scholarly journals Экстремальная функция интерполяционных формул в пространстве W2(2,0)

2021 ◽  
pp. 123-132
Author(s):  
A.K. Boltaev ◽  
Kh.M. Shadimetov ◽  
F.A. Nuraliev
Keyword(s):  
Hilbert Space ◽  
Explicit Expression ◽  
Computational Methods ◽  
Interpolation Formula ◽  
Optimal Interpolation ◽  
Computational Mathematics ◽  
Error Functional ◽  
The Error Functional ◽  
Norm Of The Error

One of the main problems of computational mathematics is the optimization of computational methods in functional spaces. Optimization of computational methods are well demonstrated in the problems of the theory of interpolation formulas. In this paper, we study the problem of constructing an optimal interpolation formula in a Hilbert space. Here, using the Sobolev method, the first part of the problem is solved, i.e., an explicit expression of the square of the norm of the error functional of the optimal interpolation formulas in the Hilbert space W2(2,0) is found. Одна из основных проблем вычислительной математики — оптимизация вычислительных методов в функциональных пространствах. Оптимизация вычислительных методов хорошо проявляется в задачах теории интерполяционных формул. В данной статье исследуется проблема построения оптимальной интерполяционной формулы в гильбертовом пространстве. Здесь с помощью метода Соболева решается первая часть задачи — явное выражение квадрата нормы функционала погрешности оптимальных интерполяционных формул в гильбертовом пространстве W2(2,0) .

scholarly journals Optimal interpolation formulas with derivative in the space L(m)2(0,1)

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5661-5675
Author(s):  
M.Kh. Shadimetov ◽  
A.R. Hayotov ◽  
F.A. Nuraliev
Keyword(s):  
Order Of Convergence ◽  
Linear Equations ◽  
Interpolation Formula ◽  
Discrete Analog ◽  
Optimal Interpolation ◽  
System Of Linear Equations ◽  
Lagrange Method ◽  
Error Functional ◽  
The Error Functional ◽  
Norm Of The Error

The paper studies the problem of construction of optimal interpolation formulas with derivative in the Sobolev space L(m)2 (0,1). Here the interpolation formula consists of the linear combination of values of the function at nodes and values of the first derivative of that function at the end points of the interval [0,1]. For any function of the space L(m)2 (0, 1) the error of the interpolation formulas is estimated by the norm of the error functional in the conjugate space L(m)* 2 (0,1). For this, the norm of the error functional is calculated. Further, in order to find the minimum of the norm of the error functional, the Lagrange method is applied and the system of linear equations for coefficients of optimal interpolation formulas is obtained. It is shown that the order of convergence of the obtained optimal interpolation formulas in the space L(m)2 (0,1) is O(hm). In order to solve the obtained system it is suggested to use the Sobolev method which is based on the discrete analog of the differential operator d2m= dx2m. Using this method in the cases m = 2 and m = 3 the optimal interpolation formulas are constructed. It is proved that the order of convergence of the optimal interpolation formula in the case m = 2 for functions of the space C4(0,1) is O(h4) while for functions of the space L(2)2 (0,1) is O(h2). Finally, some numerical results are presented.

scholarly journals TOP EVALUATION FOR THE RA TION FOR THE RATE OF FUNC TE OF FUNCTIONAL OF ERROR AL OF ERROR WEIGHT CUBATURE FORMUL TURE FORMULA IN SP A IN SPACE

2019 ◽  
Vol 3 (4) ◽  
pp. 32-37
Author(s):  
Ozodjon Isomidinovich Jalolov ◽  
◽  
Khurshidzhon Usmanovich Khayatov
Keyword(s):  
Sobolev Space ◽  
Upper Bound ◽  
Cubature Formula ◽  
Integration Formulas ◽  
Error Functional ◽  
The Error Functional ◽  
Error Weight ◽  
Norm Of The Error ◽  
Approximate Integration

An upper bound is obtained for the norm of the error functional of the weight cubature formula in the Sobolev space . The modern formulation of the problem of optimization of approximate integration formulas is to minimize the norm of the error functional of the formula on the selected normalized spaces. In these works, the problem of optimality with respect to some definite space is investigated. Most of the problems are considered in the Sobolev space

scholarly journals The order of convergence of an optimal quadrature formula with derivative in the space W(2,1)2

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3835-3844
Author(s):  
A.R. Hayotov ◽  
R.G. Rasulov
Keyword(s):  
Quadrature Formula ◽  
Order Of Convergence ◽  
Discrete Analog ◽  
Trapezoidal Rule ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
First Derivative ◽  
Error Functional ◽  
The Error Functional ◽  
Norm Of The Error

