• Corpus ID: 233207339

@inproceedings{Marchi2021QuadratureAF,
author={Stefano De Marchi and Giacomo Elefante and Emma Perracchione and Davide Poggiali},
year={2021}
}
We investigate the use of the so-called mapped bases or fake nodes approach in the framework of numerical integration. We show that such approach is able to mitigate the Gibbs phenomenon when integrating functions with steep gradients. Moreover, focusing on the optimal properties of the Chebyshev-Lobatto nodes, we are able to analytically compute the quadrature weights of the fake Chebyshev-Lobatto nodes. Such weights, quite surprisingly, coincide with the composite trapezoidal rule. Numerical…
2 Citations

## Figures from this paper

On Kosloff Tal-Ezer least-squares quadrature formulas
• Computer Science, Mathematics
ArXiv
• 2021
This work investigates the combination of the Kosloff Tal-Ezer map and Least-squares approximation for numerical quadrature and finds that some static choices of the map’s parameter improve the results of the composite trapezoidal rule, while a dynamic approach achieves larger stability and faster convergence, even when the sampling nodes are perturbed.
Stable discontinuous mapped bases: the Gibbs-Runge-Avoiding Stable Polynomial Approximation (GRASPA) method
• Mathematics, Computer Science
Computational and Applied Mathematics
• 2021
This work proposes a novel approach, termed Gibbs-Runge-Avoiding Stable Polynomial Approximation (GRASPA), where both Runge’s and Gibbs phenomena are mitigated and a theoretical analysis of the Lebesgue constant associated to the mapped nodes is provided.

## References

SHOWING 1-10 OF 27 REFERENCES
Polynomial interpolation via mapped bases without resampling
• Computer Science, Mathematics
J. Comput. Appl. Math.
• 2020
Numerical evidence confirms that such scheme can be applied to mitigate Runge and Gibbs phenomena and is referred to as a new method for univariate polynomial interpolation based on what is called mapped bases.
New Quadrature Formulas from Conformal Maps
• Computer Science, Mathematics
SIAM J. Numer. Anal.
• 2008
New nonpolynomial quadrature methods are proposed that avoid the usual ellipse of convergence to an infinite strip or another approximately straight-sided domain by conformally mapping the usual circle of convergence.
Jumping with variably scaled discontinuous kernels (VSDKs)
• Mathematics
• 2019
In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernel-based basis functions. The
On the Gibbs Phenomenon and Its Resolution
• Mathematics, Computer Science
SIAM Rev.
• 1997
The Gibbs phenomenon is reviewed from a different perspective and it is shown that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case.
Divergence (Runge Phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock-Chebyshev subset interpolation
• Mathematics, Computer Science
Appl. Math. Comput.
• 2009
The Runge Phenomenon can be completely defeated by interpolation on a ''mock-Chebyshev'' grid: a subset of (N+1) points from an equispaced grid with O(N^2) points chosen to mimic the non-uniform N+1-point ChebysheV-Lobatto grid.
A modified Chebyshev pseudospectral method with an O(N –1 ) time step restriction
• Mathematics
• 1989
Abstract The extreme eigenvalues of the Chebyshev pseudospectral differentiation operator are O(N2), where N is the number of grid points. As a result of this, the allowable time step in an explicit
A Mapped Polynomial Method for High-Accuracy Approximations on Arbitrary Grids
• Mathematics, Computer Science
SIAM J. Numer. Anal.
• 2016
A new method based on mapped polynomial approximation based on careful selection of the mapping parameter is introduced for the approximation of analytic functions on compact intervals from their pointwise values on arbitrary grids.
Interpolating functions with gradient discontinuities via Variably Scaled Kernels
In kernel–based methods, how to handle the scaling or the choice of the shape parameter is a well– documented but still an open problem. The shape or scale parameter can be tuned by the user
ON THE LEBESGUE FUNCTION FOR POLYNOMIAL INTERPOLATION
Properties of the Lebesgue function associated with interpolation at the Chebyshev nodes ${{\{ \cos [(2k - 1)\pi } {(2n)}}],\, k = 1,2, \cdots ,n\}$ are studied. It is proved that the relative
Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration
• Mathematics
• 2010
Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi-Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch