Academic literature on the topic 'Constrained mock-Chebyshev least squares'

The bias in the determination of the Hubble parameter and the Hubble constant in the modern Universe is discussed. It could appear due to the statistical processing of data on the redshifts of galaxies and the estimated distances based on some statistical relations with limited accuracy. This causes a number of effects leading to either underestimation or overestimation of the Hubble parameter when using any methods of statistical processing, primarily the least squares method (LSM). The value of the Hubble constant is underestimated when processing a whole sample; when the sample is constrained by distance, especially when constrained from above. Moreover, it is significantly overestimated due to the data selection. The bias significantly exceeds the values of the erro ofr the Hubble constant calculated by the LSM formulae. These effects are demonstrated both analytically and using Monte Carlo simulations, which introduce deviations in the velocities and estimated distances to the original dataset described by the Hubble law. The characteristics of the deviations are similar to real observations. Errors in the estimated distances are up to 20%. They lead to the fact that, when processing the same mock sample using LSM, it is possible to obtain an estimate of the Hubble constant from 96% of the true value when processing the entire sample to 110% when processing the subsample with distances limited from above. The impact of these effects can lead to a bias in the Hubble constant obtained from real data and an overestimation of the accuracy of determining this value. This may call into question the accuracy of determining the Hubble constant and can significantly reduce the tension between the values obtained from the observations in the early and modern Universes, which were actively discussed during the last year.

A method for suppressing residual vibrations in flexible systems is presented and experimentally demonstrated. The proposed method is based on the preconditioning of the inputs to the system using low-pass Finite Impulse Response (FIR) digital filters. Provided that the cutoff frequency of FIR filters is selected lower than the lowest expected natural frequency of the system and their stop-band is maximized, we show that these filters can be designed to exhibit maximally robust behavior with respect to changes of the system natural frequencies. To perform the proper design of FIR filters for robust vibration suppression, this paper introduces a series of dimensionless performance indexes and the Delay-Error-Order (DEO) curves that represent graphically the delay time introduced by the filter as a function of the remaining residual vibrations, and the filter order. Several classes of FIR filters such as: a) Parks-McClellan; b) Window-based methods (using Chebyshev window); and c) Constrained Least Squares method, are shown to present maximally robust behavior, almost identical to the theoretically predicted. Parallel, they demonstrate excellent vibration suppression while they introduce the minimum possible delay. Further advantages offered by the proposed method, is that no modeling of the flexible system is required, the method can be used in a variety of systems exhibiting vibrations, it is independent of the guidance function and it is simple to implement in practical applications. The results are experimentally verified on a flexible aluminum beam with a significantly varying mass, attached to the end-effector of a robot manipulator. The beam is rotated, using one joint of the manipulator, from an initial to a final position. It is shown that the preconditioned inputs to the flexible system induce very little amount of residual vibrations compared to the inputs with no preconditioning.

This paper proposes a rocket substage vertical landing guidance method based on the second-order Picard-Chebyshev-Newton type algorithm. Firstly, the continuous-time dynamic equation is discretized based on the natural second-order Picard iteration formulation and the Chebyshev polynomial. Secondly, the landing problem that considers terminal constraints is transformed into a nonlinear least-squares problem with respect to the terminal constraint function and solved with the Gauss-Newton method. In addition, the projection method is introduced to the iteration process of the Gauss-Newton method to realize the inequality constraints of the thrust. Finally, the closed-loop strategy for rocket substage vertical landing guidance is proposed and the numerical simulations of the rocket vertical landing stage are carried out. The simulation results demonstrate that compared with the sequential convex optimization algorithm, the proposed algorithm has higher computational efficiency.

