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The celebrated minimax principle of Yao says that for any Boolean-valued function f with finite domain, there is a distribution µ over the domain of f such that computing f to error against inputs from µ is just as hard as computing f to error on worst-case inputs. Notably, however, the distribution µ depends on the target error level1: the hard distribution which is tight for bounded error might be trivial to solve to small bias, and the hard distribution which is tight for a small bias level might be far from tight for bounded error levels. In this work, we introduce a new type of minimax theorem which can provide a hard distribution µ that works for all bias levels at once. We show that this works for randomized query complexity, randomized communication complexity, some randomized circuit models, quantum query and communication complexities, approximate polynomial degree, and approximate logrank. We also prove an improved version of Impagliazzo's hardcore lemma. Our proofs rely on two innovations over the classical approach of using Von Neumann's minimax theorem or linear programming duality. First, we use Sion's minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as "forecasting algorithms" evaluated by a certain proper scoring rule. The expected score of the forecasting version of a randomized algorithm appears to be a more fine-grained way of analyzing the bias of the algorithm. We show that such expected scores have many elegant mathematical properties--for example, they can be amplified linearly instead of quadratically. We anticipate forecasting algorithms will find use in future work in which a fine-grained analysis of small-bias algorithms is required. [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]

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 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]

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]

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]

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]

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]

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]

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 problem of low rank approximation is ubiquitous in science. Traditionally this problem is solved in unitary invariant norms such as Frobenius or spectral norm due to existence of efficient methods for building approximations. However, recent results reveal the potential of low rank approximations in Chebyshev norm, which naturally arises in many applications. In this paper we tackle the problem of building optimal rank-1 approximations in the Chebyshev norm. We investigate the properties of alternating minimization algorithm for building the low rank approximations and demonstrate how to use it to construct optimal rank-1 approximation. As a result we propose an algorithm that is capable of building optimal rank-1 approximations in Chebyshev norm for moderate matrices. [ABSTRACT FROM AUTHOR]

We address the problem of adaptive minimax density estimation on |$\mathbb{R}^{d}$| with |$L_{p}$| loss functions under Huber's contamination model. To investigate the contamination effect on the optimal estimation of the density, we first establish the minimax rate with the assumption that the density is in an anisotropic Nikol'skii class. We then develop a data-driven bandwidth selection procedure for kernel estimators, which can be viewed as a robust generalization of the Goldenshluger-Lepski method. We show that the proposed bandwidth selection rule can lead to the estimator being minimax adaptive to either the smoothness parameter or the contamination proportion. When both of them are unknown, we prove that finding any minimax-rate adaptive method is impossible. Extensions to smooth contamination cases are also discussed. [ABSTRACT FROM AUTHOR]

We fully characterize the nonasymptotic minimax separation rate for sparse signal detection in the Gaussian sequence model with |$p$| equicorrelated observations, generalizing a result of Collier, Comminges and Tsybakov. As a consequence of the rate characterization, we find that strong correlation is a blessing, moderate correlation is a curse and weak correlation is irrelevant. Moreover, the threshold correlation level yielding a blessing exhibits phase transitions at the |$\sqrt{p}$| and |$p-\sqrt{p}$| sparsity levels. We also establish the emergence of new phase transitions in the minimax separation rate with a subtle dependence on the correlation level. Additionally, we study group structured correlations and derive the minimax separation rate in a model including multiple random effects. The group structure turns out to fundamentally change the detection problem from the equicorrelated case and different phenomena appear in the separation rate. [ABSTRACT FROM AUTHOR]

