Linear Approximation Theory Research Articles

Existence of solutions for fixed point theorem

The development of a mathematical model based on diffusion has received a great dealof attention in recent years, many scientist and mathematician have tried to apply basicknowledge about the differential equation and the boundary condition to explain anapproximate the diffusion and reaction model. The subject of fractional calculus attracted much attentions and is rapidly growing area of research because of itsnumerous applications in engineering and scientific disciplines such as signal processing, nonlinear control theory, viscoelasticity, optimization theory [1], controlled thermonuclear fusion, chemistry, nonlinear biological systems, mechanics,electric networks, fluid dynamics, diffusion, oscillation, relaxation, turbulence, stochastic dynamical system, plasmaphysics, polymer physics, chemical physics, astrophysics, and economics. Therefore, it deserves an independent theoryparallel to the theory of ordinary differential equations (DEs).In the development of non-linear analysis, fixed point theory plays an important role. Also, it has been widely used in different branches of engineering and sciences. Banach fixed point theory is a essential part of mathematical analysis because of its applications in various area such as variational and linear inequalities, improvement and approximation theory. The fixed-point theorem in diffusion equations plays a significant role to construct methods to solve the problems in sciences and mathematics. Although Banach fixed point theory is a vast field of study and is capable of solving diffusion equations. The main motive of the research is solving the diffusion equations by Banach fixed point theorems and Adomian decomposition method. To analysis the drawbacks of the other fixed-point theorems and different solving methods, the related works are reviewed in this paper.

Bounding the Lebesgue constant for a barycentric rational trigonometric interpolant at periodic well-spaced nodes

A well-known result in linear approximation theory states that the norm of the operator, known as the Lebesgue constant, of polynomial interpolation on an interval grows only logarithmically with the number of nodes, when these are Chebyshev points. Results like this are important for studying the conditioning of the approximation. A cosine change of variable shows that polynomial interpolation at Chebyshev points is just the special case for even functions of trigonometric interpolation (on the circle) at equidistant points. The Lebesgue constant of the latter grows logarithmically, also for functions with no particular symmetry. In the present work, we show that a linear rational generalization of the trigonometric interpolant enjoys a logarithmically growing Lebesgue constant for more general sets of nodes, namely periodic well-spaced ones, patterned after those introduced for an interval by Bos et al. (2013) few years ago. An important special case are conformally shifted equispaced points, for which the rational trigonometric interpolant is known to converge exponentially.

Qualitative Analysis of Some Libration Points in the Restricted Four-body Problem

We consider the situation where three heavy gravitational bodies form the Lagrange configuration rotating in a fixed plane and the fourth body of negligible mass moves in this plane. We present three cases of so-called libration points and we study their stability using linear approximation and KAM theory. In some situations we prove the Lyapunov stability for generic values of some parameter of the problem.

Mission-Critical Resource Allocation With Puncturing in Industrial Wireless Networks Under Mixed Services

As the manufacturing industry gradually expands and tends to be intelligent, the industrial networks have been greatly applied and are of paramount importance. Furthermore, taking the advantages of wireless networks into account, such as flexible deployment, low cost, and easy maintenance, etc., this paper is dedicated to the research of industrial wireless networks. Facing the problem of limited radio resource, in order to make full use of frequency band, this paper proposes a resource scheduling scheme with puncturing, which guarantees the reliability of subsequent ultra-reliable and low-latency communications (URLLC) services under the conditions of existing enhanced mobile broadband (eMBB) services. By optimizing the bandwidth pre-allocation for eMBB services, puncture weight and transmit power, this scheme can effectively meet the needs of the system. Specifically, the purpose of this paper is to minimize the decoding error rate of the devices carrying URLLC service while ensuring the demand for throughput of eMBB services. The problem proposed is a nonlinear stochastic optimization problem involving complex Q-function. The closed-form expression of the objective function is obtained through linear approximation and probability theory, which is further decoupled into three sub-problems. To this end, a block coordinate descent optimization (BCDO) algorithm is proposed to obtain the optimal bandwidth allocation, puncture weight and transmit power. The simulation results verify the rationality of the theoretical analysis and the effectiveness of the proposed algorithm, and also elaborate the influence of different parameters on the decoding error rate of the URLLC devices.

A discrete fractional-order Cournot duopoly game

This paper focuses on associating the Cournot duopoly problem with long-memory effects. First of all, we generate a discrete fractional-order Cournot duopoly game by introducing the Caputo fractional-order difference calculus to the classical duopoly theory. In the fractional-order game, participants can make their decisions by taking full advantage of their historical information. Then we discuss both Nash equilibria and local stability of the game by employing the linear approximation theory. At last, we numerically validate the main results by using bifurcation diagrams, phase portraits, the largest Lyapunov exponent, and the 0–1 test algorithms.

