Nonlinear Approximation Problems Research Articles

Behavioral Modeling of Memristors under Harmonic Excitation.

Memristors are devices built on the basis of fourth passive electrical elements in nanosystems. Because of the multitude of technologies used for memristor implementation, it is not always possible to obtain analytical models of memristors. This difficulty can be overcome using behavioral modeling, which is when mathematical models are constructed according to the input-output relationships on the input and output signals. For memristor modeling, piecewise neural and polynomial models with split signals are proposed. At harmonic input signals of memristors, this study suggests that split signals should be formed using a delay line. This method produces the minimum number of split signals and, as a result, simplifies behavioral models. Simplicity helps reduce the dimension of the nonlinear approximation problem solved in behavioral modeling. Based on the proposed method, the piecewise neural and polynomial models with harmonic input signals were constructed to approximate the transfer characteristic of the memristor, in which the current dynamics are described using the Bernoulli differential equation. It is shown that the piecewise neural model based on the feedforward network ensures higher modeling accuracy at almost the same complexity as the piecewise polynomial model.

Analysis and Research on Nonlinear Complex Function Approximation Problem Based on Deep Learning

Shallow models have limited ability to express high-dimensional nonlinear complex functions. Based on deep learning, a Gaussian radial basis function neural network (RBFNN) is proposed, which is an analysis method of nonlinear complex function approximation based on the Gaussian-RBFNN model. The proposed method can approximate single-variable and binary complex nonlinear functions, and the approximation error is less than 0.1, which can achieve the ideal approximation effect. Finally, the proposed method is applied to analyze the nonlinear complex stock prediction approximation problem, and the effectiveness and practicality of the method are further verified. It can be seen that compared with the BP neural network model, this model performs better in average training time, training standard error, Ratt, RATR, and trmse For RBFNN; the training only needs 50 iterations, while the traditional BP needs 7100 training to gradually converge. In the actual function test, after 275 times of training, the convergence speed gradually tends to be stable, and the predicted value is closer to the actual value. Therefore, the model can be used to analyze practical nonlinear complex function approximation problems.

ЗАСТОСУВАННЯ ШТУЧНОГО ІНТЕЛЕКТУ ДО МОДЕЛЮВАННЯ КУРСУ БІТКОЙНА

Artificial neural networks are modern methods suitable for solving the problem of nonlinear dependency approximation, which is successfully applied in many fields. This paper compares the predictive capabilities of Back Propagation, Radial Basis Function, Extreme Learning Machine, and Long-Short Term Memory neural networks to determine which artificial intelligence algorithm is best for modeling the price of Bitcoin opening. The criterion for comparing network performance was the standard deviation, the mean absolute deviation, and the accuracy of predicting the direction of change of course. At the same time, in the study of time series, it is recommended to perform a comprehensive data analysis using appropriate networks, depending on the length of the series and the specificity of the database.

Nonlinear differential equations piecewise continuous linear approximation in computational structural analysis

This paper presents a piecewise-continuous approach to nonlinear differential equations system approximation problem. In many cases, the nonlinear differential equations right-hand sides’ linearization is extremely difficult or even impossible. Then piecewise continuous and piecewise linear approximations of nonlinear differential equations can be used. The piecewise continuous linear differential equations can improve piecewise linear differential equations by increasing the number of separate linear differential equations systems areas and thereby reducing errors on the borders of these areas. The matrices of piecewise continuous linear differential equations system for steady-state points are estimated using nonlinear model time responses, nonlinear programming and random search method. The proposed approach application results are presented.

Piecewise continuous approach to nonlinear differential equations approximation problem of computational structural mechanics

This paper presents a nonlinear differential equations system piecewise continuous approximation. The piecewise continuous approximation improves piecewise linear approximation through reducing the errors at the boundaries of different linear differential equations systems areas. The matrices of piecewise continuous differential and algebraic equations systems are estimated using nonlinear differential equations system time responses and random search method. The results of proposed approach application are presented.

Autonomous location control of a robot manipulator for live maintenance of high-voltage transmission lines

