Gaussian translation operator in a multilevel scheme

Abstract

A multilevel computation scheme for time-harmonic fields in three dimensions will be formulated with a new Gaussian translation operator that decays exponentially outside a circular cone centered on the line connecting the source and observation groups. This Gaussian translation operator is directional and diagonal with its sharpness determined by a beam parameter. When the beam parameter is set to zero, the Gaussian translation operator reduces to the standard fast multipole method translation operator. The directionality of the Gaussian translation operator makes it possible to reduce the number of plane waves required to achieve a given accuracy. The sampling rate can be determined straightforwardly to achieve any desired accuracy. The use of the computation scheme will be illustrated through a near-field scanning problem where the far-field pattern of a source is determined from near-field measurements with a known probe. Here the Gaussian translation operator improves the condition number of the matrix equation that determines the far-field pattern. The Gaussian translation operator can also be used when the probe pattern is known only in one hemisphere, as is common in practice. Also, the Gaussian translation operator will be used to solve the scattering problem of the perfectly conducting sphere.

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

“…In FMM computations one should preselect a sampling rate for each group level and determine the beam parameter for each group to group interaction to achieve this sampling rate. Future publications [13], [14] will further investigate the effectiveness of the new plane-wave expansion.…”

“…4. Its plane-wave receiving characteristic referenced to is defined so that the output voltage of the antenna is (14) when the antenna is illuminated by the plane wave , where is a constant vector with . If the receiving antenna is reciprocal, its receiving characteristic is proportional to its far-field pattern [2, p. 267].…”

Solution of large inverse sources problems

“…An important property of inverse current/sources problems is that there is usually a certain distance between the sources and the observation locations. Thus, full translations due to short translation distances are often not needed, and it is here usually of much more benefit to utilize the Gaussian translation operator than in the solution of radiation or scattering integral equations [6]. Another important observation is that inverse sources problems do often not require the explicit determination of the spatial sources distributions as involved in (2) and (3).…”

“…In the original versions of the algorithm, these operations were performed with numerical samples located on the complete Ewald sphere. By introducing the Gaussian translation operator as proposed by Thorkhild Hansen in [4], [5], [6], the translations can be restricted to considerably fewer plane wave samples resulting in a considerable speed-up of the numerical computations, where, however, the complete algorithm including aggregations, translations, disaggregations, and testing needs to be upgraded in order to be able to benefit from the reduced number of plane wave samples. As the directivity of the Gaussian translation operator is better for larger translation distances, it can be of benefit to perform the necessary translations on finer levels than in the case of using the full translation operator.…”

“…Recently, FIAFTA has been upgraded for a more efficiently uitlization of the data structures and in order to support the Gaussian translation operator as proposed by Thorkhild Hansen in [4], [5], [6]. The properties of FIAFTA are deomonstrated by a large-scale antenna NFFFT problem, where the near-field measurements have been synthetically generated by a large-scale method of moments (MoM) solution of the forward radiation problem.…”