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result(s) for
"Chebyshev approximation"
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Chebyshev Approximation of Multivariable Functions by a Constrained Rational Expression
The authors propose a method for constructing the Chebyshev approximation of multivariable functions by the rational expression with the interpolation condition. The idea of the method is based on constructing the limiting power mean approximation by a rational expression with an interpolation condition in the norm of space L
p
as p → ∞. To construct such an approximation, an iterative scheme based on the least squares method with two variable weight functions is used. One weight function ensures the construction of a power mean approximation with the interpolation condition, and the second one specifies the parameters of the rational expression according to its linearization scheme. The convergence of the method is provided by the original method of sequential specification of the values of weight functions, which takes into account the approximation results at previous iterations. The results of test examples confirm the fast convergence of the proposed method of constructing the Chebyshev approximation by a rational expression with a condition.
Journal Article
The Chebyshev Approximation by the Logarithm of a Rational Expression
The authors propose a method for constructing the Chebyshev approximation with an absolute error by the logarithm of a rational expression. It intends to build the intermediate Chebyshev approximation with a relative error of the exponent of the approximated function by a rational expression. The approximation by the rational expression is calculated as the boundary of mean-power approximation using an iterative scheme based on the least squares method with two variable weight functions. The results of solving the test examples confirm the fast convergence of the method.
Journal Article
Moduli of continuity of functions in Holder’s class by First kind Chebyshev Wavelets and Its Applications in the Solution of Lane-Emden Differential Equations
In this paper , two new moduli of continuityand two estimators E2k−1,0 and,E2k−1,M of a functions f in H¨older’s class Hα,2ωk [0, 1) by First kind Chebyshev Wavelets have been determined. These moduli of continuity and estimators are new and best possible in wavelet analysis. Applying this technique ,Lane -Emden differential equations have been solved by first kind Chebyshev wavelet method.These solutions obtained by first kind Chebyshev wavelet method are approximately coincided with their exact solutions. This is a significant achievement of this research paper in wavelet analysis.
Journal Article
Chebyshev Approximation of Multivariable Functions by a Power Expression
The paper proposes the method for constructing Chebyshev approximation of multivariable functions with a relative error using a power expression. It involves building an intermediate Chebyshev approximation of values rooted to the corresponding degree of the approximated function by a polynomial with a relative error. Parameters for the polynomial approximation are computed as the boundary mean-power approximation through an iterative scheme using the least squares method with a variable weight function. The authors provide test examples, confirming the fast convergence of the method for constructing the Chebyshev approximation for the functions of one, two, and three variables using power expression.
Journal Article
Chebyshev Approximation of Multivariable Functions by a Logarithmic Expression
The method for constructing the Chebyshev approximation of multivariable functions by a logarithmic expression with an absolute error is proposed. It implies constructing an intermediate Chebyshev approximation of the values of the exponent of an approximated function by a polynomial with the relative error. The construction of the Chebyshev approximation by a polynomial is based on calculating the boundary mean-power approximation by an iterative scheme based on the least squares method with properly formed values of variable weight function. The presented results of test examples’ solving confirm the fast convergence of the method in calculating the parameters of the Chebyshev approximation of the functions of one, two, and three variables by the logarithmic expression.
Journal Article
Chebyshev Approximation of a Multivariable Function with Reproducing the Values of the Function and Its Partial Derivatives
The method for constructing the Chebyshev approximation of a discrete multivariable function with reproducing the values of the function and its partial derivatives at given points is proposed. The method is based on constructing a boundary mean-power approximation with appropriate interpolation conditions. The authors use an iterative scheme based on the least squares method with a variable weight function for constructing the mean-power approximation. The results of approximating the one-variable function confirm the fulfillment of the characteristic feature of the Chebyshev approximation with the reproduction of the function and its derivative values at given points. The test examples instantiate the fast convergence of the proposed method.
Journal Article
Optimal Error Estimates for Chebyshev Approximations of Functions with Endpoint Singularities in Fractional Spaces
In this paper, we introduce some new definitions and more general results of fractional spaces in order to deal with functions with endpoint singularities. Based on this theoretical framework, we derive optimal decay rates for Chebyshev expansion coefficients by applying the uniform upper bounds of generalized Gegenbauer functions of fractional degree (GGF-Fs). This enables us to further present the optimal
L
∞
-estimates and
L
2
-estimates of the Chebyshev polynomial approximations. In particular, we provide point-wise error estimates and the precise upper and lower bounds for
u
(
x
)
=
(
1
+
x
)
α
,
α
>
0
on
Ω
¯
=
[
-
1
,
1
]
in
L
∞
-norm. Moreover, we also discuss the extension of our main results to optimal error estimates of the related Chebyshev interpolation and quadrature measured in various norms at Chebyshev–Gauss points. Numerical results demonstrate the perfect coincidence with the error estimates. Indeed, the analysis techniques can enrich the theoretical foundation of
p
and
hp
methods for singular problems.
Journal Article
A Hybrid Method of Adaptive Cross Approximation Algorithm and Chebyshev Approximation Technique for Fast Broadband BCS Prediction Applicable to Passive Radar Detection
A hybrid method combining the adaptive cross approximation method (ACA) and the Chebyshev approximation technique (CAT) is presented for fast wideband BCS prediction of arbitrary-shaped 3D targets based on non-cooperative radiation sources. The incident and scattering angles can be computed by using their longitudes, latitudes and altitudes according to the relative positions of the satellite, the target and the passive bistatic radar. The ACA technique can be employed to reduce the memory requirement and computation time by compressing the low-rank matrix blocks. By exploiting the CAT into ACA, it is only required to calculate the currents at several Chebyshev–Gauss frequency sampling points instead of direct point-by-point simulations. Moreover, a wider frequency band can be obtained by using the Maehly approximation. Three numerical examples are presented to validate the accuracy and efficiency of the hybrid ACA-CAT method.
Journal Article
Chebyshev spectral approximation-based physics-informed neural network for solving higher-order nonlinear differential equations
Physics-informed neural networks (PINNs) typically involve higher-order partial derivatives with respect to their inputs, which are too costly to compute and store by using automatic differentiation (AD) even for relatively small neural networks. This paper presents an improved kind of PINN, named CD-PINN, where a Chebyshev spectral method is introduced to replace the AD method to accelerate the computation of higher-order derivatives in PDE residual, while the AD method is still used to compute the derivatives of the PDE residual with respect to weights and biases. This method directly approximates the input-output relationship of the neural network in small subdomains by using the Chebyshev series expansion. Benefiting from the superb properties of the derivatives of the Chebyshev polynomials, the higher-order derivatives of neural networks can be computed very quickly. A series of nonlinear differential equations are solved by the CD-PINN and AD-PINN simultaneously. The results show that the CD-PINN can achieve comparable accuracy to the AD-PINN and simultaneously reduce computation time significantly. To achieve the same accuracy, the time needed by the CD-PINN is much less than that of the AD-PINN. The larger the size of the neural network, the more layers it has, and the higher the derivative order in the PDEs, the greater the advantage of the CD-PINN. This research demonstrates an alternative way to improve the training speed of PINNs and a promising approach for blending traditional numerical methods with PINNs.
Journal Article