Momentum-accelerated randomized geometric iterative methods for curve and surface approximation

Improving geometric iterative approximation methods using local approximations

Newton Geometric Iterative Method for B-Spline Curve and Surface Approximation

Geometric iterative methods, including progressive iterative approximation and geometric interpolation methods, are efficient for fitting a given data set. With the development of big data technology, the number of fitting data points has become massive, and the progressive iterative approximation for least-squares fitting (LSPIA) is generally applied to fit mass data. Combining the Schulz iterative method for calculating the Moore–Penrose generalized inverse matrix with the traditional LSPIA method, this paper presents an accelerated LSPIA method for tensor product surfaces and shows that the corresponding iterative surface sequence converged to the least-squares fitting surface of the given data set. The iterative format is that of a non-stationary iterative method, and the convergence rate increased rapidly as the iteration number increased. Some numerical examples are provided to illustrate that the proposed method has a faster convergence rate.

Background. There are a large number of neural networks that have their advantages and disadvantages, for example, simple, fast and easy to use single-stranded perceptrons are suitable for linear and linearized regression tasks, and more complicated neural networks are expendable in training and prediction time. Therefore, the problem arises for the development of fast and efficient algorithms for training artificial neural networks. An additional factor for researching new methods for training neural networks is finding the smallest training and prediction errors.Objective. The aim of the paper is to search and analyze the properties of the most effective method of training artificial neural networks using a combined approximation of the response surface. Another step is to perform computational experiments on proposed artificial neural networks and compare the results of experiments with known and developed methods.Methods. Analysis of known methods of combined approximation of the response surface was used. New algorithms for training neural networks, based on clustering of data using k-means method were developed. The algorithm with the smallest errors of artificial neural network learning and data prediction is chosen.Results. The results of research of different methods of training of artificial neural networks are given. Peculiarities of the methods of combined approximation of the response surface are analyzed. It is shown that the two methods of combined approximation of the response surface for training of artificial neural networks and prediction confirm the effectiveness of the proposed approach. Combined approximation algorithm is selected, which provides the lowest learning and forecasting errors.Conclusions. It was investigated that developed methods of combined approximation of the response surface allow training neural networks and predicting data with less error than when using autoregressive model with moving average, multilayer perceptron or artificial neural networks of models of geometric transformations without additional data processing.

Massive multiple-input multiple-output provides improved energy efficiency and spectral efficiency in 5G. However it requires large-scale matrix computation with tremendous complexity, especially for data detection and precoding. Recently, many detection and precoding methods were proposed using approximate iteration methods, which meet the demand of precision with low complexity. In this paper, we compare these approximate iteration methods in precision and complexity, and then improve these methods with iteration refinement at the cost of little complexity and no extra hardware resource. By derivation, our proposal is a combination of three approximate iteration methods in essence and provides remarkable precision improvement on desired vectors. The results show that our proposal provides 27%–83% normalized mean-squared error improvement of the detection symbol vector and precoding symbol vector. Moreover, we find the bit-error rate is mainly controlled by soft-input soft-output Viterbi decoding when using approximate iteration methods. Further, only considering the effect on soft-input soft-output Viterbi decoding, the simulation results show that using a rough estimation for the filter matrix of minimum mean square error detection to calculating log-likelihood ratio could provide enough good bit-error rate performance, especially when the ratio of base station antennas number and the users number is not too large.