This paper focuses on the constrained minimax location problem with the closest distance. Some properties concerning the existence and uniqueness of the optimal solution are provided. To achieve these results, we apply non-smooth approach that allow us to give the explicit solution structure of the constrained problem. Moreover, we develop an effective algorithm for solving this class of problems and we provide its convergence under some mild assumptions. At the end, some computational test problems are provided to illustrate the effectiveness of the method and certify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, I consider a doubly stochastic Poisson process with intensity λt=qXt$$ {\lambda}_t=q\left({X}_t\right) $$ where X$$ X $$ is a continuous Itô semi‐martingale. Both processes are observed continuously over a fixed period 0,1$$ \left[0,1\right] $$. I propose a local polynomial estimator for the function q$$ q $$ on a given interval. Next, I propose a method to select the bandwidth in a nonasymptotic framework that leads to an oracle inequality. Considering the asymptotic n$$ n $$, and q=nq˜$$ q=n\tilde{q} $$, the accuracy of the proposed estimator over the Hölder class of order β$$ \beta $$ is n−β2β+1$$ {n}^{\frac{-\beta }{2\beta +1}} $$ if the degree of the chosen polynomial is greater than ⌊β⌋$$ \left\lfloor \beta \right\rfloor $$ and it is optimal in the minimax setting. I apply those results to data on French temperature and electricity spot prices from which I infer the intensity of electricity spot spikes as a function of the temperature. [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]

In this paper, we study zeroth-order algorithms for minimax optimization problems that are nonconvex in one variable and strongly-concave in the other variable. Such minimax optimization problems have attracted significant attention lately due to their applications in modern machine learning tasks. We first consider a deterministic version of the problem. We design and analyze the Zeroth-Order Gradient Descent Ascent (ZO-GDA) algorithm, and provide improved results compared to existing works, in terms of oracle complexity. We also propose the Zeroth-Order Gradient Descent Multi-Step Ascent (ZO-GDMSA) algorithm that significantly improves the oracle complexity of ZO-GDA. We then consider stochastic versions of ZO-GDA and ZO-GDMSA, to handle stochastic nonconvex minimax problems. For this case, we provide oracle complexity results under two assumptions on the stochastic gradient: (i) the uniformly bounded variance assumption, which is common in traditional stochastic optimization, and (ii) the Strong Growth Condition (SGC), which has been known to be satisfied by modern over-parameterized machine learning models. We establish that under the SGC assumption, the complexities of the stochastic algorithms match that of deterministic algorithms. Numerical experiments are presented to support our theoretical results. [ABSTRACT FROM AUTHOR]

Nonconvex minimax problems have attracted significant interest in machine learning and many other fields in recent years. In this paper, we propose a new zeroth-order alternating randomized gradient projection algorithm to solve smooth nonconvex-linear problems and its iteration complexity to find an ε -first-order Nash equilibrium is O ε - 3 and the number of function value estimation per iteration is bounded by O d x ε - 2 . Furthermore, we propose a zeroth-order alternating randomized proximal gradient algorithm for block-wise nonsmooth nonconvex-linear minimax problems and its corresponding iteration complexity is O K 3 2 ε - 3 and the number of function value estimation is bounded by O d x ε - 2 per iteration. The numerical results indicate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The design of a low-pass-frequency filter with the electronic change of the approximation characteristics of resulting responses is presented. The filter also offers the reconnection-less reconfiguration of the order (1st-, 2nd-, 3rd- and 4th-order functions are available). Furthermore, the filter offers the electronic control of the cut-off frequency of the output response. The feature of the electronic change in the approximation characteristics is investigated for the Butterworth, Bessel, Elliptic, Chebyshev and Inverse Chebyshev approximations. The design is verified by PSpice simulations and experimental measurements. The results are also supported by the transient domain response (response to the square waveform), comparison of the group delay, sensitivity analysis and implementation feasibility based on given approximation. The benefit of the proposed electronic change in the approximation characteristics feature (in general signal processing or for sensors in particular) is presented and discussed for an exemplary scenario. [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]

