Approximation Properties Of Functions Research Articles

Application of neural network in integration of shape from shading and stereo

In this paper, a simple and efficient approach is presented for the reconstruction of 3-D surfaces using the integration of shape from shading (SfS) and stereo. First, a new SfS algorithm is derived to obtain the depth-map of a 3-D surface using linear and generalized Lambertian reflectance model. Later, the accuracy of the depth-map is improved by integrating stereo depth data. The stereo sparse depth data are obtained at the points which have higher similarity score in the rectified pair of stereo images. A feed-forward neural network is used to integrate the SfS and stereo depth data due to its strong nonlinear function approximation property. The integration process is based on the correction of 3-D visible surface obtained from SfS using the stereo data. The experiments have been performed on real and synthetic images to demonstrate the usability and accuracy of the approach.

On the instability of the Riemann hypothesis for curves over finite fields

We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) an analog of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. No number theory is required to understand the theorems and their proofs, for it is known that the zeta-functions of curves over finite fields are very explicit meromorphic functions. We study the approximation properties of these meromorphic functions.

Neural networks based airfoil generation for a given [formula omitted] using Bezier–PARSEC parameterization

Determining the airfoil geometry from a given Cp-distribution is an inverse problem of paramount importance specially in the context of variable geometry aerodynamic platforms. This work describes the implementation of artificial neural nets for the airfoil geometry determination. Instead of using full coordinates of the airfoil, Bezier–PARSEC 3434 parameters have been used to describe an airfoil. Some of these parameters have been determined using a Genetic Algorithm. In the second stage Cp-distribution in terms of cl, cd and cm for 10 angles of attack has been input into three different neural nets for learning and then estimating the corresponding BP3434 parameters. Feed-forward backpropagation, Generalized regression and Radial basis neural nets have been trained and then compared in terms of performance and regression statistics. The work establishes the superiority of feed-forward backpropagation neural nets. The result is partly due to good function approximation properties of the neural architecture and partly due to the use of Bezier–PARSEC 3434 parameterization scheme.

Approximation and regularization properties of augmented penalty functions in convex programming

The possibilities of using augmented penalty functions for the regularization and optimal correction of convex programming problems are investigated. Convergence conditions are formulated for the corresponding methods and an iteration algorithm for a linear optimization problem is proposed.

Analysis of spectral approximations using prolate spheroidal wave functions

In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of non-periodic functions in Sobolev spaces. These results serve as an indispensable tool for the analysis of PSWF spectral methods. A PSWF spectral-Galerkin method is proposed and analyzed for elliptic-type equations. Illustrative numerical results consistent with the theoretical analysis are also presented.

An approximation property of exponential functions

We solve the inhomogeneous linear first order differential equations of the form y′(x) − λy(x) = Σm=0∞am(x − c)m, and prove an approximation property of exponential functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equations of the form y′(x) = λy(x) in a special class of analytic functions.

Quelques propriétés d'approximation des éléments finis de Nédélec, application à l'analyse a posteriori

We prove some approximation properties of nonsmooth functions in the space constructed from Nédélec finite elements of order 1. They rely either on the Nédélec operator or on a Clément type regularization operator linked to these elements. The main application of these results is the a posteriori analysis of the error when the discretization involves this space, we present a basic example. To cite this article: C. Bernardi, F. Hecht, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

A NEW VERSION OF ELMAN NEURAL NETWORKS FOR DYNAMIC SYSTEMS MODELING AND CONTROL

Elman network is a class of recurrent neural networks used for function approximation. The main problem of this class is that its structure has a set of global sigmoid functions at its hidden layer. That means that if the operating conditions of a process be identified, are changed the function approximation property of the network is degraded. This paper introduces a new version of the Elman network named Elman Recurrent Wavelet Neural Network (ERWNN). It merges the multi- resolution property of the wavelets and the learning capabilities of the Elman neural network to inherit the advantages of the two paradigms and to avoid their drawbacks. Stability and convergence property is proven for the proposed network. The paper also develops a model reference control scheme using the proposed ERWNN. The proposed scheme belongs to indirect adaptive control schemes. The dynamic back propagation (DBP) algorithm is employed to train both the two networks structured for the indirect control scheme. This paper derives also the plant sensitivity for adjusting the parameters of the developed controller. The advantages of this new version of ERWNN in modeling and controlling time intensive dynamic processes, are reflected in our simulation results.