Integrating capacity analysis with high-speed railway timetabling: A minimum cycle time calculation model with flexible overtaking constraints and intelligent enumeration

ABSTRACTWe present a fast approximate method for three‐dimensional low frequency controlled source electro‐magnetic modeling. We apply the method to a synthetic model in a typical marine controlled source electromagnetic scenario, where conductivity and permittivity are different from the known background medium. For 3D configurations, fast computational methods are relevant for both forward and inverse modelling studies. Since this problem involves a large number of unknowns, it has to be solved efficiently to obtain results in a timely manner, without compromising accuracy. For this reason, the Born approximation, extended Born approximation and iterative extended Born approximation are implemented and compared with the full solution of the conjugate gradient fast Fourier transformation method. These methods are based on an electric field domain integral equation formulation. It is shown here how well the iterative extended Born approximation method performs in terms of both accuracy and speed with different configurations and different source positions. The improved accuracy comes at virtually no additional computational cost. With the help of this method, it is now possible to perform sensitivity analysis using 3D modelling in a timely manner, which is vital for controlled source electromagnetic applications. For forward modeling the solution at the sea‐bottom is of interest, because that is where the receivers are usually located. For inverse modeling, the accuracy of the solution in the target zone is important to obtain reasonably accurate conductivity values from the inversion using this approximate solution method. Our modelling studies show that the iterative extended Born approximation method is fast and accurate for both forward and inverse modelling. Sensitivity analysis as a function of the source position and different reservoir sizes validate the accuracy of the iterative extended Born approximation.

Force-based multiphysics coupling methods have become popular since they provide a simple and efficient coupling mechanism, avoiding the difficulties in formulating and implementing a consistent coupling energy. They are also the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. However, the development of efficient and reliable iterative solution methods for the force-based approximation presents a challenge due to the non-symmetric and indefinite structure of the linearized force-based quasicontinuum approximation, as well as to its unusual stability properties. In this paper, we present rigorous numerical analysis and computational experiments to systematically study the stability and convergence rate for a variety of linear stationary iterative methods.

On the convergence of high-order Gargantini–Farmer–Loizou type iterative methods for simultaneous approximation of polynomial zeros

Simulation of stationary non-Gaussian stochastic vector processes using an eigenvalue-based iterative translation approximation method

Formal solutions are given for the integral equations governing the mass or unidirectional magnetic dipole densities on equivalent strata. The densities are given in terms of infinite series in the derivatives of the potential fields observed on planes parallel to the strata. In the magnetic case, the formal solution applies to dipoles with moments normal to the strata. A truncation of the series leads to an approximation method that is applicable to the longwave components of the densities. An iterative approximation method given for the case of oblique field intensities and moments is applicable when the field and source vectors deviate little from the vertical.

In this paper, we present and define second kind linear mixed voltera-fredholm-integro differential equation (MVF-IDE-SK) and use two iterative approximate methods like Successive approximation method (SAM) and Adomian decomposition method (ADM), to solve this problem and algorithms of procedure for methods. The numerical problems were solved by mention methods, the outcomes are compared with the exact solution. The study confirm that techniques are valid, and can be generally applied to solve MVF-IDE-SK. For additional illustration. The convergence and uniqueness solution of the problems are stated and proved. The comparison between approximate and exact solutions, the error analysis and the computational efficacy, respectively.

Numerical estimates of the measured electromotive force (EMF) for the electric field excited by a multi-turn solenoidal coil depend on the way the field source is approximated. The simple representation of the source as a point dipole is correct for measurements in the far field. For measurements in the vicinity of the field source, this method of approximation may give unreliable results. This paper proposes two ways of approximating a multi-turn solenoidal coil: surface approximation or single-coil approximation. Here we present the numerical simulation of the downhole logging procedure. As a result, we show the three-dimensional electromagnetic field, an electromotive force (EMF) induced in the receiver coils, and the phase difference of EMF depending on the method of source approximation. To calculate the three-dimensional electromagnetic field, the vector finite element method on the tetrahedral unstructured partition was used. The difference in the electromagnetic field configuration for different source approximation methods is shown. The effect of the number of cores on the time of solving a finite element system of linear algebraic equations is demonstrated. The use of a larger number of processor cores makes it possible to significantly reduce the time of solving the problem.KeywordsVector finite element methodSLAEElectromagnetic fieldSolenoidal source approximationHelmholtz equation

LSPIA, (stochastic) gradient descent, and parameter correction

On the convergence of Gander’s type family of iterative methods for simultaneous approximation of polynomial zeros