The differential microphone array, or differential beamformer, has attracted much attention for its frequency-invariant beampattern, high directivity factor and compact size. In this work, the design of differential beamformers with small inter-element spacing planar microphone arrays is concerned. In order to exactly control the main lobe beamwidth and sidelobe level and obtain minimum main lobe beamwidth with a given sidelobe level, we design the desired beampattern by applying the Chebyshev polynomials at first, via exploiting the structure of the frequency-independent beampattern of a theoretical Nth-order differential beamformer. Next, the so-called null constrained and least square beamformers, which can obtain approximately frequency-invariant beampattern at relatively low frequencies and can be steered to any direction without beampattern distortion, are proposed based on planar microphone arrays to approximate the designed desired beampatterns. Then, for dealing with the white noise amplification at low-frequency bands and beampattern divergence problems at high-frequency bands of the null constrained and least square beamformers, the so-called minimum norm and combined solutions are deduced, which can compromise among the white noise gain, directivity factor and beampattern distortion flexibly. Preliminary simulation results illustrate the properties and advantages of the proposed differential beamformers.

In this paper, a new path planning method is proposed to resolve the problem of two-dimensional terrain following flight of flying robots in mountainous regions. The performance criteria considered for this mission design could include either the minimum vertical acceleration or the minimum flying time. To impose the terrain following/terrain avoidance constraints, various approaches such as least square method, Fourier series method, Gaussian estimation method, and Chebyshev orthogonal polynomial are explored. The resulting optimal control problem is discretized by employing a numerical technique namely direct collocation and then transformed into a nonlinear programming problem. The efficacy of the proposed method is demonstrated by extensive simulations, and particularly, it has been verified that this method is able to produce a solution that satisfies all hard constraints of the underlying problem.

This paper addresses the structural-parametric synthesis and kinematic analysis of the RoboMech class of parallel mechanisms (PM) having two sliders. The proposed methods allow the synthesis of a PM with its structure and geometric parameters of the links to obtain the given laws of motions of the input and output links (sliders). The paper outlines a possible application of the proposed approach to design a PM for a cold stamping technological line. The proposed PM is formed by connecting two sliders (input and output objects) using one passive and one negative closing kinematic chain (CKC). The passive CKC does not impose a geometric constraint on the movements of the sliders and the geometric parameters of its links are varied to satisfy the geometric constraint of the negative CKC. The negative CKC imposes one geometric constraint on the movements of the sliders and its geometric parameters are determined on the basis of the Chebyshev and least-square approximations. Problems of positions and analogues of velocities and accelerations of the considered PM are solved to demonstrate the feasibility and effectiveness of the proposed formulations and case of study.

Un problème très courant en science computationnelle est la détermination d'une approximation, dans un intervalle fixe, d'une fonction dont les évaluations ne sont connues que sur un ensemble fini de points. Une approche courante pour résoudre ce problème repose sur l'interpolation polynomiale. Un cas d'un grand intérêt pratique est celui où ces points suivent une distribution équidistante dans l'intervalle considéré. Dans ces hypothèses, un problème lié à l'interpolation polynomiale est le phénomène de Runge, caractérisé par une augmentation de l'erreur d'interpolation près des extrémités de l'intervalle. En 2009, J. Boyd et F. Xu ont démontré que le phénomène de Runge pouvait être éliminé en interpolant la fonction que sur un sous-ensemble approprié formé par les noeuds les plus proches des noeuds de Chebyshev-Lobatto, communément appelés noeuds de mock-Chebyshev.Cependant, cette stratégie implique de ne pas utiliser presque toutes les données disponibles. Afin d'améliorer la précision de la méthode proposée par Boyd et Xu, tout en tirant pleinement parti des données disponibles, S. De Marchi, F. Dell'Accio et M. Mazza ont introduit une nouvelle technique appelée constrained mock-Chebyshev least squares approximation. Dans cette méthode, le rôle du polynôme nodal, est crucial. Son extension au cas bivarié nécessite cependant des approches alternatives. La procédure développée par F. Dell'Accio, F. Di Tommaso et F. Nudo, utilisant la méthode des multiplicateurs de Lagrange, permet également de définir l'approximation des moindres carrés de mock-Chebyshev sur une grille uniforme de points. Cette technique innovante, équivalente à la méthode univariée précédemment introduite en termes analytiques, se révèle également plus précise en termes numériques. La première partie de la thèse est consacrée à l'étude de cette nouvelle technique et à son application à des problèmes de quadrature et de différenciation numérique.Dans la deuxième partie de cette thèse, nous nous concentrons sur le développement d'un cadre unifié et général pour l'enrichissement de l'élément fini linéaire triangulaire standard en deux dimensions et de l'élément fini linéaire simplicial standard en dimensions supérieures. La méthode des éléments finis est une approche largement adoptée pour résoudre numériquement les équations aux dérivées partielles qui se posent en ingénierie et en modélisation mathématique [55]. Sa popularité est attribuable en partie à sa polyvalence pour traiter diverses formes géométriques. Cependant, les approximations produites par cette méthode s'avèrent souvent inefficaces pour résoudre des problèmes présentant des singularités. Pour surmonter ce problème, diverses approches ont été proposées, l'une des plus célèbres reposant sur l'enrichissement de l'espace d'approximation des éléments finis en ajoutant des fonctions d'enrichissement appropriées. Un des éléments finis le plus simple est l'élément fini triangulaire linéaire standard, largement utilisé dans les applications. Dans cette thèse, nous introduisons un enrichissement polynomial de l'élément fini triangulaire linéaire standard et utilisons ce nouvel élément fini pour introduire une amélioration de l'opérateur triangulaire de Shepard. Ensuite, nous introduisons une nouvelle classe d'éléments finis en enrichissant l'élément triangulaire linéaire standard avec des fonctions d'enrichissement qui ne sont pas nécessairement polynomiales, mais qui satisfont la condition d'annulation aux sommets du triangle.Nous généralisons les résultats présentés dans le cas bidimensionnel au cas de l'élément fini simplicial linéaire standard, en utilisant également des fonctions d'enrichissement qui ne satisfont pas la condition d'annulation aux sommets du simplexe.Enfin, nous appliquons ces nouvelles stratégies d'enrichissement pour définir une version plus généralede l'enrichissement de l'élément fini linéaire vectoriel simplicial développé par Bernardi et Raugel
A very common problem in computational science is the determination of an approximation, in a fixed interval, of a function whose evaluations are known only on a finite set of points. A common approach to solving this problem relies on polynomial interpolation, which consists of determining a polynomial that coincides with the function at the given points. A case of great practical interest is the case where these points follow an equispaced distribution within the considered interval. In these hypotheses, a problem related to polynomial interpolation is the Runge phenomenon, which consists in increasing the magnitude of the interpolation error close to the ends of the interval. In 2009, J. Boyd and F. Xu demonstrated that the Runge phenomenon could be eliminated by interpolating the function only on a proper subset formed by nodes closest to the Chebyshev-Lobatto nodes, the so called mock-Chebyshev nodes.However, this strategy involves not using almost all available data. In order to improve the accuracy of the method proposed by Boyd and Xu, while making full use of the available data, S. De Marchi, F. Dell'Accio, and M. Mazza introduced a new technique known as the constrained mock-Chebyshev least squares approximation. In this method, the role of the nodal polynomial, essential for ensuring interpolation at mock-Chebyshev nodes, is crucial. Its extension to the bivariate case, however, requires alternative approaches. The recently developed procedure by F. Dell'Accio, F. Di Tommaso, and F. Nudo, employing the Lagrange multipliers method, also enables the definition of the constrained mock-Chebyshev least squares approximation on a uniform grid of points. This innovative technique, equivalent to the previously introduced univariate method in analytical terms, also proves to be more accurate in numerical terms. The first part of the thesis is dedicated to the study of this new technique and its application to numerical quadrature and differentiation problems.In the second part of this thesis, we focus on the development of a unified and general framework for the enrichment of the standard triangular linear finite element in two dimensions and the standard simplicial linear finite element in higher dimensions. The finite element method is a widely adopted approach for numerically solving partial differential equations arising in engineering and mathematical modeling [55]. Its popularity is partly attributed to its versatility in handling various geometric shapes. However, the approximations produced by this method often prove ineffective in solving problems with singularities. To overcome this issue, various approaches have been proposed, with one of the most famous relying on the enrichment of the finite element approximation space by adding suitable enrichment functions. One of the simplest finite elements is the standard linear triangular element, widely used in applications. In this thesis, we introduce a polynomial enrichment of the standard triangular linear finite element and use this new finite element to introduce an improvement of the triangular Shepard operator. Subsequently, we introduce a new class of finite elements by enriching the standard triangular linear finite element with enrichment functions that are not necessarily polynomials, which satisfy the vanishing condition at the vertices of the triangle.Later on, we generalize the results presented in the two-dimensional case to the case of the standard simplicial linear finite element, also using enrichment functions that do not satisfy the vanishing condition at the vertices of the simplex.Finally, we apply these new enrichment strategies to extend the enrichment of the simplicial vector linear finite element developed by Bernardi and Raugel

