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

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) .

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

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

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

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 .

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

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

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

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.

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

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.

The Modified Trapezoidal Rule for Computing Hypersingular Integral on Interval

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.

QUADRANT-SEARCHING: A NEW TECHNIQUE TO REDUCE THE SEARCH SPACE OF THE INVERSE PROBLEM OF EIT USING BEM TO THE DIRECT PROBLEM

The image reconstruction using the EIT (Electrical Impedance Tomography) technique is a nonlinear and ill-posed inverse problem which demands a powerful direct or iterative method. A typical approach for solving the problem is to minimize an error functional using an iterative method. In this case, an initial solution close enough to the global minimum is mandatory to ensure the convergence to the correct minimum in an appropriate time interval. The aim of this paper is to present a new, simple and low cost technique (quadrant-searching) to reduce the search space and consequently to obtain an initial solution of the inverse problem of EIT. This technique calculates the error functional for four different contrast distributions placing a large prospective inclusion in the four quadrants of the domain. Comparing the four values of the error functional it is possible to get conclusions about the internal electric contrast. For this purpose, initially we performed tests to assess the accuracy of the BEM (Boundary Element Method) when applied to the direct problem of the EIT and to verify the behavior of error functional surface in the search space. Finally, numerical tests have been performed to verify the new technique.

Characterisation of Composite Material Properties by an Inverse Technique

An inverse technique based on vibration tests to characterise isotropic, orthotropic and viscoelastic material properties of advanced composites is developed. An optimisation using the planning of experiments and response surface technique to minimise the error functional is applied to decrease considerably computational expenses. The inverse technique developed is tested on aluminium plates and applied to characterise orthotropic material properties of laminated composites and viscoelastic core material properties of sandwich composites.