Approximation Of Arbitrary Functions Research Articles

Inverse differential quadrature method: mathematical formulation and error analysis.

Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach by direct approximation of system variables can potentially incur considerable error due to high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques which can better satisfy the requirements of numerical accuracy desirable in solution of high-order systems. To this end, a novel inverse differential quadrature method (iDQM) is proposed for approximation of engineering systems. A detailed formulation of iDQM based on integration and DQM inversion is developed separately for approximation of arbitrary low-order functions from higher derivatives. Error formulation is further developed to evaluate the performance of the proposed method, whereas the accuracy through convergence, robustness and numerical stability is presented through articulation of two unique concepts of the iDQM scheme, known as Mixed iDQM and Full iDQM. By benchmarking iDQM solutions of high-order differential equations of linear and nonlinear systems drawn from heat transfer and mechanics problems against exact and DQM solutions, it is demonstrated that iDQM approximation is robust to furnish accurate solutions without losing computational efficiency, and offer superior numerical stability over DQM solutions.

Designing optimal models of nonlinear MIMO systems based on orthogonal polynomial neural networks

ABSTRACT This paper presents a new method for modelling of dynamic systems by using specially designed orthogonal polynomial neural networks. These networks utilize the feature that the basis made of orthogonal functions can be used for approximation of arbitrary function, while their property of orthogonality enables optimal performances in the sense of both convergence time and approximation error. In this regard, generalized quasi-orthogonal polynomials, specifically tailored for the application in the modelling of complex dynamic systems with time-varying behaviour, are considered. Adaptivity of the designed model is achieved by using variable factors inside the orthogonal basis. The designed orthogonal neural network is applied in modelling of laboratory twin-rotor aero-dynamic system as a representative of nonlinear multiple input-multiple output systems. Detailed comparative analysis is performed for a different number of polynomials in expansion with the purpose of finding the optimal model in the sense of trade-off between model accuracy and complexity.

Approximately Self-Consistent Ensemble Density Functional Theory: Toward Inclusion of All Correlations.

Recent theory developments in ensemble density functional theory (EDFT) promise to bring decades of work for ground states to the practical resolution of excited states, provided newly discovered "density-driven correlations" can be dealt with and adequate effective potentials can be found. This Letter introduces simple theories for both and shows that EDFT using these theories outperforms ΔSCF DFT and time-dependent DFT for low-lying gaps in most of the small atoms and molecules tested, even when all use the same density functional approximations. It thus establishes EDFT as a promising tool for low-cost studies of low-lying excited states and provides a clear route to practical EDFT implementation of arbitrary functional approximations.

ALGORITHMS FOR MANAGING THE LOGICAL STRUCTURE OF THE DATABASE USING A PARAMETRIC MODEL OF COMPETITIVE ACCESS TO QUERIES BASED ON THE RANDOM FOREST METHOD

In article discusses the approach to development of mathematical software for support the process of managing the data schema in relational DBMS in terms of processing of parallel queries stream that compete for data in the hierarchy of the DBMS core memory. The necessity of the formation of a parametric model of queries competitive access. Briefly discusses methods of machine learning, allowing to solve the problem of regression recovery. The use of the random forest method as the most universal method of approximation of arbitrary functions is substantiated. A method of forming a parametric model of competitive access based on the random forest method, as well as an approach with the ensemble of sets of decision trees, which allows to provide the required generalizing ability and stability of the model to partial features and diversity of all types of queries received at the input of the DBMS. The stages of the developed algorithms are presented: ranking query parameters by total execution time and automatic data distribution, allowing you to go from approximating the target system with linear-continuous functions to a set of logical data schema objects, ordered by their effect on time, total query execution time, reducing multi-criteria optimization task to a task optimization by one criterion.

Local Approximation of Arbitrary Functions by Solutions of Nonlocal Equations

We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function is locally s-caloric, up to a small error. The case of non-elliptic and non-parabolic operators is taken into account as well.

Prediction of shear and tensile strength of the diffusion bonded AA5083 and AA7075 aluminium alloy using ANN

Diffusion bonding is a pressure welding technique to establish bonds by inter diffusion of atoms. Bonding characteristics were generated by varying the significant process conditions such as the bonding temperature, the pressing load and the duration of pressure while bonding the aluminium alloys AA5083 and AA7075. Deriving analytical correlation with the process variables to weld strength is quite involved due to the non-linear dependency of the process variables with the mechanical strength of the joints. An arbitrary function approximation mechanism, the artificial neural network (ANN) is therefore employed to develop the models for predicting the mechanical properties of the bonded joints. Back propagation technique, which alters the network weights to minimize the mean square error was used to develop the ANN models. The models were tested, validated and found to be satisfactory with good prediction accuracy.