신경회로망을 이용한 적응 고차조화제어 기법 연구

In this paper, adaptive higher harmonic control technique using Neural Networks (NN) is proposed. First, linear transfer function is estimated to relate the input harmonics and output harmonics, then NN which has the universal function approximation property is applied to expand application range of the transfer function. Optimal control gain matrix computed from the transfer function is used to train NN weights. Online weight adaptation laws are derived from Lyapunov`s direct method to guarantee internal stability. Results of the simulation of 6-input 2-output nonlinear system show that adaptive HHC is applicable to the system with uncertain transfer function.㠌눀؀㌵㔮㐻⠀灵扬楣⁡摭楮楳瑲慴楯渠☠浩汩瑡特⁳捩敮捥

Fully tensorial nodal and modal shape functions for triangles and tetrahedra

This paper presents nodal and modal shape functions for triangle and tetrahedron finite elements. The functions are constructed based on the fully tensorial expansions of one-dimensional polynomials expressed in barycentric co-ordinates. The nodal functions obtained from the application of the tensorial procedure are the standard h-Lagrange shape functions presented in the literature. The modal shape functions use Jacobi polynomials and have a natural global C0 inter-element continuity. An efficient Gauss–Jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices. An example illustrates the approximation properties of the modal functions. Copyright © 2005 John Wiley & Sons, Ltd.

Groundwater Pollution Source Identification and Simultaneous Parameter Estimation Using Pattern Matching by Artificial Neural Network

Pollution of groundwater often occurs because of unknown disposal of toxic wastes, especially from industrial sites, or due to undetected leakage from pipes, waste storage containers, or underground tanks. The determination of pollution sources by using only concentration measurement data in the aquifer is analogous to reconstructing the history of events that has occurred in the aquifer over a time horizon. Identification of unknown groundwater pollution sources becomes more difficult when the hydrogeologic flow and transport parameters are also unknown. A trained artificial neural network (ANN) can be utilized to simultaneously solve the problems of estimating unknown groundwater pollution sources and estimating unknown hydrogeologic parameters (hydraulic conductivity, porosity, and dispersivities). In this article, the universal function approximation property of a multilayer, feed-forward ANN was utilized to estimate temporally and spatially varying unknown pollution sources, as well as to provide a reliable estimation of unknown flow and transport parameters. ANN was trained on patterns of simulated data using a back-propagation algorithm. A set of source fluxes and temporally varying simulated concentration measurements constituted the pattern for training. This article also describes a potential applicability of this methodology by using an illustrative example. Additionally, the methodology performance is evaluated under varying concentration measurement errors. The limited performance evaluations show that the proposed methodology performs reasonably well even with large measurement errors.

Approximation properties of wavelets and relations among scaling moments II

Abstract A new orthonormality condition for scaling functions is derived. This condition shows a close connection between orthonormality and relations among discrete scaling moments. This new condition in connection with certain approximation properties of scaling functions enables to prove new relations among discrete scaling moments and consequently the same relations for continuous scaling moments.

신경회로망을 이용한 헬리콥터 적응 비선형 제어

본 논문에서는 광범위한 비선형 함수 근사 성질을 갖고 있는 온라인 적응 신경회로망을 이용하여 헬리콥터 비행 제어 시스템을 설계하였다. 기존의 시스템 모델링 오차를 보상하는 방식과는 달리, 시스템의 입출력 정보를 통해 피드백 선형화 기법에서 필요한 두 개의 비선형 함수를 신경회로망을 이용하여 대체하는 방법을 적용하였다. 두 개의 비선형 함수를 신경회로망으로 대체하여 구성된 폐회로 시스템의 추적 성능과 내부 안정성을 보장하기 위하여 신경회로망의 가중치 학습 방법을 리야프노프 함수를 이용하여 유도하였다. 그리고 헬리콥터 저속 비행 모드에 대한 수치 시뮬레이션 결과를 통해 신경회로망을 적용한 제어 시스템의 성능을 검증하였다. In this paper, the helicopter flight control system using online adaptive neural networks which have the universal function approximation property is considered. It is not compensation for modeling errors but approximation two functions required for feedback linearization control action from input/output of the system. To guarantee the tracking performance and the stability of the closed loop system replaced two nonlinear functions by two neural networks, weight update laws are provided by Lyapunov function and the simulation results in low speed flight mode verified the performance of the control system with the neural networks.

