congestion has become increasingly serious, and research on traffic system control has become a new hotspot. Studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable support points can alleviate traffic congestion from a new perspective. This article improves the full speed differential model considering strong wind models from the perspective of bifurcation control to adjust traffic flow. This article theoretically proves the existence conditions of Hopf bifurcation and saddle node bifurcation in the model and finds the stability mutation point of the transportation system stability. A nonlinear system feedback controller was designed for unstable bifurcation points using Chebyshev polynomial approximation and random feedback control methods. Without changing the system equilibrium point, the advance, delay, and elimination of Hopf bifurcation were achieved, and the abrupt behavior of the transportation system was controlled, thereby alleviating traffic congestion. This article explains the changes in the stability of complex transportation systems from the perspective of bifurcation analysis, which can better capture the characteristics of traffic flow. By adjusting the control parameters in the feedback controller, the influence of boundary conditions on the stability of the transportation system is fully described, and the influence of unstable focal points and saddle points on the system is suppressed, thereby slowing down the traffic flow. In addition, unstable bifurcation points can be eliminated, and the Hopf bifurcation can be controlled to advance, delay, and disappear, thereby achieving control over the stable behavior of the transportation system. This helps alleviate traffic congestion and also helps describe actual traffic phenomena. [ABSTRACT FROM AUTHOR]

The Chebyshev Rational Approximation Method (CRAM) has become one of the dominant methods for solving the Bateman equations for nuclear fuel depletion analysis. Since its introduction over a decade ago, several improvements have been made to CRAM improving its accuracy and reducing its run time. We analyzed its run time using two previously published methods for solving the CRAM system of equations, direct matrix inversion (DMI) and sparse Gaussian elimination (SGE), for depletion systems of varying numbers of nuclides to see how the two methods perform relative to one another. In addition to these two methods, we introduced the Gauss-Seidel (GS) method for solving the CRAM system of equations and compared its performance relative to DMI and SGE for depletion systems with varying numbers of nuclides. We demonstrated that for practical purposes, GS is faster than SGE and DMI and achieves a practical level of accuracy. All testing was performed using the CRAM implementation in the Griffin reactor physics analysis application. [ABSTRACT FROM AUTHOR]

Purpose: Highway traffic systems are complex and variable, and studying the bifurcation characteristics of traffic flow systems and designing control schemes for unstable bifurcation points can alleviate traffic congestion from a new perspective. Bifurcation analysis is used to explain the changes in system stability, identify the unstable bifurcation points of the system, and design feedback controllers to realize the control of the unstable bifurcation points of the traffic system. It helps to control the sudden changes in the stable behavior of the traffic system and helps to alleviate traffic congestion, which is of great practical significance. Design/methodology/approach: In this paper, we improve the macroscopic traffic flow model by integrating severe weather factors such as rainfall, snowfall, and dust. We use traveling wave transform to convert it into a traffic flow stability model suitable for branching analysis, thus converting the traffic flow problem into a system stability analysis problem. First, this paper derives the existence conditions of the model Hopf bifurcation and saddle-node bifurcation for the improved macroscopic model, and finds the stability mutation point of the system. Secondly, the connection between the stability mutation points and bifurcation points of the traffic system is analyzed. Finally, for the unstable bifurcation point, a nonlinear system feedback controller is designed using Chebyshev polynomial approximation and stochastic feedback control method. Findings: The Hopf bifurcation is delayed and completely eliminated without changing the equilibrium point of the system, thus controlling the abrupt behavior of the traffic system. Originality/value: Currently there are fewer studies to explain the changes in the stability of the transportation system through bifurcation analysis, in this paper; we design a feedback controller for the unstable bifurcation point of the system to realize the control of the transportation system. It is a new research method that helps to control the sudden change of the stable behavior of the traffic system and helps to alleviate traffic congestion, which is of great practical significance. [ABSTRACT FROM AUTHOR]

