Numerical Properties of a Gaussian Translation Operator for the 2D FMM

Abstract

A diagonal Gaussian translation operator for the time-harmonic fast multipole method (FMM) in two dimensions is examined numerically. The Gaussian translation operator depends on a beam parameter that determines its sharpness. When the beam parameter is set to zero, the Gaussian translation operator reduces to the standard FMM translation operator. The sampling rate can be determined straightforwardly to achieve any desired accuracy. The directionality of the Gaussian translation operator makes it possible to reduce the number of plane waves required to achieve a given accuracy. The required number of plane waves depends strongly on the actual source-receiver locations, not just on the diameter of the source and receiver regions. Index Terms-Electromagnetic propagation, electromagnetic theory.0018-926X

On Sparsity Related to the Integral Equations of 2D Electromagnetic Scattering

“…In future work one could consider complex observation points (Hansen, 2013(Hansen, , 2014. This approach would be more complicated to implement but if doable the kernel would be modified independently of the basis functions.…”

Exact Plane-Wave Expansion With Directional Spectrum: Application to Transmitting and Receiving Antennas

“…Here, is a Gaussian-beam parameter [5], is the spherical Hankel function of the first kind, and is the Legendre polynomial [10]. The integral over the unit sphere of a function is defined as (7) with and (8) (6) and (9) hold when , where is the interior of a body of revolution that is centered at the origin with axis parallel to . A cross section of is shown.…”

“…The two-dimensional analog of the plane-wave representation in [5] was derived in [6] and examined numerically in [7]. Numerous references to both 2-D and 3-D Gaussian-beam expansions and plane-wave expansions can be found in [5]- [8].…”

Gaussian translation operator in a multilevel scheme

“…Moreover, the Gaussian translation operator requires only that the far‐field pattern of the source be sampled over the unit sphere. The paper [ Hansen , ] derives the corresponding two‐dimensional Gaussian translation operator, which is numerically examined in Hansen []. In contrast, to the quasi‐planar configuration used in the 3‐D steepest‐descent FMM [ Chew et al , , chapter 17], the plane wave expansion of the present paper is designed to reduce the number of required plane waves for source‐receiver geometries, where the receivers lie inside a circular cone when observed from each point in the source region.…”