Cluster Sets and Approximation Properties of Quasi-Continuous and Cliquish Functions

Cluster Sets and Approximation Properties of Quasi-Continuous and Cliquish Functions

An approximation property of harmonic functions in Lipschitz domains and some of its consequences

An extension of an inequality of J. B. Garnett (1979), with improvements by B. E. J. Dahlberg (1980), on an approximation property of harmonic functions is proved. The weighted inequality proved here was suggested by the work of J. Pipher (1993) and it implies an extension of a result of S. Y. A. Chang, J. Wilson and T. Wolff (1985) and C. Sweezy (1991) on exponential square integrability of the boundary values of solutions to second-order linear differential equations in divergence form. This implies a solution of a problem left open by R. Bañuelos and C. N. Moore (1989) on sharp estimates for the area integral of harmonic functions in Lipschitz domains.

An approximation property of elastic Herglotz wave functions and its application in the linear sampling method

Herglotz wave functions play an important role in a class of reconstruction methods for inverse scattering problems known as linear sampling methods. We here consider these functions in the setting of linearized elasticity and derive representations in terms of eigenfunctions to the Navier operator in two spatial dimensions. We then show the important property that the elastic Herglotz Wave functions are dense in the space of solutions to the Navier equation with respect to the [H1 (D)]2 norm for any bounded Lipschitz domain D. The proof of this property in three-dimensions, not essentially different from the 2D argument, is also outlined. The paper is concluded with an application of the approximation property in the mathematical foundation of the linear sampling method for the reconstruction of rigid obstacles from the knowledge of the far field operator.

An approximation property of elastic Herglotz wave functions and its application in the linear sampling method

Herglotz wave functions play an important role in a class of reconstruction methods for inverse scattering problems known as linear sampling methods. We here consider these functions in the setting of linearized elasticity and derive representations in terms of eigenfunctions to the Navier operator in two spatial dimensions. We then show the important property that the elastic Herglotz Wave functions are dense in the space of solutions to the Navier equation with respect to the [H1 (D)]2 norm for any bounded Lipschitz domain D. The proof of this property in three-dimensions, not essentially different from the 2D argument, is also outlined. The paper is concluded with an application of the approximation property in the mathematical foundation of the linear sampling method for the reconstruction of rigid obstacles from the knowledge of the far field operator.

APPROXIMATION BY PIECEWISE CONSTANT FUNCTIONS IN A BV METRIC

We study the approximation properties of piecewise constant functions with respect to triangular and rectangular finite elements in a metric defined on functions of bounded variation. We apply our results to a thin film model for martensitic crystals and to the approximation of deformations with microstructure.

Q-Moduli of Continuity in Hp([formula omitted]), p>0, and an Inequality of Hardy and Littlewood

Some aspects of the interplay between approximation properties of analytic functions and the smoothness of its boundary values are discussed. One main result describes the equivalence of a special q-modulus of continuity and an intrinsic K-functional. Further, a generalization of a theorem due to G. H. Hardy and J. E. Littlewood (1932, Math. Z.34, 403–439) on the growth of fractional derivatives is deduced with the help of this K-functional.

Adaptive output feedback control of nonlinear systems using neural networks

A direct adaptive output feedback control design procedure is developed for highly uncertain nonlinear systems, that does not rely on state estimation. The approach is also applicable to systems of unknown, but bounded dimension. In particular, we consider single-input/single-output nonlinear systems, whose output has known, but otherwise arbitrary relative degree. This includes systems with both parameter uncertainty and unmodeled dynamics. The result is achieved by extending the universal function approximation property of linearly parameterized neural networks to model unknown system dynamics from input/output data. The network weight adaptation rule is derived from Lyapunov stability analysis, and guarantees that the adapted weight errors and the tracking error are bounded. Numerical simulations of an output feedback controlled van der Pol oscillator, coupled with a linear oscillator, is used to illustrate the practical potential of the theoretical results.