Recently, max-min fuzzy relation inequalities (FRIs) have been used to model a (peer-topeer) P2P network system. Any feasible scheme in the P2P network system is reflected by a solution of the max-min FRIs. One of the objectives of system managers is to decrease network congestion. To satisfy this objective, we attempt to minimize a weighted minimax function motivated by existing research. As a consequence, we establish a weighted minimax programming model in which the constraint is the max-min FRIs. Our goal in this work is to develop an effective algorithm to obtain the optimal solution of the optimization model. The so-called SCP-based algorithm is proposed to find the optimal solution. A numerical example shows the efficiency of our proposed SCP-based algorithm. [ABSTRACT FROM AUTHOR]

This article presents an active realisation of an electronically controlled FO capacitor or a constant phase element (CPE) and an FO inductor (FOI) in the form of an integrated circuit. The realisation is demonstrated using an OTA-C structure in AMS 0.35 μm C35B4C3 technology. The same core is used for both realisations of CPE and FOI, and the angles can be realised in all four quadrants. The realisation of active constant-phase elements using OTAs with MOS transistors in the saturation region is proposed. The operating frequency is in the high range of 7–350 kHz, with a centre frequency of 50 kHz. A tuning method is proposed using different bias currents of the OTAs, which in turn are digitally controlled to obtain the desired parameters such as impedance and angle of an element. The linearisation of the individual OTAs is achieved by source degeneration. The newly introduced minimax approximation is used to design three non-integer orders of 1/3, 1/2, and 2/3. The integrated circuit was designed with a total area of 710 × 1127 µm2. The power consumption of the entire system is 12.37 mW. [ABSTRACT FROM AUTHOR]

Population games are games with a finite set of available strategies and an infinite number of players, in which the reward for choosing a given strategy is a function of the distribution of players over strategies. The paper shows that, in a certain class of maxmin optimization problems, it is possible to associate a population game to a given maxmin problem in such a way that solutions to the optimization problem are found from Nash equilibria of the associated game. Iterative solution methods for maxmin optimization problems can then be derived from systems of differential equations whose trajectories are known to converge to Nash equilibria. In particular, we use a discrete-time version of the celebrated replicator equation of evolutionary game theory, also known in machine learning as the exponential multiplicative weights algorithm. The resulting algorithm can be viewed as a generalization of the Iteratively Reweighted Least Squares (IRLS) method, which is well known in numerical analysis as a useful technique for solving Chebyshev function approximation problems on a finite grid. Examples are provided to show the use of the generalized IRLS method in collective investment and in decision making under model uncertainty. [ABSTRACT FROM AUTHOR]

In the presented paper, we investigate efficient solutions to optimization problems with multiple criteria and bounded trade-offs. A nonlinear optimization problem to find the relationships between the upper bound for trade-offs and objective functions is presented. Due to this problem, we determine some properly efficient points that are closer to the ideal point. To this end, we apply the extended form of the generalized Tchebycheff norm. Note that all the presented results work for general problems and no convexity assumption is needed. [ABSTRACT FROM AUTHOR]

Outliers are inconsistent values that commonly occur in every experimental setting and damage the designs' properties to be explored. Based on the orthogonal uniform composite designs (OUCDs) proposed by Zhang et al., this article introduces a new class of second-order designs called orthogonal array composite minimax outlying designs by using the minimax outlying criterion that minimizes the maximum outlying effect of all design points. The proposed designs' efficiencies and generalized scaled standard deviations are compared with central composite design, small composite designs, orthogonal array composite designs, orthogonal array composite minimax loss designs, and OUCDs. The proposed designs are more robust to outlier than the existing OUCDs. These new designs are more optimal in terms of D-, E-, G- and T- efficiencies in most cases than the existing composite designs. [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]

We propose a distributed-memory parallel algorithm for computing some of the algebraically smallest eigenvalues (and corresponding eigenvectors) of a large, sparse, real symmetric positive definite matrix pencil that lie within a target interval. The algorithm is based on Chebyshev interpolation of the eigenvalues of the Schur complement (over the interface variables) of a domain decomposition reordering of the pencil and accordingly exposes two dimensions of parallelism: one derived from the reordering and one from the independence of the interpolation nodes. The new method demonstrates excellent parallel scalability, comparing favorably with PARPACK, and does not require factorization of the mass matrix, which significantly reduces memory consumption, especially for 3D problems. Our implementation is publicly available on GitHub. [ABSTRACT FROM AUTHOR]

