Least-squares orthogonal polynomial approximation in several independent variables

1993 ◽  
Vol 7 (3) ◽  
pp. 336 ◽  
Author(s):  
Robert S. Caprari
Keyword(s):  
Orthogonal Polynomial ◽  
Least Squares ◽  
Polynomial Approximation ◽  
Independent Variables

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.

Thermal Conductivity of Heavy Water (D2O)

1971 ◽  
Vol 13 (3) ◽  
pp. 205-216 ◽  
Author(s):  
J. E. S. Venart ◽  
N. Mani
Keyword(s):  
Thermal Conductivity ◽  
Orthogonal Polynomial ◽  
Least Squares ◽  
Heavy Water ◽  
Reduced Pressure ◽  
Pure Fluid ◽  
Thermal Conductivity Λ

Existing thermal conductivity (λ) data for heavy water (D2O) between 0 and 600°C at pressures from atmospheric to 1000 bar are correlated as simple functions of reduced pressure and temperature, utilizing orthogonal polynomial least squares analysis. Correlations in the vapour and supercritical states are made from a knowledge of the ratio λd2o/λh2o. A table of best values for the liquid, vapour and supercritical states of the pure fluid is also provided. Using these values, the thermal conductivity of heavy water can be estimated to between ±2 per cent and ±10 per cent from values of pressure and temperature only.

On the effect of sunshine on wheat yield at Rothamsted

1926 ◽  
Vol 16 (2) ◽  
pp. 159-165 ◽  
Author(s):  
L. H. C. Tippett
Keyword(s):  
Orthogonal Polynomial ◽  
Least Squares ◽  
Wheat Yield ◽  
Polynomial Functions ◽  
Partial Regression ◽  
Image Position

The method used in this paper to reduce sunshine data is that developed by Fisher(1). It consists, briefly, in fitting to the distribution for each year, a curvewhere T0, T1T2, etc., are orthogonal polynomial functions of zero, first, second, etc., order in time. The constants s0, s1, s2, … etc., are found by least squares, and are correlated with similar rainfall constants (r0, r1, r2 etc.) and with the crop.The regression of the wheat yield on rainfall has already been found (1), so a method has been devised, whereby those results can be used in order to find the partial regression of wheat yield on the sunshine sequence, eliminating all rainfall effect.

Inverse Problems With Errors in the Independent Variables Errors-in-Variables, Total Least Squares, and Bayesian Inference

2008 ◽  
Author(s):  
A. F. Emery
Keyword(s):  
Bayesian Inference ◽  
Maximum Likelihood ◽  
Inverse Problems ◽  
Least Squares ◽  
Total Least Squares ◽  
Errors In Variables ◽  
Independent Variables ◽  
Error In Variables ◽  
Best Estimates ◽  
Normally Distributed

Most practioners of inverse problems use least squares or maximum likelihood (MLE) to estimate parameters with the assumption that the errors are normally distributed. When there are errors both in the measured responses and in the independent variables, or in the model itself, more information is needed and these approaches may not lead to the best estimates. A review of the error-in-variables (EIV) models shows that other approaches are necessary and in some cases Bayesian inference is to be preferred.

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