PurposeThis paper aims to develop a robot for tightening charged bolt to solve the shortcomings of high labor intensity, low efficiency, high risk and poor reliability in artificially tightening drainage board bolt of strain clamp for high voltage transmission line. Realizing bolt-nut capture and location by manipulator is a critical process to complete the whole working task. To solve such key technology, an autonomous location control method for N-joint robot manipulator based on kinematics was proposed.Design/methodology/approachThrough D-H kinematics analysis under flexible working environment of transmission line, the autonomous location control of double manipulators can be abstracted as a nonlinear approximation problem based on joint inverse kinematics. In addition, regarding the complex coupling relationship among different joint angles and the complex decoupling process which leads to the non-uniqueness of inverse solution, an improved backpropagation (BP) network was proposed based on the combination of dynamic adaptive adjustment of learning rate and variable momentum factor, so that the inverse kinematics of manipulator can be solved and the optimization evaluation mechanism of inverse solution can be presented. The proposed autonomous location control method is of adaptability to flexible environment and structural parameters of different drainage boards. The simulation results verified the effectiveness of the proposed method. Compared with the other location control, this method can achieve faster location speed, higher precision and lower hardware cost. Finally, the field operation test further validated that such autonomous location control method was of strong engineering practicability.FindingsThe proposed autonomous location control method is adaptable to a flexible environment and to the structural parameters of different types of drainage board. Simulation results confirm the effectiveness of the proposed method, which, in comparison with other approaches to location control, can achieve faster location, higher precision and lower hardware cost. Finally, a field test further confirms the engineering practicability of the proposed autonomous location control method.Originality/valueThe proposed method can achieve faster location speed, higher precision which meet the requirement of real-time control relative to the standard BP algorithm. Moreover, it is of strong adaptability to flexible environment and structural parameters for different drainage board. Field operation experiment further validated the engineering practicability of the method.

Piecewise-linear and bilinear approaches to nonlinear differential equations approximation problem of computational structural mechanics

This paper presents a bilinear approach to nonlinear differential equations system approximation problem. Sometimes the nonlinear differential equations right-hand sides linearization is extremely difficult or even impossible. Then piecewise-linear approximation of nonlinear differential equations can be used. The bilinear differential equations allow to improve piecewise-linear differential equations behavior and reduce errors on the border of different linear differential equations systems areas. The matrices of bilinear differential equations system are estimated through nonlinear programming. The results of proposed approach application are presented.

Solving a Uniform Nonlinear Approximation Problem Using Continuous Genetic Algorithm

Solving a Uniform Nonlinear Approximation Problem Using Continuous Genetic Algorithm

Modelling sulphur dioxide levels of Konya city using artificial intelligent related to ozone, nitrogen dioxide and meteorological factors

Increasing industrial developments increased the environmental pollution problems in many cities of the world. Air quality modelling and indexes are used to introduce the information on local air quality indicators in polluted regions. Estimation and monitoring of air quality in the city centres are important due to environmental health and comfort of human-related topics. Air quality approximation is a complicate subject that artificial intelligent techniques are successfully used for modelling the complicated and nonlinear approximation problems. In present study, artificial neural network and an adaptive neuro-fuzzy logic method developed to approximate the impact of certain environmental conditions on air quality and sulphur dioxide pollution level and used with this study in Konya city centre. Data of sulphur dioxide concentrations were collected from 15 selected points of Konya city for prediction of air quality. Using air quality standards, air quality was discussed by considering the sulphur dioxide concentration as independent variables with meteorological parameters. Different meteorological parameters were used for investigation of pollution relation. One of the important modelling tools, adaptive network-based fuzzy inference system model, was used to assess performance by a number of checking data collected from different sampling stations in Konya. The outcomes of adaptive network-based fuzzy inference system model was evaluated by fuzzy quality charts and compared to the results obtained from Turkey and Environmental Protection Agency air quality standards. From the present results, fuzzy rule-based adaptive network-based fuzzy inference system model is a valuable tool prediction and assessment of air quality and tends to propagate accurate results.

Infinite Horizon Optimal Control Problems in the Light of Convex Analysis in Hilbert Spaces

In this paper a class of linear-quadratic infinite horizon optimal control problems is considered. Problems of this type are not only of practical interest. They also appear as an approximation of nonlinear problems. The key idea is to introduce weighted Sobolev spaces as state space and weighted Lebesgue spaces as control spaces into the problem setting. We investigate the question of existence of an optimal solution in these spaces and establish a Pontryagin type Maximum Principle as a necessary optimality condition including transversality conditions.

E-optimal designs for second-order response surface models

$E$-optimal experimental designs for a second-order response surface model with $k\geq1$ predictors are investigated. If the design space is the $k$-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1 (1977a) 121-132] determined optimal designs in a restricted class of designs (defined by the multiplicity of the minimal eigenvalue) and stated their universal optimality as a conjecture. In this paper, we prove this claim and show that these designs are in fact $E$-optimal in the class of all approximate designs. Moreover, if the design space is the unit ball, $E$-optimal designs have not been found so far and we also provide a complete solution to this optimal design problem. The main difficulty in the construction of $E$-optimal designs for the second-order response surface model consists in the fact that for the multiplicity of the minimum eigenvalue of the "optimal information matrix" is larger than one (in contrast to the case $k=1$) and as a consequence the corresponding optimality criterion is not differentiable at the optimal solution. These difficulties are solved by considering nonlinear Chebyshev approximation problems, which arise from a corresponding equivalence theorem. The extremal polynomials which solve these Chebyshev problems are constructed explicitly leading to a complete solution of the corresponding $E$-optimal design problems.

On nonlinear simultaneous approximation problems

This paper deals with best nonlinear simultaneous approximation problems to two continuous functions under a general monotonic norm on . Characterizations results of Kolmogorov type and of alternation type and uniqueness results for the best simultaneous approximation are established. Applications are given to the set of rational functions.