In two‐phase materials, each phase having a non‐local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [‐1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [‐1,1] and, to simplify the analysis, Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [‐1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov‐type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time‐dependent boundary potentials are suitably tailored. © 2022 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for non-parametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses |${\textbf Y} = (Y_1,\ldots ,Y_n)$| onto a subspace spanned by certain eigenvectors of a neighbourhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density |$p$|⁠ , PCR-LE achieves the optimal rates for both estimation (where the optimal rate in squared |$L^2$| norm is known to be |$n^{-2s/(2s + d)}$|⁠) and goodness-of-fit testing (⁠|$n^{-4s/(4s + d)}$|⁠). We also consider the situation where the design is supported on a manifold of small intrinsic dimension |$m$|⁠ , and give upper bounds establishing that PCR-LE achieves the faster minimax estimation (⁠|$n^{-2s/(2s + m)}$|⁠) and testing (⁠|$n^{-4s/(4s + m)}$|⁠) rates of convergence. Interestingly, these rates are almost always much faster than the known rates of convergence of graph Laplacian eigenvectors to their population-level limits; in other words, for this problem regression with estimated features appears to be much easier, statistically speaking, than estimating the features itself. We support these theoretical results with empirical evidence. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new framework to construct confidence sets for a |$d$| -dimensional unknown sparse parameter |${\boldsymbol \theta }$| under the normal mean model |${\boldsymbol X}\sim N({\boldsymbol \theta },\sigma ^{2}\bf{I})$|⁠. A key feature of the proposed confidence set is its capability to account for the sparsity of |${\boldsymbol \theta }$|⁠ , thus named as sparse confidence set. This is in sharp contrast with the classical methods, such as the Bonferroni confidence intervals and other resampling-based procedures, where the sparsity of |${\boldsymbol \theta }$| is often ignored. Specifically, we require the desired sparse confidence set to satisfy the following two conditions: (i) uniformly over the parameter space, the coverage probability for |${\boldsymbol \theta }$| is above a pre-specified level; (ii) there exists a random subset |$S$| of |$\{1,...d\}$| such that |$S$| guarantees the pre-specified true negative rate for detecting non-zero |$\theta _{j}$| 's. To exploit the sparsity of |${\boldsymbol \theta }$|⁠ , we allow the confidence interval for |$\theta _{j}$| to degenerate to a single point 0 for any |$j\notin S$|⁠. Under this new framework, we first consider whether there exist sparse confidence sets that satisfy the above two conditions. To address this question, we establish a non-asymptotic minimax lower bound for the non-coverage probability over a suitable class of sparse confidence sets. The lower bound deciphers the role of sparsity and minimum signal-to-noise ratio (SNR) in the construction of sparse confidence sets. Furthermore, under suitable conditions on the SNR, a two-stage procedure is proposed to construct a sparse confidence set. To evaluate the optimality, the proposed sparse confidence set is shown to attain a minimax lower bound of some properly defined risk function up to a constant factor. Finally, we develop an adaptive procedure to the unknown sparsity. Numerical studies are conducted to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The inherent non-smoothness of the vibroimpact system leads to complex behaviors and a strong sensitivity to parameter changes. Unfortunately, uncertainties and errors in system parameters are inevitable in mechanical engineering. Therefore, investigations of dynamical behaviors for vibroimpact systems with stochastic parameters are highly essential. The present study aims to analyze the dynamical characteristics of the three-degree-of-freedom vibroimpact system with an uncertain parameter by means of the Chebyshev polynomial approximation method. Specifically, the vibroimpact system model considered is one with unilateral constraint. Firstly, the three-degree-of-freedom vibroimpact system with an uncertain parameter is transformed into an equivalent deterministic form using the Chebyshev orthogonal approximation. Then, the ensemble means responses of the stochastic vibroimpact system are derived. Numerical simulations are performed to verify the effectiveness of the approximation method. Furthermore, the periodic and chaos motions under different system parameters are investigated, and the bifurcations of the vibroimpact system are analyzed with the Poincaré map. The results demonstrate that both the restitution coefficient and the random factor can induce the appearance of the periodic bifurcation. It is worth noting that the bifurcations fundamentally differ between the stochastic and deterministic systems. The former has a bifurcation interval, while the latter occurs at a critical point. [ABSTRACT FROM AUTHOR]