Adaptive Neural Network Fuzzy Control Plane Double Inverted Pendulum

Double inverted pendulum system is nonlinear and unstable. Fuzzy control uses some expert's experience knowledge and learns approximate reasoning algorithm. For it does not depend on the mathematical model of controlled object, it has been widely used for years. In practical engineering applications, most systems are nonlinear time-varying parameter systems. As the fuzzy control theory lacks of on-line self-learning and adaptive ability, it can not control the controlled object effectively. In order to compensate for these defects, it introduced adaptive, self-organizing, self-learning functions of neural network algorithm. We called it adaptive neural network fuzzy inference system (ANFIS). ANFIS not only takes advantage of the fuzzy control theory of abstract ability, the nonlinear processing ability, but also makes use of the autonomous learning ability of neural network, the arbitrary function approximation ability. The controller was applied to double inverted pendulum system and the simulation results showed that this method can effectively control the double inverted pendulum system.

An Approach for Short Term Traffic Flow Forecasting Based on Genetic Neural Network

In this paper, genetic neural network is applied to forecast the short-term traffic flow and traffic guidance. Because of the factors of time correlation and spatial correlation, we construct the short-term traffic flow forecasting model using back-propagation neural network that has the function of arbitrary nonlinear function approximation. In order to find proper initial values of the neural network weights and threshold quickly, a combination of neural network prediction method is presented. This method utilizes genetic algorithm to choose the initial weights and threshold, and uses L-M algorithm to train sample, which can enhance the global convergence rate. Trained network is used for short-term traffic flow prediction with mean square error as the forecast performance evaluation. The results show that the performance of genetic neural network is better than a separate BP neural network for short-term traffic flow prediction.

Finite Mixture Estimation Algorithm for Arbitrary Function Approximation

The paper considers a new prospect of the arbitrary continuous function approximation from a limited set of input data with the REBMIX algorithm, developed for the finite mixture density estimation. Since the REBMIX estimates the unknown parameters with the unique semiparametric method, it is assumed that it could be used also for the estimation of the unknown parameters in the fields that are not directly connected to density function estimation. For the approximation of the arbitrary continuous function with the REBMIX algorithm, the required procedure is developed in the paper. The results gained by the proposed procedure and by the radial basis function network for three different datasets are compared by calculating the RMSE values between estimated and test output values. The adequacy of the proposed procedure is estimated by using both univariate and bivariate datasets. It can be concluded that with the developed procedure, the REBMIX algorithm can be applied successfully for the continuous function approximation.

Evolutionary selection extreme learning machine optimization for regression

Neural network model of aggression can approximate unknown datasets with the less error. As an important method of global regression, extreme learning machine (ELM) represents a typical learning method in single-hidden layer feedforward network, because of the better generalization performance and the faster implementation. The “randomness” property of input weights makes the nonlinear combination reach arbitrary function approximation. In this paper, we attempt to seek the alternative mechanism to input connections. The idea is derived from the evolutionary algorithm. After predefining the number L of hidden nodes, we generate original ELM models. Each hidden node is seemed as a gene. To rank these hidden nodes, the larger weight nodes are reassigned for the updated ELM. We put L/2 trivial hidden nodes in a candidate reservoir. Then, we generate L/2 new hidden nodes to combine L hidden nodes from this candidate reservoir. Another ranking is used to choose these hidden nodes. The fitness-proportional selection may select L/2 hidden nodes and recombine evolutionary selection ELM. The entire algorithm can be applied for large-scale dataset regression. The verification shows that the regression performance is better than the traditional ELM and Bayesian ELM under less cost gain.

Optimal Piecewise Affine Approximations of Nonlinear Functions Obtained from Measurements

The paper describes a two-stage procedure for obtaining piecewise affine approximations of static nonlinearities obtained from measured data. In the first step we search for a suitable function which fits the data while minimizing the fitting error. Subsequently we show how to approximate, in an optimal fashion, the nonlinear fitting function by a piecewise affine function of pre-specified complexity. We illustrate that approximation of arbitrary nonlinear functions boils down to a series of one-dimensional approximations, rendering the procedure efficient from a computational point of view.

Substructural Time-Varying Parameter Identification Using Wavelet Multiresolution Approximation

Identifying properties of civil engineering structures is an important task for their condition assessment, damage diagnosis, maintenance and repair, and life-cycle management. These structures usually contain a large number of degrees of freedom and exhibit some time-varying or nonlinear behavior, especially under extreme excitation or when damaged. In this study, an offline substructure method based on wavelet multiresolution approximation (WMRA) is proposed for the identification of arbitrary time-varying parameters in a shear-beam building. Assuming that the possible damage location of the building can be identified a priori, a substructural model containing both interface and internal restoring forces can be formulated. The WMRA can then be used to approximate the time-varying damping and stiffness parameters and convert a time-varying parametric identification problem into a time-invariant coefficient estimation problem. To obtain accurate estimation and minimize the computational effort, Akaike’s information criterion is used for the selection of a resolution level and the pruning of insignificant terms. Numerical and experimental results show that the proposed technique can locate and quantify the time-varying parameters in the substructure. The method inherits the wavelet’s excellent capability in the approximation of arbitrary functions and provides a flexible approach for identifying various types of time-varying parameters in shear-beam buildings.

