Polynomial interpolation via mapped bases without resampling

@article{Marchi2020PolynomialIV,
  title={Polynomial interpolation via mapped bases without resampling},
  author={Stefano De Marchi and Francesco Marchetti and Emma Perracchione and Davide Poggiali},
  journal={J. Comput. Appl. Math.},
  year={2020},
  volume={364}
}
Abstract In this work we propose a new method for univariate polynomial interpolation based on what we call mapped bases. As theoretically shown, constructing the interpolating function via the mapped bases, i.e. in the mapped space, turns out to be equivalent to map the nodes and then construct the approximant in the classical form without the need of resampling. In view of this, we also refer to such mapped points as “fake” nodes. Numerical evidence confirms that such scheme can be applied to… 
Multivariate approximation at fake nodes
TLDR
This work proposes an effective method for interpolating via mapped bases in the multivariate setting as Fake Nodes Approach (FNA), and the theoretical results are confirmed by various numerical experiments devoted to point out the robustness of the proposed scheme.
Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-Gibbs algorithm
TLDR
This work focuses on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in De Marchi et al. (2020) and shows that it yields an accurate approximation of discontinuously functions.
Stable discontinuous mapped bases: the Gibbs-Runge-Avoiding Stable Polynomial Approximation (GRASPA) method
TLDR
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.
Mapped polynomials and discontinuous kernels for Runge and Gibbs phenomena
In this paper, we present recent solutions to the problem of approximating functions by polynomials for reducing in a substantial manner two well-known phenomena: Runge and Gibbs. The main idea is to
Jumping with variably scaled discontinuous kernels (VSDKs)
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
Quadrature at fake nodes
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
Generalizations of the constrained mock-Chebyshev least squares in two variables: Tensor product vs total degree polynomial interpolation
TLDR
The univariate constrained mock-Chebyshev least squares interpolation is extended to the bivariate case in two different ways, relying on the tensor product interpolation and on the interpolation at the mock-Padua nodes.
RBF-based tensor decomposition with applications to oenology
As usually claimed, meshless methods work in any dimension and are easy to implement. However in practice, to preserve the convergence order when the dimension grows, they need a huge number of
A linear barycentric rational interpolant on starlike domains
TLDR
The present article makes use of the fact that linear rational barycentric interpolants converge rapidly toward analytic and several times differentiable functions to interpolate on two-dimensional starlike domains parametrized in polar coordinates and introduces a variant of a tensor-product interpolant of the above two schemes.
On Kosloff Tal-Ezer least-squares quadrature formulas
TLDR
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.
...
1
2
...

References

SHOWING 1-10 OF 34 REFERENCES
Jumping with variably scaled discontinuous kernels (VSDKs)
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
Mappings and accuracy for Chebyshev pseudo-spectral approximations
Abstract The effect of mappings on the approximation, by Chebyshev collocation, of functions which exhibit localized regions of rapid variation is studied. A general strategy is introduced whereby
Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval
Abstract Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions. Following [1], we suggest to
On the Lebesgue constant of Berrut's rational interpolant at equidistant nodes
TLDR
It is proved by proving that the Lebesgue constant of Berrut's rational interpolant grows only logarithmically in the number of interpolation nodes, and the numerical results suggest that theLebesgue constant behaves similarly for interpolation at Chebyshev as well as logarithsmically distributed nodes.
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
Analysis of the Gibbs phenomenon in stationary subdivision schemes
TLDR
The conditions are applied to non-negative masks and the Gibbs phenomenon in classical and recent subdivision schemes like B-splines, Deslauriers and Dubuc interpolation subdivision schemes and the schemes proposed in Siddiqi and Ahmad (2008).
A Mapped Polynomial Method for High-Accuracy Approximations on Arbitrary Grids
TLDR
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.
New Quadrature Formulas from Conformal Maps
TLDR
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.
Spectral filtering for the reduction of the Gibbs phenomenon for polynomial approximation methods on Lissajous curves with applications in MPI
Polynomial interpolation and approximation methods on sampling points along Lissajous curves using Chebyshev series is an effective way for a fast image reconstruction in Magnetic Particle Imaging.
A note on the Gibbs phenomenon with multiquadric radial basis functions
Any global or high order approximation method suffers from the Gibbs phenomenon if the approximant has a jump discontinuity in the given domain. In this note, we present a numerical study of the
...
1
2
3
4
...