Wildfires are an important and heretofore contributor to the climate, especially in some special country,for example southeastern Australia. A large population live in this region, so the wildfires would be dangerous in the extreme. In this paper, we will study how to stop wildfires as quickly as possible. First, The fuzzy clustering analysis is used to classify forest coverage, wind velocity and population distribution. Then In the analysis model level, the elements we considered are the weight of above three factors, made up of some combination of them. In adapting to the changes that may occur in extreme fire events in the next decade, we use the Braess paradox to verify the models with the impact of climate. We supposed the scan cover of the UAV is circular function related with the flying altitude, so linear programming equation could be established at different altitude. We also use this idea to constrain some models of different terrain. It showed that the model has good adaptability with a quick convergence speed by computer simulation. The model was adopted for least-squares data fitting to process experimental data, and a high correlation coefficient was obtained.

Conformation of a manufactured feature to the applied geometric tolerances is done by analyzing the point cloud that is measured on the feature. To that end, a geometric feature is fitted to the point cloud and the results are assessed to see whether the fitted features lies within the specified tolerance limits or not. Coordinate Measuring Machines (CMMs) use feature fitting algorithms that incorporate least square estimates as a basis for obtaining minimum, maximum, and zone fits. However, a comprehensive set of algorithms addressing the fitting procedure (all datums, targets) for every tolerance class is not available. Therefore, a Library of algorithms is developed to aid the process of feature fitting, and tolerance verification. This paper addresses linear, planar, circular, and cylindrical features only. This set of algorithms described conforms to the international Standards for GD&T. In order to reduce the number of points to be analyzed, and to identify the possible candidate points for linear, circular and planar features, 2D and 3D convex hulls are used. For minimum, maximum, and Chebyshev cylinders, geometric search algorithms are used. Algorithms are divided into three major categories: least square, unconstrained, and constrained fits. Primary datums require one sided unconstrained fits for their verification. Secondary datums require one sided constrained fits for their verification. For size and other tolerance verifications we require both unconstrained and constrained fits.

Coordinate Measuring Machines (CMMs) collect a sampling of points on measured features for use in dimensional metrology. Conformance to specified geometric tolerances is done by analyzing the point cloud to fit the corresponding feature to the point cloud to determine if the simulated feature lies within the specified tolerance limits. Different types of feature fitting algorithms are needed: nominal, minimal/maximal, circumscribing/inscribing, and zone. Studies have shown that the same point cloud data sent to different vendors CMM software, produces different results. It is suspected that some of these algorithms may be inconsistent with the tolerance class definitions in tolerance standards and, in some cases, with shop floor conventional practices. We have previously reported on the development of normative algorithms and a feature fitting library that could be used by all CMMs. This paper gives a summary of those algorithms and then reports on methods used for verification. Three different types of verification methods were used to validate the algorithms developed. The scope of the current work is limited to linear, planar, circular, and cylindrical features. This set of algorithms described conforms to the international Standards for GD&T. In order to reduce the number of points to be analyzed, and to identify the possible candidate points for linear, circular and planar features, 2D and 3D convex hulls are used. For minimum, maximum, and Chebyshev cylinders, geometric search algorithms are used. Algorithms are divided into three major categories: least square, unconstrained, and constrained fits. Primary datums require one sided unconstrained fits for their verification. Secondary datums require one sided constrained fits for their verification. For size and other tolerance verifications, we require both unconstrained and constrained fits. Use of three different methods has validated the robustness, efficiency and accuracy of the algorithms.