A novel self-constructing Radial Basis Function Neural-Fuzzy System

This paper proposes a novel self-constructing least-Wilcoxon generalized Radial Basis Function Neural-Fuzzy System (LW-GRBFNFS) and its applications to non-linear function approximation and chaos time sequence prediction. In general, the hidden layer parameters of the antecedent part of most traditional RBFNFS are decided in advance and the output weights of the consequent part are evaluated by least square estimation. The hidden layer structure of the RBFNFS is lack of flexibility because the structure is fixed and cannot be adjusted effectively according to the dynamic behavior of the system. Furthermore, the resultant performance of using least square estimation for output weights is often weakened by the noise and outliers.This paper creates a self-constructing scenario for generating antecedent part of RBFNFS with particle swarm optimizer (PSO). For training the consequent part of RBFNFS, instead of traditional least square (LS) estimation, least-Wilcoxon (LW) norm is employed in the proposed approach to do the estimation. As is well known in statistics, the resulting linear function by using the rank-based LW norm approximation to linear function problems is usually robust against (or insensitive to) noises and outliers and therefore increases the accuracy of the output weights of RBFNFS. Several nonlinear functions approximation and chaotic time series prediction problems are used to verify the efficiency of self-constructing LW-GRBFNIS proposed in this paper. The experimental results show that the proposed method not only creates optimal hidden nodes but also effectively mitigates the noise and outliers problems.

Two approaches to the nonlinear functions approximation problem.

Two approaches to the nonlinear functions approximation problem.

On nonlinear simultaneous approximation problems

In this paper we prove that if Y is a reflexive subspace of a Banach space X, then L ∞(μ, Y) is simultaneously proximinal in L ∞(μ, X). Furthermore if X is reflexive and μ 0 is the restriction of μ to a sub-σ-algebra, then L ∞(μ 0, X) is simultaneously proximinal in L ∞(μ, X).

Nonlinear Approximation by Sums of Exponentials and Translates

In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem. Let $h$ be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands if finitely many perturbed, uniformly sampled data of $h$ are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let $\varphi$ be a given 1-periodic window function as defined in section 4. Further let $f$ be a linear combination of translates of $\varphi$. Determine all shift parameters, all coefficients, and the number of translates if finitely many perturbed, uniformly sampled data of $f$ are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.

Approximation using scattered shifts of a multivariate function

The approximation of a general d-variate function f by the shifts �(� − �), � ∈ � ⊂ R d , of a fixed functionoccurs in many applications such as data fit- ting, neural networks, and learning theory. When � = hZ d is a dilate of the integer lattice, there is a rather complete understanding of the approximation problem (6, 18) using Fourier techniques. However, in most applications the center set � is either given, or can be chosen with complete freedom. In both of these cases, the shift-invariant setting is too restrictive. This paper studies the approximation problem in the caseis arbitrary. It establishes approximation theorems whose error bounds reflect the local density of the points in �. Two different settings are analyzed. The first is when the setis prescribed in advance. In this case, the theorems of this paper show that, in analogy with the classical univariate spline ap- proximation, improved approximation occurs in regions where the density is high. The second setting corresponds to the problem of non-linear approximation. In that setting the setcan be chosen using information about the target function f. We discuss how to 'best' make these choices and give estimates for the approximation error.

Best m-term one-sided trigonometric approximation of some function classes defined by a kind of multipliers

In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm Lp (1 ≤ p ≤ ∞) of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results.

Analytical Approximations for a Class of Strongly Nonlinear Forced Vibration Systems

A strongly cubic nonlinear forced vibration system with single degree of freedom is investigated by means of homotopy analysis method(HAM).Different from other approximate computational method,the HAM is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.The HAM provides us with a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The HAM adjusts and controls progression solution convergence region and the convergence rate through the introduction of auxiliary parameters and auxiliary functions.Therefore it opens up a new approach to the solution of analytical approximation of non-linear problems,and is especially suitable for strong non-linear problems.The computation results indicate that this method can not only solve the steady state solution but also calculate the unsteady state solution,and has good computational accuracy.

An Adaptive Learning Method for the Generation of Fuzzy Inference System from Data

Designing a fuzzy inference system (FIS) from data can be divided into two main phases: structure identification and parameter optimization. First, starting from a simple initial topology, the membership functions and system rules are defined as specific structures. Second, to speed up the convergence of the learning algorithm and lighten the oscillation, an improved descent method for FIS generation is developed. Furthermore, the convergence and the oscillation of the algorithm are systematically analyzed. Third, using the information obtained from the previous phase, it can be decided in which region of the input space the density of fuzzy rules should be enhanced and for which variable the number of fuzzy sets that used to partition the domain must be increased. Consequently, this produces a new and more appropriate structure. Finally, the proposed method is applied to the problem of nonlinear function approximation.