255 research outputs found
Stability of barycentric interpolation formulas
The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or "first barycentric" formula dating to Jacobi in 1825. This difference in stability confirms the theory published by N. J. Higham in 2004 (IMA J. Numer. Anal., v. 24) and has practical consequences for computation with rational functions
Conformally Mapped Polynomial Chaos Expansions for Maxwell's Source Problem with Random Input Data
Generalized Polynomial Chaos (gPC) expansions are well established for
forward uncertainty propagation in many application areas. Although the
associated computational effort may be reduced in comparison to Monte Carlo
techniques, for instance, further convergence acceleration may be important to
tackle problems with high parametric sensitivities. In this work, we propose
the use of conformal maps to construct a transformed gPC basis, in order to
enhance the convergence order. The proposed basis still features orthogonality
properties and hence, facilitates the computation of many statistical
properties such as sensitivities and moments. The corresponding surrogate
models are computed by pseudo-spectral projection using mapped quadrature
rules, which leads to an improved cost accuracy ratio. We apply the methodology
to Maxwell's source problem with random input data. In particular, numerical
results for a parametric finite element model of an optical grating coupler are
given
Regularity theory and high order numerical methods for the (1D)-fractional Laplacian
This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight times a ``regular´´ unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein Ellipse, analyticity in the same Bernstein Ellipse is obtained for the ``regular´´ unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the Fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babu{s}ka and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nystr"om numerical method for the Fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.Fil: Acosta, Gabriel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Borthagaray, Juan Pablo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Bruno, Oscar Ricardo. California Institute Of Technology; Estados UnidosFil: Maas, Martín Daniel. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentin
A matrix method for fractional Sturm-Liouville problems on bounded domain
A matrix method for the solution of direct fractional Sturm-Liouville
problems on bounded domain is proposed where the fractional derivative is
defined in the Riesz sense. The scheme is based on the application of the
Galerkin spectral method of orthogonal polynomials. The order of convergence of
the eigenvalue approximations with respect to the matrix size is studied. Some
numerical examples that confirm the theory and prove the competitiveness of the
approach are finally presented
Is Gauss quadrature better than Clenshaw-Curtis?
We consider the question of whether Gauss quadrature, which is very famous, is more powerful than the much simpler Clenshaw-Curtis quadrature, which is less well-known. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following Elliott and O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of in the complex plane. Gauss quadrature corresponds to Pad\'e approximation at . Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at is only half as high, but which is nevertheless equally accurate near
On Bernstein type inequalities and a weighted Chebyshev approximation problem on ellipses
A classical inequality due to Bernstein which estimates the norm of polynomials on any given ellipse in terms of their norm on any smaller ellipse with the same foci is examined. For the uniform and a certain weighted uniform norm, and for the case that the two ellipses are not too close, sharp estimates of this type were derived and the corresponding extremal polynomials were determined. These Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. Some new results were also presented for a weighted approximation problem of this type
- …