Path-based clustering algorithms typically generate clusters by optimizing a benchmark function. Most optimization methods in clustering algorithms often offer solutions close to the general optimal value. This study achieves the global optimum value for the criterion function in a shorter time using the minimax distance, Maximum Spanning Tree "MST", and meta-heuristic algorithms, including Genetic Algorithm "GA" and Particle Swarm Optimization "PSO". The Fast Path-based Clustering "FPC" algorithm proposed in this paper can find cluster centers correctly in most datasets and quickly perform clustering operations. The FPC does this operation using MST, the minimax distance, and a new hybrid meta-heuristic algorithm in a few rounds of algorithm iterations. This algorithm can achieve the global optimal value, and the main clustering process of the algorithm has a computational complexity of O (k² × n). However, due to the complexity of the minimum distance algorithm, the total computational complexity is O (n²). Experimental results of FPC on synthetic datasets with arbitrary shapes demonstrate that the algorithm is resistant to noise and outliers and can correctly identify clusters of varying sizes and numbers. In addition, the FPC requires the number of clusters as the only parameter to perform the clustering process. A comparative analysis of FPC and other clustering algorithms in this domain indicates that FPC exhibits superior speed, stability, and performance. [ABSTRACT FROM AUTHOR]

New upper bounds for the weighted Chebyshev functional under various conditions, including those of Steffensen type, are given. The obtained results are used to establish some new bounds for the Jensen functional. [ABSTRACT FROM AUTHOR]

Composite minimization involves a collection of functions which are aggregated in a nonsmooth manner. It covers, as a particular case, smooth approximation of minimax games, minimization of max-type functions, and simple composite minimization problems, where the objective function has a nonsmooth component. We design a higher-order majorization algorithmic framework for fully composite problems (possibly nonconvex). Our framework replaces each component with a higher-order surrogate such that the corresponding error function has a higher-order Lipschitz continuous derivative. We present convergence guarantees for our method for composite optimization problems with (non)convex and (non)smooth objective function. In particular, we prove stationary point convergence guarantees for general nonconvex (possibly nonsmooth) problems and under Kurdyka–Lojasiewicz (KL) property of the objective function we derive improved rates depending on the KL parameter. For convex (possibly nonsmooth) problems we also provide sublinear convergence rates. [ABSTRACT FROM AUTHOR]

We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension of the solution set and show that nonuniqueness is generic for ill-posed problems on discrete domains. Moreover, given a prescribed set of points of minimal and maximal deviation we construct a function for which the dimension of the set of best approximating polynomials is maximal for any choice of domain. We also present several examples that illustrate the aforementioned phenomena, demonstrate practical application of our results and propose a number of open questions. [ABSTRACT FROM AUTHOR]

In this paper, a computational method based on parameterizing state and control variables is presented for solving Stochastic Optimal Control (SOC) problems. By using Chebyshev wavelets with unknown coefficients, state and control variables are parameterized, and then a stochastic optimal control problem is converted to a stochastic optimization problem. The expected cost functional of the resulting stochastic optimization problem is approximated by sample average approximation thereby the problem can be solved by optimization methods more easily. For facilitating and guaranteeing convergence of the presented method, a new theorem is proved. Finally, the proposed method is implemented based on a newly designed algorithm for solving one of the well-known problems in mathematical finance, the Merton portfolio allocation problem in finite horizon. The simulation results illustrate the improvement of the constructed portfolio return. [ABSTRACT FROM AUTHOR]

This article presents a new DIRECT-type SCABALL (scattering balls) algorithm with a new partition method for derivation-free optimization problems. It does not focus on dividing the region of interest into specific geometric shapes, but rather scatters several balls to cover it. In SCABALL, several potential optimal regions are selected at each iteration, and they are covered by smaller balls sequentially. In this way, the SCABALL ensures the everywhere dense convergence. The center points and radii of the scattered balls significantly influence the efficiency of SCABALL; therefore, the minimax designs are used in the initial and sequential stages to obtain better coverage. The SCABALL parameters, including the number of balls and their radii, were analyzed by numerical investigation. We provided the empirical choices for those parameters and found that the balls' radii can be contracted to balance efficiency and global convergence. Numerical experiments show that the SCABALL algorithm is locally biased and robust. [ABSTRACT FROM AUTHOR]