The present work is devoted to extension of the trapezoidal rule in the space W(2,1)2. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of an integrand. Using the discrete analog of the operator d2/dx2-1 the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set F = span{1,x,ex,e-x}. Finally, in the space W(2,1) 2 the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space L(2)2 .

scholarly journals On Error Estimates in Sp for Cubature Formulas Exact for Haar Polynomials

2020 ◽  
pp. 398-413
Author(s):  
Kirill A. Kirillov
Keyword(s):  
Error Estimates ◽  
Dimensional Case ◽  
Cubature Formulas ◽  
Error Functional ◽  
The Error Functional ◽  
Norm Of The Error

On the spaces Sp, an upper and lower estimates for the norm of the error functional cubature formulas possessing the Haar d-property are obtained for the n-dimensional case

Improving the Generalization Properties of Radial Basis Function Neural Networks

1991 ◽  
Vol 3 (4) ◽  
pp. 579-588 ◽  
Author(s):  
Chris Bishop
Keyword(s):  
Neural Networks ◽  
Radial Basis Function ◽  
Basis Function ◽  
Learning Algorithm ◽  
Error Functional ◽  
Radial Basis ◽  
The Error Functional ◽  
Network Mapping ◽  
Nonlinear Mappings

An important feature of radial basis function neural networks is the existence of a fast, linear learning algorithm in a network capable of representing complex nonlinear mappings. Satisfactory generalization in these networks requires that the network mapping be sufficiently smooth. We show that a modification to the error functional allows smoothing to be introduced explicitly without significantly affecting the speed of training. A simple example is used to demonstrate the resulting improvement in the generalization properties of the network.

scholarly journals Learning Precise Spike Train–to–Spike Train Transformations in Multilayer Feedforward Neuronal Networks

2016 ◽  
Vol 28 (5) ◽  
pp. 826-848 ◽  
Author(s):  
Arunava Banerjee
Keyword(s):  
Spike Train ◽  
Gradient Descent ◽  
Probabilistic Models ◽  
Spike Timing ◽  
Inhibitory Neurons ◽  
Closed Form Solutions ◽  
Learning Framework ◽  
Error Functional ◽  
The Error Functional ◽  
Quantities Of Interest

We derive a synaptic weight update rule for learning temporally precise spike train–to–spike train transformations in multilayer feedforward networks of spiking neurons. The framework, aimed at seamlessly generalizing error backpropagation to the deterministic spiking neuron setting, is based strictly on spike timing and avoids invoking concepts pertaining to spike rates or probabilistic models of spiking. The derivation is founded on two innovations. First, an error functional is proposed that compares the spike train emitted by the output neuron of the network to the desired spike train by way of their putative impact on a virtual postsynaptic neuron. This formulation sidesteps the need for spike alignment and leads to closed-form solutions for all quantities of interest. Second, virtual assignment of weights to spikes rather than synapses enables a perturbation analysis of individual spike times and synaptic weights of the output, as well as all intermediate neurons in the network, which yields the gradients of the error functional with respect to the said entities. Learning proceeds via a gradient descent mechanism that leverages these quantities. Simulation experiments demonstrate the efficacy of the proposed learning framework. The experiments also highlight asymmetries between synapses on excitatory and inhibitory neurons.

scholarly journals An optimal interpolation formula

1977 ◽  
Vol 20 (3) ◽  
pp. 264-274 ◽  
Author(s):  
B.D Bojanov ◽  
V.G Chernogorov
Keyword(s):  
Interpolation Formula ◽  
Optimal Interpolation

scholarly journals The Modified Trapezoidal Rule for Computing Hypersingular Integral on Interval

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jin Li ◽  
Xiuzhen Li
Keyword(s):  
Error Function ◽  
Quadrature Rule ◽  
Posteriori Error ◽  
Trapezoidal Rule ◽  
Boundary Element Methods ◽  
Posteriori Error Estimate ◽  
Order Of Accuracy ◽  
Hypersingular Integrals ◽  
Error Functional ◽  
The Error Functional

The modified trapezoidal rule for the computation of hypersingular integrals in boundary element methods is discussed. When the special function of the error functional equals zero, the convergence rate is one order higher than the general case. A new quadrature rule is presented and the asymptotic expansion of error function is obtained. Based on the error expansion, not only do we obtain a high order of accuracy, but also a posteriori error estimate is conveniently derived. Some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms.

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