Advanced Mathematical Methods for Scientists and Engineers

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

A. Dedekind Sums

New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets

A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

The Filon–Clenshaw–Curtis method (FCC) for the computation of highly oscillatory integrals is known to attain surprisingly high precision. Yet, for large values of frequency $$\omega $$
 it is not competitive with other versions of the Filon method, which use high derivatives at critical points and exhibit high asymptotic order. In this paper we propose to extend FCC to a new method, FCC
 $$+$$
 , which can attain an arbitrarily high asymptotic order while preserving the advantages of FCC. Numerical experiments are provided to illustrate that FCC
 $$+$$
 shares the advantages of both familiar Filon methods and FCC, while avoiding their disadvantages.

A generalization of Filon–Clenshaw–Curtis quadrature for highly oscillatory integrals

We investigate the impact of adding inner nodes for a Filon-type method for highly oscillatory quadrature. The error of Filon-type method is composed of asymptotic and interpolation errors, and the interplay between the two varies for different frequencies. We are particularly concerned with two strategies for the choice of inner nodes: Clenshaw–Curtis points and zeros of an appropriate Jacobi polynomial. Once the frequency $$\omega $$
 is large, the asymptotic error dominates, but the situation is altogether different when $$\omega \ge 0$$
 is small. In the first regime, our optimal error bounds indicate that Clenshaw–Curtis points are always marginally better, but this is reversed for small $$\omega $$
 ; then, Jacobi points enjoy an advantage. The main tool in our analysis is the Peano Kernel theorem. While the main part of the paper addresses integrals without stationary points, we indicate how to extend this work to the case when stationary points are present. Numerical experiments are provided to illustrate theoretical analysis.

/pdf/error-analysis-of-the-extended-filon-type-method-for-highly-47bgksivs1.pdf

Error analysis of the extended Filon-type method for highly oscillatory integrals

We address the evaluation of highly oscillatory integrals, with power-law and logarithmic singularities. Such problems arise in numerical methods in engineering. Notably, the evaluation of oscillatory integrals dominates the run-time for wave-enriched
boundary integral formulations for wave scattering, and many of these exhibit singularities. We show that the asymptotic behaviour of the integral depends on the
integrand and its derivatives at the singular point of the integrand, the stationary
points and the endpoints of the integral. A truncated asymptotic expansion achieves
an error that decays faster for increasing frequency. Based on the asymptotic analysis,
a Filon-type method is constructed to approximate the integral. Unlike an asymptotic expansion, the Filon method achieves high accuracy for both small and large
frequency. Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function. Numerical results indicate that the
complex-valued Gaussian quadrature achieves the highest accuracy when the three
methods are compared. However, while it achieves higher accuracy for the same number of function evaluations, it requires significant additional cost of computation of
orthogonal polynomials and their zeros.

/pdf/quadrature-methods-for-highly-oscillatory-singular-integrals-50sf04sppd.pdf

Jing Gao

Papers

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

A generalization of Filon–Clenshaw–Curtis quadrature for highly oscillatory integrals

Error analysis of the extended Filon-type method for highly oscillatory integrals

Quadrature methods for highly oscillatory singular integrals.

Spectral computation of highly oscillatory integral equations in laser theory