New representation of the remainder in the Bernstein approximation

This short note presents a new representation of the remainder in the Bernstein approximation based on divided differences and some immediate applications. It is the only known representation of the remainder in the Bernstein approximation of arbitrary functions as a convex combination of divided differences of second order on known knots. As an application we obtain sharp inequalities for functions possessing bounded divided differences of second order and a new proof of the classical Weierstrass approximation theorem.

A CLASS OF ALMOST ORTHOGONAL FILTERS

In this paper we define a new class of the almost orthogonal filters. These filters are a generalization of the classical orthogonal filters commonly used in the circuit theory, control system theory, signal processing and process identification. Almost orthogonal filters generate the series of almost orthogonal Legendre functions on the interval (0, ∞). They can be successfully used for the analysis, synthesis and designing of imperfect technical systems. They can also be used for approximation of arbitrary functions on (0, ∞). A new method for obtaining the models of imperfect systems is presented as well. Simulations were performed in order to validate the theoretical results and the comparison of new filters with classical filters is given.

A numerical algorithm for extrapolation of transfer functions

A numerical approach for extrapolation of frequency-domain transfer functions is presented. The algorithm is based on a result of Krein and Nudel'man on approximation of arbitrary functions by minimum-energy transfer functions. Regularization is included yielding a mechanism for adjusting the tradeoff between accuracy and stability. The algorithm is verified by several numerical examples using noise-free, noisy, and experimental input data. The extrapolation method is also applied to the synthesis of optimal finite-bandwidth filters.

L/sup p/ approximation of Sigma-Pi neural networks

A feedforward Sigma-Pi neural network with a single hidden layer of m neurons is given by mSigma(j=1) cjg (nPi(k=1) xk-thetak(j)/lambdak(j)) where cj, thetak(j), lambdak are elements of R. In this paper, we investigate the approximation of arbitrary functions f: Rn-->R by a Sigma-Pi neural network in the Lp norm. An Lp locally integrable function g(t) can approximate any given function, if and only if g(t) can not be written in the form Sigma(j=1)n Sigma(k=0)m alphajk(ln/t/)(j-1)tk.

Universal approximation in p-mean by neural networks

A feedforward neural net with d input neurons and with a single hidden layer of n neurons is given by g(x 1, …, x d)= ∑ j=1 n a jσ ∑ i=1 d w jix i−θ j , where a j , θ j , w ji ∈ R. In this paper we study the approximation of arbitrary functions f: R d → R by a neural net in an L p (μ) norm for some finite measure μ on R d . We prove that under natural moment conditions, a neural net with non-polynomial function can approximate any given function.

Missile Control Using Fuzzy Cerebellar Model Arithmetic Computer Neural Networks

Anovel fuzzyneuralnetworkcontrolsystem architecture,calledthefuzzycerebellarmodelarithmeticcomputer (fuzzy CMAC) neural network, is proposed. This fuzzy neural network architecture exploits a synergism between the original CMAC neural network and the theory of fuzzy logic controller. The fuzzy CMAC is able to perform arbitrary function approximation and can achieve a higher learning speed and greater accuracy than the original CMAC. It also has the capability of conveniently incorporating human knowledge into the system and processing information based on fuzzy inference rules. We evaluated the use of fuzzy CMAC in the design of advanced missile control systems. The method is demonstrated by missile interception simulations, and promising results are obtained.

Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method

The Lasry-Lions regularization method is extended to arbitrary functions. In particular, to any proper lower semicontinuous function f: X→ ℝ ∪{ + ∞} defined on a Hilbert space X and which is quadratically minorized (i. e. f(x)≥−−c(1+‖x‖2)) for some c≧0), is associated a sequence of differentiable functions with Lipschitz continuous derivatives which approximate f from below. Some variants of the method are considered including the case of non quadratic kernels.

On pointwise approximation of arbitrary functions by countable families of continuous functions

The following problem is considered. Given a real-valued function f defined on a topological space X, when can one find a countable familyfn:n∈ω of continuous real-valued functions on X that approximates f on finite subsets of X? That is, for any finite set F⊂X and every real number e>0 one can choosen∈ω such that ∥f(x)−fn(x)∥<e for everyx∈F. It will be shown that the problem has a positive solution if and only if X splits. A space X is said to split if, for any A⊂X, there exists a continuous mapfA:X→Rω such that A=fA−1(A). Splitting spaces will be studied systematically.