The research on traffic congestion control has seen rapid development in recent years. Investigating the bifurcation characteristics of traffic flow and designing control schemes for unstable bifurcation points can offer new methods for alleviating traffic congestion. This paper focuses on studying the bifurcation characteristics and nonlinear control of traffic flow based on the continuous model and the taillight effect. Firstly, the traffic flow model is transformed into a stability model suitable for branching analysis through the use of the traveling wave transform. This transformation facilitates the analysis of stability that reflects unstable traffic characteristics such as congestion. Based on this stability model, the existence condition of Hopf bifurcation is proved and some bifurcation points of the traffic system are identified. Secondly, the congestion and stability mutation behaviors near equilibrium and branching points are studied to understand the formation mechanism of traffic congestion. Finally, control schemes are designed using Chebyshev polynomial approximation and stochastic feedback control to delay or eliminate unstable bifurcation points and relieve traffic congestion. This improved traffic flow model helps explain changes in system stability through bifurcation analysis and identify unstable bifurcation points. It can also effectively manage these points by designing a feedback controller. It is beneficial for controlling sudden changes in traffic system stability behavior and mitigating traffic congestion, with important theoretical significance and practical application value. [ABSTRACT FROM AUTHOR]

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained. Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship. • Propose a fast and accurate vertical line time-domain Green function. • Use Chebyshev approximation and asymptotic expansion expressions for small and large parameters in proposed method. • Develop a new hybrid time-domain method coupling 1D Gaussian integral-based vertical line source with TEBEM for seakeeping. [ABSTRACT FROM AUTHOR]

Dielectric elastomers (DE), a novel class of actuators, are increasingly employed in fields such as soft robotics and vibration control. Most existing studies investigating the dynamic characteristics of DE structures neglect the viscoelasticity and strain-stiffening effects of materials, with only a few offering approximate analytical solutions. This paper establishes the governing equation for the dynamics of a DE balloon, utilizing the principle of virtual work and considering material viscoelasticity and strain-stiffening effects through linear damping and the Gent hyperelastic model, respectively. The nonlinear governing equations are simplified using the Chebyshev polynomial approximation. The multiple scales method is applied to derive approximate solutions for the structure's free vibration under static voltage excitation and parametrically excited vibration under harmonic voltage excitation. The accuracy of these approximate analytical solutions is validated by comparison with numerical solutions. Additionally, the paper analyzes stability and bifurcation phenomena through time-domain response curves, amplitude-frequency curves, and phase trajectories. Furthermore, the influence of the stretching limit, voltage, and damping on the nonlinear vibration of the balloon structure is studied, providing a valuable theoretical foundation for the future design and utilization of dielectric elastomers. • Dynamic equations with damping and strain hardening are formulated. • Combination of Chebyshev and multiple scales yield innovative analytic solutions. • Damping and strain hardening mitigate the degree of response softening. • Adjust bifurcation and stability by varying damping, stretching limit, and voltage. [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]

In this article, we study the inconsistency of systems of max-min fuzzy relational equations. We first compute by an explicit analytical formula the Chebyshev distance (defined by the L-infinity norm) between the second member of an inconsistent system and the set of second members of consistent systems defined with the same matrix: that of the inconsistent system. Then we study the set of Chebyshev approximations of the second member of the inconsistent system, which are the second members of the consistent systems defined with the same matrix as the inconsistent system, whose distance to the second member of the inconsistent system is equal to the computed Chebyshev distance. This set is non-empty and associated to the approximate solutions set of the inconsistent system in the following sense: an approximate solution is a solution vector of a consistent system defined with the same matrix as the inconsistent system, whose second member is a Chebyshev approximation of the second member of the inconsistent system. As main results, we describe both the structure of the approximate solutions set and that of the set of the Chebyshev approximations of the second member of the inconsistent system. As an application of our results, we show that our work can help to find approximate values for the rule parameters of a possibilistic rule-based system. [ABSTRACT FROM AUTHOR]