scholarly journals Numerical approximation to ODEs using the error functional

2012 ◽  
Vol 140 (12) ◽  
pp. 4295-4308 ◽  
Author(s):  
L. Bayón ◽  
J. M. Grau ◽  
M. M. Ruiz ◽  
P. M. Suárez
Keyword(s):  
Numerical Approximation ◽  
Error Functional ◽  
The Error Functional

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 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 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.

Implementation, study and calibration of a modified ASM2d for the simulation of SBR processes

2001 ◽  
Vol 43 (3) ◽  
pp. 69-76 ◽  
Author(s):  
s. Marsili Libelli ◽  
P. Ratini ◽  
A. Spagni ◽  
G. Bortone
Keyword(s):  
Process Model ◽  
Block Diagram ◽  
Basic Mechanism ◽  
Calibration Procedure ◽  
Sensitivity Study ◽  
Trajectory Sensitivity ◽  
Error Functional ◽  
Calibration Accuracy ◽  
Static Sensitivity ◽  
The Error Functional

An enhanced process model for SBRs has been developed. Though the basic mechanism largely draws on the Activated Sludge Model n. 2d, its new features are the splitting of the nitrification stage in a two-step process, according to the well known Nitrosomonas - Nitrobacter oxidation sequence, and an improved XPAO dynamics, involved in the anaerobic/aerobic phosphorus removal process. The model was implemented through the DLL technique allowing complied C++ modules to be linked to an ordinary Simulink block diagram. The static sensitivity study revealed that if the parameter vector is partitioned into subsets of biologically related parameters and calibrated separately, the calibration procedure does not present particularly difficult aspects. Trajectory sensitivity showed also to which extent data collection could be optimised in order to improve calibration accuracy. The study of the shape of the error functional generated by parameters couples allows a much more effective calibration strategy.

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.

Characterisation of Composite Material Properties by an Inverse Technique

2007 ◽  
Vol 345-346 ◽  
pp. 1319-1322 ◽  
Author(s):  
Evgeny Barkanov ◽  
Andris Chate ◽  
Sandris Ručevskis ◽  
Eduards Skukis
Keyword(s):  
Material Properties ◽  
Core Material ◽  
Inverse Technique ◽  
Error Functional ◽  
Viscoelastic Core ◽  
Response Surface Technique ◽  
The Error Functional ◽  
Vibration Tests ◽  
Planning Of Experiments ◽  
Surface 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.

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