Better Approximations of High Dimensional Smooth Functions by Deep Neural Networks with Rectified Power Units

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU network for functions in Sobolev space and Korobov space have recently been made by [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H. Montanelli and Q. Du, SIAM J Math. Data Sci., 1:78-92, 2019], etc. In this paper, we show that deep networks with rectified power units (RePU) can give better approximations for smooth functions than deep ReLU networks. Our analysis bases on classical polynomial approximation theory and some efficient algorithms proposed in this paper to convert polynomials into deep RePU networks of optimal size with no approximation error. Comparing to the results on ReLU networks, the sizes of RePU networks required to approximate functions in Sobolev space and Korobov space with an error tolerance $\varepsilon$, by our constructive proofs, are in general $\mathcal{O}(\log\frac{1}{\varepsilon})$ times smaller than the sizes of corresponding ReLU networks constructed in most of the existing literature. Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. Math. 1:61-80, 1993], our constructions use less number of activation functions and numerically more stable, they can be served as good initials of deep RePU networks and further trained to break the limit of linear approximation theory. The functions represented by RePU networks are smooth functions, so they naturally fit in the places where derivatives are involved in the loss function.

A full spectrum k-distribution based non-gray radiative property model for unburnt char

By using the concept of weighted sum of four gray particles and spectrum k-distribution (WSGP-SK), a non-gray radiative property model for unburnt char particles is developed. Based on the carbon burnout kinetic model for structure during oxidation, and the linear mixed approximation theory for complex index of refraction, spectral radiative properties of unburnt char particles are first calculated as function of the burnout ratio by Mie theory. Referring to the full spectrum k-distribution model, k-distribution is applied to reorder absorption and scattering efficiencies of particles. Then, weighting factors and efficiency factors of the non-gray radiative property model are directly obtained from Gaussian integral points of k-distribution. The model is validated against the benchmark solutions of line-by-line (LBL) model. Maximum relative errors of this model are 3% and 15% for radiative heat fluxes and source terms in non-isothermal inhomogeneous particulate media, respectively. The assumption of linearly varying radiative properties with burnout ratio (Lockwood et al. 1986) will result in a predicted deviation of 53% for radiative source terms. Results also show that this non-gray model is remarkably better than the Planck mean method. Moreover, a satisfactory comparison with LBL solutions is achieved in the gas and particle mixture by combining the non-gray WSGG-SK model (Guo et al. 2015). As a radiation sub-model, this non-gray radiative property model can significantly improve prediction accuracy of radiative heat transfer in oxy-fuel combustion.

Entropies of Overcomplete Kernel Dictionaries

In signal analysis and synthesis, linear approximation theory considers a linear decomposition of any given signal in a set of atoms, collected into a so-called dictionary. Relevant sparse representations are obtained by relaxing the orthogonality condition of the atoms, yielding overcomplete dictionaries with an extended number of atoms. More generally than the linear decomposition, overcomplete kernel dictionaries provide an elegant nonlinear extension by defining the atoms through a mapping kernel function (e.g., the gaussian kernel). Models based on such kernel dictionaries are used in neural networks, gaussian processes and online learning with kernels. The quality of an overcomplete dictionary is evaluated with a diversity measure the distance, the approximation, the coherence and the Babel measures. In this paper, we develop a framework to examine overcomplete kernel dictionaries with the entropy from information theory. Indeed, a higher value of the entropy is associated to a further uniform spread of the atoms over the space. For each of the aforementioned diversity measures, we derive lower bounds on the entropy. Several definitions of the entropy are examined, wth an extensive analysis in both the input space and the mapped feature space.

Hyperbolic cross approximation in infinite dimensions

We give tight upper and lower bounds of the cardinality of the index sets of certain hyperbolic crosses which reflect mixed Sobolev–Korobov-type smoothness and mixed Sobolev-analytic-type smoothness in the infinite-dimensional case where specific summability properties of the smoothness indices are fulfilled. These estimates are then applied to the linear approximation of functions from the associated spaces in terms of the ε-dimension of their unit balls. Here, the approximation is based on linear information. Such function spaces appear for example for the solution of parametric and stochastic PDEs. The obtained upper and lower bounds of the approximation error as well as of the associated ε-complexities are completely independent of any parametric or stochastic dimension. Moreover, the rates are independent of the parameters which define the smoothness properties of the infinite-variate parametric or stochastic part of the solution. These parameters are only contained in the order constants. This way, linear approximation theory becomes possible in the infinite-dimensional case and corresponding infinite-dimensional problems get tractable.

