9 research outputs found
New cubature formulae and hyperinterpolation in three variables
A new algebraic cubature formula of degree for the product Chebyshev
measure in the -cube with nodes is established. The
new formula is then applied to polynomial hyperinterpolation of degree in
three variables, in which coefficients of the product Chebyshev orthonormal
basis are computed by a fast algorithm based on the 3-dimensional FFT.
Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis
type cubature formula in the 3-cube
Polynomial (chaos) approximation of maximum eigenvalue functions: efficiency and limitations
This paper is concerned with polynomial approximations of the spectral
abscissa function (the supremum of the real parts of the eigenvalues) of a
parameterized eigenvalue problem, which are closely related to polynomial chaos
approximations if the parameters correspond to realizations of random
variables.
Unlike in existing works, we highlight the major role of the smoothness
properties of the spectral abscissa function. Even if the matrices of the
eigenvalue problem are analytic functions of the parameters, the spectral
abscissa function may not be everywhere differentiable, even not everywhere
Lipschitz continuous, which is related to multiple rightmost eigenvalues or
rightmost eigenvalues with multiplicity higher than one.
The presented analysis demonstrates that the smoothness properties heavily
affect the approximation errors of the Galerkin and collocation-based
polynomial approximations, and the numerical errors of the evaluation of
coefficients with integration methods. A documentation of the experiments,
conducted on the benchmark problems through the software Chebfun, is publicly
available.Comment: This is a pre-print of an article published in Numerical Algorithms.
The final authenticated version is available online at:
https://doi.org/10.1007/s11075-018-00648-
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)
ON CUBATURE RULES ASSOCIATED TO WEYL GROUP ORBIT FUNCTIONS
The aim of this article is to describe several cubature formulas related to the Weyl group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems. The diagram containing the relations among the special functions associated to the Weyl group orbit functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are summarized for all simple Lie algebras and their properties simultaneously tested on model functions. The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie algebra C2
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 theframework of Lissajous sampling for MPI (Magnetic Particle Imaging)
Numerical methods for Fredholm integral equations based on Padua points
The numerical solution of two-dimensional Fredholm integral equations on the square by Nyström and collocation methods based on the Padua points is investigated. The convergence, stability and well conditioning of the methods are proved in suitable subspaces of continuous functions of Sobolev type. Some numerical examples illustrate the efficiency of the methods. A comparison with the tensorial approximation methods, of Nyström and collocation type, based on Legendre zeros, is given
Nontensorial Clenshaw-Curtis cubature ∗
We extend Clenshaw-Curtis quadrature to the square in a nontensorial way, by using Sloan’s hyperinterpolation theory and two families of points recently studied in the framework of bivariate (hyper)interpolation, namely the Morrow-Patterson-Xu points and the Padua points. The construction is an application of a general approach to product-type cubature, where we prove also a relevant stability theorem. The resulting cubature formulas turn out to be competitive on nonentire integrands with tensorproduct Clenshaw-Curtis and Gauss-Legendre formulas, and even with the few known minimal formulas