Fast Iterative Solution of Integral Equations With Method of Moments and Matrix Decomposition Algorithm – Singular Value Decomposition

Abstract

The multilevel matrix decomposition algorithm (MLMDA) was originally developed by Michielsen and Boag for 2-D TMz scattering problems and later implemented in 3-D by Rius et al. The 3-D MLMDA was particularly efficient and accurate for piece-wise planar objects such as printed antennas. However, for arbitrary 3-D problems it was not as efficient as the multilevel fast multipole algorithm (MLFMA) and the matrix compression error was too large for practical applications. This paper will introduce some improvements in 3-D MLMDA, like new placement of equivalent functions and SVD postcompression. The first is crucial to have a matrix compression error that converges to zero as the compressed matrix size increases. As a result, the new MDA-SVD algorithm is comparable with the MLFMA and the adaptive cross approximation (ACA) in terms of computation time and memory requirements. Remarkably, in high-accuracy computations the MDA-SVD approach obtains a matrix compression error one order of magnitude smaller than ACA or MLFMA in less computation time. Like the ACA, the MDA-SVD algorithm can be implemented on top of an existing MoM code with most commonly used Green's functions, but the MDA-SVD is much more efficient in the analysis of planar or piece-wise planar objects, like printed antennas. Index Terms-Fast integral equation methods, method of moments (MoM), multilayer Green's function, printed antennas.

A simple and effective microwave invisibility cloak based on frequency selective surfaces

“…The entire cylindrical cloak structure has been meshed and simulated in FIESTA-3D (Fast Integral Equation Solver for scaTterers and Antennas in 3D) [7], a Method of Momentsbased software developed at the AntennaLAB of the Universitat Politècnica de Catalunya.…”

Compressed representation of matrix decomposition algorithm-singular value decomposition for full-wave analysis microstrip problems

“…Many redundancies are included in U mp and ðZ qp Þ À1 Á V qn which can be removed by QR decomposition and SVD. [16] Thus, Z iÂj l be expressed as…”

“…In practice, the MLMDA has a computational overload due to the generation and positioning of so many sets of equivalent functions, so the MLMDA results more efficient than the MDA only for very large boxes (sides > 8 wavelength). For arbitrary 3-D problems, MDA-SVD was presented by Rius et al [16], which is a new and improved formulation of MLMDA, and comparable to MLFMA and ACA in terms of computation time and memory usages.…”

RCS Computation Using a Parallel in-Core and Out-of-Core Direct Solver

“…It is worth mentioning the use of higher order discretizations [16][17][18], fast techniques such as Fast Multipole Method (FMM) [19], grid approaches based on Fast Fourier Transform (FFT) (e.g., [20][21][22][23][24]) and those based on matrix compression. The matrix compression approaches can be further subdivided into pure algebraic approaches (e.g., based on QR factorization [25], Adaptive Cross Approximation -ACA -, [26,27], and, in general, algorithms related to hierarchical matrices [28,29]) and approaches in which the compression is performed through the use of block basis functions based on the physics of the specific problem to solve (e.g., Matrix Decomposition Algorithm -MDA - [30,31], Macro Basis Functions -MBF - [32], Characteristic Basis Functions Method -CBFM - [33,34]). Also it is worth noting the use of wavelet basis and multiresolution analysis (e.g., [35,36]).…”