V. Dominguez

V. Dominguez, I. G. Graham, T. Kim

In this paper we propose and analyse composite Filon-Clenshaw-Curtis quadrature rules for integrals of the form $I_{k}^{[a,b]}(f,g) := \int_a^b f(x) \exp(\mathrm{i}kg(x)) \rd x $, where $k \geq 0$, $f$ may have integrable singularities and $g$ may have stationary points. Our composite rule is defined on a mesh with $M$ subintervals and requires $MN+1$ evaluations of $f$. It satisfies an error estimate of the form $C_N k^{-r} M^{-N-1 + r}$, where $r$ is determined by the strength of any singularity in $f$ and the order of any stationary points in $g$ and $C_N$ is a constant which is independent of $k$ and $M$, but depends on $N$. The regularity requirements on $f$ and $g$ are explicit in the error estimates. For fixed $k$, the rate of convergence of the rule as $M \rightarrow \infty$ is the same as would be obtained if $f$ was smooth. Moreover, the quadrature error decays at least as fast as $k \rightarrow \infty$ as does the original integral $I_{k}^{[a,b]}(f,g)$. For the case of nonlinear oscillators $g$, the algorithm requires the evaluation of $g^{-1}$ at non-stationary points. Numerical results demonstrate the sharpness of the theory. An application to the implementation of boundary integral methods for the high-frequency Helmholtz equation is given.

V. Dominguez, M. Ganesh

We propose, analyze, and implement interpolatory approximations and Filon-type cubature for efficient and accurate evaluation of a class of wideband generalized Fourier integrals on the sphere. The analysis includes derivation of (i) optimal order Sobolev norm error estimates for an explicit discrete Fourier transform type interpolatory approximation of spherical functions; and (ii) a wavenumber explicit error estimate of the order $\mathcal{O}(κ^{-\ell} N^{-r_\ell})$, for $\ell = 0, 1, 2$, where $κ$ is the wavenumber, $N$ is the number of interpolation/cubature points on the sphere and $r_\ell$ depends on the smoothness of the integrand. Consequently, the cubature is robust for wideband (from very low to very high) frequencies and very efficient for highly-oscillatory integrals because the quality of the high-order approximation (with respect to quadrature points) is further improved as the wavenumber increases. This property is a marked advantage compared to standard cubature that require at least ten points per wavelength per dimension and methods for which asymptotic convergence is known only with respect to the wavenumber subject to stable of computation of quadrature weights. Numerical results in this article demonstrate the optimal order accuracy of the interpolatory approximations and the wideband cubature.

V. Dominguez, C. Turc

We present superalgebraic compatible Nyström discretizations for the four Helmholtz boundary operators of Calderón's calculus on smooth closed curves in 2D. These discretizations are based on appropriate splitting of the kernels combined with very accurate product-quadrature rules for the different singularities that such kernels present. A Fourier based analysis shows that the four discrete operators converge to the continuous ones in appropriate Sobolev norms. This proves that Nyström discretizations of many popular integral equation formulations for Helmholtz equations are stable and convergent. The convergence is actually superalgebraic for smooth solutions.

V. Dominguez, M. Ganesh

The fast Fourier transform (FFT) based matrix-free ansatz interpolatory approximations of periodic functions are fundamental for efficient realization in several applications.In this work we design, analyze, and implement similar constructive interpolatory approximations of spherical functions, using samples of the unknown functions at the poles and at the uniform spherical-polar grid locations. The spherical matrix-free interpolation operator range space consists of a selective subspace of two dimensional trigonometric polynomials which are rich enough to contain all spherical polynomials of degree less than $N$. The spherical interpolatory approximation is efficiently constructed by applying the FFT techniques with only ${\mathcal{O}}(N^2 \log N)$ complexity. We describe the construction details using the FFT operators and provide complete convergence analysis of the interpolatory approximation in the Sobolev space framework. We prove that the rate of spectrally accurate convergence of the interpolatory approximations in Sobolev norms (of order zero and one) are similar (up to a log term) to that of the best approximation in the finite dimensional ansatz space. Efficient interpolatory quadratures on the sphere are important for several applications including radiation transport and wave propagation computer models. We use our matrix-free interpolatory approximations to construct robust FFT-based quadrature rules for a wide class of non-, mildly-, and strongly-oscillatory integrands on the sphere. We provide numerical experiments to demonstrate fast evaluation of the algorithm and various theoretical results presented in the article.