Divergence (Runge Phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock–Chebyshev subset interpolation

2009 ◽  
Vol 210 (1) ◽  
pp. 158-168 ◽  
Author(s):  
John P. Boyd ◽  
Fei Xu
Keyword(s):  
Least Squares ◽  
Polynomial Approximation ◽  
Runge Phenomenon

scholarly journals Multilevel weighted least squares polynomial approximation

2020 ◽  
Vol 54 (2) ◽  
pp. 649-677 ◽  
Author(s):  
Abdul-Lateef Haji-Ali ◽  
Fabio Nobile ◽  
Raúl Tempone ◽  
Sören Wolfers
Keyword(s):  
Least Squares ◽  
Polynomial Approximation ◽  
Weighted Least Squares ◽  
A Priori ◽  
Computational Cost ◽  
Discretization Error ◽  
Logarithmic Factor ◽  
Complexity Bounds ◽  
Single Level ◽  
Random Samples

Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.

scholarly journals Uniform partition and the best least-squares piecewise polynomial approximation

1991 ◽  
Vol 44 (2) ◽  
pp. 279-283
Author(s):  
François Dubeau
Keyword(s):  
Least Squares ◽  
Polynomial Approximation ◽  
Piecewise Polynomial ◽  
Degree Polynomial ◽  
Piecewise Polynomial Approximation ◽  
Uniform Partition

It is shown that the best least-squares piecewise n degree polynomial approximation of xn+1 over [a, b] is obtained for a uniform partition. Moreover the approximation is continuous for n odd and discontinuous, with equal stepsizes at the nodes, for n even.

Digital Filters

1996 ◽  
Author(s):  
Ilya Polyak
Keyword(s):  
Time Series ◽  
Theoretical Analysis ◽  
Least Squares ◽  
Polynomial Approximation ◽  
Digital Filters ◽  
Direct Consequence ◽  
Time Lags ◽  
Two Dimensional ◽  
Fourier Approximation ◽  
Two Dimensional Filtering

In this chapter, several systems of digital filters are presented. The first system consists of regressive smoothing filters, which are a direct consequence of the least squares polynomial approximation to equally spaced observations. Descriptions of some particular univariate cases of these filters have been published and applied (see, for example, Anderson, 1971; Berezin and Zhidkov, 1965; Kendall and Stuart, 1963; Lanczos, 1956), but the study presented in this chapter is more general, more elaborate in detail, and more fully illustrated. It gives exhaustive information about classical smoothing, differentiating, one- and two-dimensional filtering schemes with their representation in the spaces of time, lags, and frequencies. The results are presented in the form of algorithms, which can be directly used for software development as well as for theoretical analysis of their accuracy in the design of an experiment. The second system consists of harmonic filters, which are a direct consequence of a Fourier approximation of the observations. These filters are widely used in the spectral and correlation analysis of time series. The foundation for developing regressive filters is the least squares polynomial approximation (of equally spaced observations), a principal notion that will be considered briefly.

Springback of Wire Products Considering Natural Strain

1973 ◽  
Vol 95 (3) ◽  
pp. 809-814
Author(s):  
A. N. Palazotto ◽  
D. A. Seccombe
Keyword(s):  
Least Squares ◽  
Strain Hardening ◽  
Polynomial Approximation ◽  
Copper Alloys ◽  
Stress Strain ◽  
Natural Strain ◽  
Strain Relationship ◽  
Experimental Values ◽  
Springback Angle

This paper makes use of the Ramberg-Osgood stress-strain relationship to evaluate the springback angle for wire products. Natural strain is present in the analysis. In order to incorporate the natural strain expression into the equations, a least-squares polynomial approximation is developed primarily to transform the Ramberg-Osgood equation. Results are compared with experimental values for various radii and angle of bends using several copper alloys with different strain hardening characteristics.

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