4,336 research outputs found
On the constrained mock-Chebyshev least-squares
The algebraic polynomial interpolation on uniformly distributed nodes is
affected by the Runge phenomenon, also when the function to be interpolated is
analytic. Among all techniques that have been proposed to defeat this
phenomenon, there is the mock-Chebyshev interpolation which is an interpolation
made on a subset of the given nodes whose elements mimic as well as possible
the Chebyshev-Lobatto points. In this work we use the simultaneous
approximation theory to combine the previous technique with a polynomial
regression in order to increase the accuracy of the approximation of a given
analytic function. We give indications on how to select the degree of the
simultaneous regression in order to obtain polynomial approximant good in the
uniform norm and provide a sufficient condition to improve, in that norm, the
accuracy of the mock-Chebyshev interpolation with a simultaneous regression.
Numerical results are provided.Comment: 17 pages, 9 figure
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
Trivariate polynomial approximation on Lissajous curves
We study Lissajous curves in the 3-cube, that generate algebraic cubature
formulas on a special family of rank-1 Chebyshev lattices. These formulas are
used to construct trivariate hyperinterpolation polynomials via a single 1-d
Fast Chebyshev Transform (by the Chebfun package), and to compute discrete
extremal sets of Fekete and Leja type for trivariate polynomial interpolation.
Applications could arise in the framework of Lissajous sampling for MPI
(Magnetic Particle Imaging)
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