High-order integral equation methods for problems of scattering by bumps and cavities on half-planes

This paper presents high-order integral equation methods for evaluation of electromagnetic wave scattering by dielectric bumps and dielectric cavities on perfectly conducting or dielectric half-planes. In detail, the algorithms introduced in this paper apply to eight classical scattering problems, namely: scattering by a dielectric bump on a perfectly conducting or a dielectric half-plane, and scattering by a filled, overfilled or void dielectric cavity on a perfectly conducting or a dielectric half-plane. In all cases field representations based on single-layer potentials for appropriately chosen Green functions are used. The numerical far fields and near fields exhibit excellent convergence as discretizations are refined--even at and around points where singular fields and infinite currents exist.Comment: 25 pages, 7 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)

Surface Spline Approximation on SO(3)

The purpose of this article is to introduce a new class of kernels on SO(3) for approximation and interpolation, and to estimate the approximation power of the associated spaces. The kernels we consider arise as linear combinations of Green's functions of certain differential operators on the rotation group. They are conditionally positive definite and have a simple closed-form expression, lending themselves to direct implementation via, e.g., interpolation, or least-squares approximation. To gauge the approximation power of the underlying spaces, we introduce an approximation scheme providing precise L_p error estimates for linear schemes, namely with L_p approximation order conforming to the L_p smoothness of the target function.Comment: 22 pages, to appear in Appl. Comput. Harmon. Ana