On the Korovkin approximation theorem and Volkov-type theorems

In this short paper, we give a generalization of the classical Korovkin approximation theorem (Korovkin in Linear Operators and Approximation Theory, 1960), Volkov-type theorems (Volkov in Dokl. Akad. Nauk SSSR 115:17-19, 1957), and a recent result of

Research on Energy-Saving Operation of Screw Air Compressor

Based on analysis of a screw air compressor volumetric efficiency under different discharge pressure conditions, this study establishes the mathematic model of the adiabatic compression power consumption. Under load/unload conditions, to change the offline pressure with step of 0.01 MPa, the power consumption floats up and down with the change of unloading and loading and then the screw compressor power consumption is simulated in MATLAB. The results shows that the optimal offline pressure exists at the given air consumption and meanwhile power consumption is minimal with the optimal offline pressure. It also reveals that the required optimal offline pressures will vary by air consumptions. Then, based on dynamic mechanical analysis for fluid motion in the pipe, considering that there is a causal relationship between pressure variation and gas flow variation, a method for measuring flow rate of one dimensional unsteady flow dynamically is proposed based on theory of linear approximation. In order to lower the online pressure and further reduce the energy consumption of screw air compressor, we propose a method is to calculate the rate of pressure drop and predict times for the discharge pressure dropping to the online pressure. And the further optimal control method of screw air compressor is given. It is correct and feasible, which proved by experiments.

Establishing improved convergence and robustness properties for the repetitive learning control

A novel learning control scheme is designed for a class of nonlinear systems. Not only global asymptotic tracking is achieved but also sufficient conditions for the asymptotic “input learning” are derived. The robustness with respect to a finite memory implementation of the control algorithm (which is based on the piecewise linear approximation theory) is guaranteed in the closed loop. The proposed approach allows for the solution of global output tracking problems: (i) for relative degree one systems with output dependent uncertainties; (ii) for nonlinear systems with matching uncertainties.

A global robust iterative learning position control for current-fed permanent magnet step motors

The tracking control problem via state feedback for uncertain current-fed permanent magnet step motors with non-sinusoidal flux distribution and uncertain position-dependent load torque is addressed: a periodic reference signal (of known period) for the rotor position is required to be tracked. A robust iterative learning control algorithm is designed which, for any motor initial condition and without requiring any resetting procedure, guarantees, despite system uncertainties: exponential convergence of the rotor position tracking error to a residual ball (centered at the origin) whose radius can be made arbitrarily small by properly setting the learning gain; asymptotic convergence of the rotor position tracking error to zero. A sufficient condition for the asymptotic estimation of the uncertain reference input achieving, for compatible initial conditions, perfect tracking is derived. Robustness with respect to a finite memory implementation of the control algorithm based on the piecewise linear approximation theory is shown to be guaranteed; satisfactory performances of a discrete-time implementation of the control algorithm are obtained in realistic simulations for the full-order voltage-fed motor.

The mechanism of noise propagation in typical building blocks of biochemical reaction networks

Intracellular processes of live organisms, which can be described by sets of biochemical reactions, are inherently stochastic, where the fluctuations in molecule abundance inside the cell play a crucial role in the cellular growth and development. For typical building blocks of biochemical reaction networks （including signal transduction networks and metabolic networks）, this paper first gives a unified formulation and then presents a general mechanism of noise propagation by applying the linear noise approximation theory. The main results show that: there is noise propagation in signaling transduction pathways and the noise intensity satisfies noise sum rule, and there is no noise propagation in metabolic pathways and the distributions of the steady state molecules mutually independent. The analytical results lay a theoretical foundation for understanding the intracellular processes.

Approximations with Real Linear Modules

Real linear approximation theory is developed further by regarding ℂ n as a module over the ring of circlets. By introducing a concept of orthogonality together with the respective Gram–Schmidt orthogonalization process, improved approximations upon the standard complex Hilbert space techniques follow. Related hierarchical bases are devised leading to a new family of rapidly constructible family of unitary matrices. With circlets, so-called oplets are introduced for approximation to improve the singular value decomposition of real matrices. Complex matrix approximation is also considered through finding the nearest real matrix in small rank perturbations.