The intensive use of lamination technology for the decoration of printing prints made by various printing techniques, as well as a wide range of films, require solving an urgent problem – optimization of influencing factors to ensure the quality of laminated products. The objects of research were prints of inkjet and laser printing on paper, photo paper, and cardboard. A glossy film based on bio-oriented polypropylene BOPP was used for lamination. Studies of densitometric indicators of prints after lamination confirmed an increase in optical density by 0.28 units. This is especially true for inkjet prints. However, after lamination, all digital prints showed a tendency to improve colorimetric characteristics, namely an increase in the brightness of CMYK colors. The relationship between the technological regimes of lamination (speed and temperature) and the strength of the laminates, estimated by breaking and pressing forces, was established. To describe the dependence of the laminate strength on the temperature and speed of lamination, polynomial models were built that reproduce the values of the breaking and pushing forces of the laminate with satisfactory accuracy. The selected algorithm for calculating the parameter values of these models is based on the application of the Chebyshev approximation of the functions of many variables. The optimization models built will allow, if necessary, for making corrections in the technological process of laminating printing prints to ensure the predicted required quality and strength of laminates. [ABSTRACT FROM AUTHOR]

Traditionally, analysts solve the Bateman depletion equations to calculate the nuclide number density (NND) of each nuclide since these densities impact other reactor parameters, such as reactivity, as they change. Many quantities of interest, such as radiation damage, are calculated using simple integration methods, assuming that the NNDs are constant over a given depletion interval. However, the NNDs are time dependent, which can be accurately represented only by the Bateman depletion equations. We propose that these quantities can be calculated simultaneously with the NNDs within the Bateman depletion equations, preserving the coupled nature of these quantities to the time-dependent NNDs. We implemented this functionality in Griffin, demonstrating that only minor code modifications were necessary in order to accommodate an evaluation of these quantities in the Bateman depletion equations. The Chebyshev Rational Approximation Method was used to successfully solve for these additional quantities in the Bateman depletion equations. For radiation damage, the results calculated by Griffin were very accurate, differing by less than 2.5% from an analytical benchmark. For other quantities, the discrepancy between quantities calculated by the Bateman depletion equations versus those calculated by the Forward Euler method exceeded 10% for decay energy and 2% for fissions per initial heavy metal atom and kinetic energy released per unit mass when few depletion intervals were used. As the number of depletion intervals increased, both methods began to converge as expected. [ABSTRACT FROM AUTHOR]

The full-range debonding process of a fiber-reinforced polymer (FRP)–generic material substrate interface with mechanical end anchorages requires a better understanding. In this study, we developed a nonlinear semianalytical model based on a cohesive crack model to predict the full-range debonding process of such an interface in a single-shear mode, considering the relatively deformed region in the interface as a cohesive zone. Specifically, we produced a double exponential bond–slip constitutive model with residual bonding shear stress and boundary conditions of slip constraints at both anchoring ends. Implicit solutions for the relative slips, bonding shear stresses, and axial forces in the FRP along the bond length, as well as the load–slip responses at the loading end and effective bond length, were obtained using the Chebyshev polynomial approximation strategy. The accuracy of the proposed model was verified using experimental results available in the literature. Then, the evolution of full-range interfacial bonding behavior and load–slip responses for different bond lengths was discussed. Finally, a parametric study highlighting the effect of geometry and material properties on the load–slip response at the loading end was presented. [ABSTRACT FROM AUTHOR]

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In this paper, we proposed an adaptive QP-free method without a penalty function or a filter for minimax optimization. In each iteration, solved two linear systems of equations constructed from Lagrange multipliers and KKT-conditioned NCP functions. Based on the work set, the computational scale is further reduced. Instead of the filter structure, we adopt a nonmonotonic equilibrium mechanism with an adaptive parameter adjusted according to the result of each iteration. Feasibility of the algorithm are given, and the convergence under some assumptions is demonstrated. Numerical results and practical application are reported at the end. [ABSTRACT FROM AUTHOR]

The method for constructing the Chebyshev approximation of a discrete multivariable function with reproducing the values of the function and its partial derivatives at given points is proposed. The method is based on constructing a boundary mean-power approximation with appropriate interpolation conditions. The authors use an iterative scheme based on the least squares method with a variable weight function for constructing the mean-power approximation. The results of approximating the one-variable function confirm the fulfillment of the characteristic feature of the Chebyshev approximation with the reproduction of the function and its derivative values at given points. The test examples instantiate the fast convergence of the proposed method. [ABSTRACT FROM AUTHOR]