Martijn C. van Beurden

On the use of small 2d convolutions on GPUs

Computing many small 2D convolutions using FFTs is a basis for a large number of applications in many domains in science and engineering, among them electromagnetic diffraction modeling in physics. The GPU architecture seems to be a suitable architecture to accelerate these convolutions, but reaching high application performance requires substantial development time and non-portable optimizations. In this work, we present the techniques, performance results and considerations to accelerate small 2D convolutions using CUDA, and compare performance to a multi-threaded CPU implementation. To improve programmability and performance of applications that make heavy use of small convolutions, we argue that two improvements to software and hardware are needed: FFT libraries must be extended with a single convolution function and communication bandwidth between CPU and GPU needs to be drastically improved.

The Gabor frame as a discretization for the 2D transverse-electric scattering-problem domain integral equation

We apply the Gabor frame as a projection method to numerically solve a 2D ransverse-electric-polarized domain-integral equation for a homogeneous medium. Since the Gabor frame is spatially as well as spectrally very well convergent, it is convenient to use for solving a domain integral equation. The mixed spatial and spectral nature of the Gabor frame creates a natural and fast way to Fourier transform a function. In the spectral domain we employ a coordinate scaling to smoothen the branchcut found in the Green function. We have developed algorithms to perform multiplication and convolution efficiently, scaling as O(N log N ) on the number of Gabor coefficients, yielding an overall algorithm that also scales as O(N log N ).

Computational aspects of 2D-quasi-periodic-green-function computations for scattering by dielectric objects via surface integral eEquations

We describe a surface integral-equation (SIE) method suitable for computation of electromagnetic fields scattered by 2D-periodic high-permittivity and plasmonic scatterers. The method makes use of fast evaluation of the 2D-quasi-periodic Green function (2D-QPGF) and its gradient using a tabulation technique in combination with tri-linear interpolation. In particular we present a very efficient technique to create the look-up tables for the 2D-QPGF and its gradient where we use to our advantage that it is very effective to simultaneously compute the QPGF and its gradient, and to simultaneously compute these values for the case in which the role of source and observation point are interchanged. We use the Ewald representation of the 2D-QPGF and its gradient to construct the tables with pre-computed values. Usually the expressions for the Ewald representation of the 2D-QPGF and its gradient are presented in terms of the complex complementary error function but here we give the expressions in terms of the Faddeeva function enabling efficient use of the dedicated algorithms to compute the Faddeeva function. Expressions are given for both lossy and lossless medium parameters and it is shown that the expression for the lossless case can be evaluated twice as fast as the expression for the lossy case. Two case studies are presented to validate the proposed method and to show that the time required for computing the method of moments (MoM) integrals that require evaluation of the 2D-QPGF becomes comparable to the time required for computing the MoM integrals that require evaluation of the aperiodic Green function.

The 2D TM scattering problem for finite dielectric objects in a dielectric stratified medium employing Gabor frames in a domain integral equation

We present a method to simulate two-dimensional scattering by dielectric objects embedded in a dielectric layered medium with transverse magnetic polarization through a domain integral equation formulation. A mixed spatial-spectral discretization is employed with both a spatial and a spectral representation along the direction of the layer interfaces. In the spectral domain, a discretization on a path through the complex plane is used on which the Green function is well behaved. To calculate the field-material interaction in the spatial domain, an auxiliary field is employed similar to the Li factorization rules. Numerical results show that this auxiliary-field formulation significantly improves accuracy, compared to a formulation that directly employs the electric field.

Higher-order statistics for stochastic electromagnetic interactions: applications to a thin-wire frame

Uncertainties in an electromagnetic observable, that arise from uncertainties in geometric and electromagnetic parameters of an interaction configuration, are here characterized by combining computable higher-order moments of the observable with higher-order Chebychev inequalities. This allows for the estimation of the range of the observable by rigorous confidence intervals. The estimated range is then combined with the maximum-entropy principle to arrive at an efficient and reliable estimation of the probability density function of the observable. The procedure is demonstrated for the case of the induced voltage of a thin-wire frame that has a random geometry, is connected to a random load, and is illuminated by a random incident field.

An efficient spatial spectral integral-equation method for EM scattering from finite objects in layered media

We propose a mixed spatial spectral method aimed directly at aperiodic, finite scatterers in a layered medium. By using a Gabor frame to discretize the problem a straightforward and fast way to Fourier transform is available. The poles and branchcuts in the spectral-domain Green function can be avoided by representing the induced currents and Green function on a path deformed into the complex spectral domain.

A pseudo-spectral longitudinal expansion in a spectral domain integral equation for scattering by periodic dielectric structures

We explore several options to introduce a pseudo-spectral expansion along the longitudinal direction in a spectral-domain integral equation for scattering by periodic dielectric structures. To this end we first simplify the integral equation to the formulation for a one-dimensional dielectric slab and consider the computational efficiency, convergence, and conditioning of several schemes. One scheme has been implemented in the domain integral equation and it exhibits exponential convergence with respect to the number of expansion functions in the longitudinal direction.

Parametric modeling of doubly periodic dielectric structures via a spectral-domain integral equation

A spectral-domain volume integral equation for 2D-periodic structures needs improved material-interface conditions for enhanced numerical convergence. We explain how continuous parametric changes in the scattering setup can be incorporated in these boundary conditions in a semi-analytical way, such that trends and sensitivities of geometrical changes can be computed in a reliable way.

A 3D spatial-spectral integral equation method for electromagnetic scattering from finite objects in a layered medium

The generalization of a two-dimensional spatial spectral volume integral equation to a three-dimensional spatial spectral integral equation formulation for electromagnetic scattering from dielectric objects in a stratified dielectric medium is explained. In the spectral domain, the Green function, contrast current density, and scattered electric field are represented on a complex integration manifold that evades the poles and branch cuts that are present in the Green function. In the spatial domain, the field-material interactions are reformulated by a normal-vector field approach, which obeys the Li factorization rules. Numerical evidence is shown that the computation time of this method scales as