Hydrodynamic Design of Integrated Bulbous Bowlsonar Dome for Naval Ships

Recently, the idea of bulbous bow has been extended from the commercial ships to the design of an integrated bow that houses a sonar dome for naval ships. In the present study, a design method for a particular set of requirements consisting of a narrow range of input parameters is presented. The method uses an approximate linear theory with sheltering effect for resistance estimation and pressure distribution, and correlation with statistical analysis from the existing literature and the tank-test results available in the public domain. Though the optimisation of design parameters has been done for the design speed, but the resistance performance over the entire speed range has been incorporated in the design. The bulb behaviour has been discussed using the principle of minimisation of resistance and analysis of flow pattern over the bulb and near the sonar dome. It also explores the possible benefits arising out of new design from the production, acoustic, and hydrodynamic point of view. The results of this study are presented in the form of design parameters (for the bulbous bow) related to the main hull parameters for a set of input data in a narrow range. Finally, the method has been used to design the bulbous bow for a surface combatant vessel.

Practical Hydrodynamic Design of Bulbous Bows for Ships

Although in modern times, bulbous bows have become an integral part of commercial ships, the hydrodynamic design of bulbous bows is still difficult because of costly tank tests and patent-driven knowledge banks. The design of a bulbous bow is needed even at the preliminary design level to predict speed and power. In this work, a design method is presented that combines and extends two famous theories i.e., Kracht (1978a) and Yim (1980) for a particular set of requirements within a narrow range of parameters. The method uses a reanalysis of an approximate linear theory with sheltering effect for resistance estimation, and re-correlation with statistical analysis via a non-linear multivariate regression analysis from existing literature and tank test results available in the public domain. The optimization of design parameters has been done for the design speed. The effect of change in the speed has been discussed and suitably incorporated in the design process. In the present work, the effect of production constraints on the design of bulb parameters has also been briefly examined. The results of this study are presented in the form of design parameters related to main hull parameters for a set of input data in a narrow range. The first six parameters have been derived by re-correlation with statistical analysis and the seventh parameter by reanalysis of an approximate linear theory with sheltering effect for resistance estimation. Finally, a design example, which includes tank test results, of an additive bulbous bow for a container ship has been presented.

The effects of buoyant crust on the gravitational instability of thickened mantle lithosphere at zones of intracontinental convergence

Using both numerical experiments and approximate linear theory, we examine the stability of a 2-D, three-layered viscous system that represents crust over mantle lithosphere, overlying asthenosphere. If horizontal shortening of lithosphere occurs, as in mountain building, the development of gravitational instability may follow one of two distinct paths. Localized downwelling occurs beneath the region of horizontal shortening and crustal thickening, if the layer representing crust has a large viscosity coefficient, is relatively dense or is thin, relative to the underlying layer that represents mantle lithosphere. Conversely, for a low viscosity ratio or buoyant crust, downwelling flow develops on the flanks of the zone of convergence, and the mantle lithosphere layer thins beneath the central region. Thinning in such a setting would facilitate volcanism and high-temperature metamorphism during the development of a mountain belt. A series of numerical experiments with different density and viscosity structures, including runs with non-Newtonian viscosity, show that this general description is insensitive to minor differences in the depth distributions of density and viscosity. The tendency of structures with relatively thick crustal layers (or relatively thin lithospheric mantle) to favour paired, marginal downwellings may account for sustained volcanism during intracontinental deformation where lithosphere initially is warm, thin and weak. The western part of the Tien Shan in Asia and the region including and surrounding the Alboran Sea illustrate some of the features associated with paired, marginal downwellings. The Western Transverse Ranges of California and the Southern Alps of New Zealand share features associated with single downwelling flow beneath the axes of the belts.

On the importance of combining wavelet-based nonlinear approximation with coding strategies

This paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory. Contrasting linear and nonlinear approximation spaces, we show that there are interesting classes of functions/random processes which are much more compactly represented by wavelet-based nonlinear approximation. These classes include locally smooth signals that have singularities, and provide a model for many signals encountered in practice, in particular for images. However, we also show that nonlinear approximation results do not always translate to efficient compress on strategies in a rate-distortion sense. Based on this observation, we construct compression techniques and formulate the family of functions/stochastic processes for which they provide efficient descriptions in a rate-distortion sense. We show that this family invariably leads to Besov spaces, yielding a natural relationship among Besov smoothness, linear/nonlinear approximation order, and compression performance in a rate-distortion sense. The designed compression techniques show similarities to modern high-performance transform codecs, allowing us to establish relevant rate-distortion estimates and identify performance limits.

CONVEX OPTIMAL CONTROL DESIGN VIA PIECEWISE LINEAR APPROXIMATION

In convex optimization techniques for optimal control design, the main challenge is to manage an infinite dimensional Youla parameter. To make the problem tractable, a finite basis has to be defined. While delay basis has been shown efficient for discrete LTI plants, common bases for continuous applications are generally inefficient and numerically bad conditioned. This paper presents a straightforward functional basis derived from piecewise linear approximation theory. Several associated results on LTI systems, related with convolution product and Laplace transformation, are developed.