Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1 M Unknowns

A Fast Epile+fbsa Method Combined With Adaptive Cross Approximation for the Scattering From a Target Above a Large Ocean-Like Surface

“…As already discussed in the literature, the compression ratio of a coupling matrix (off-diagonal block-block or region-region sub-matrix) increases as the interaction distance increases: the farther the interactions are, the weaker the coupling is and the more efficient the compression from ACA is [12]. Then, ACA is obviously not efficient for (self-) impedance matrices (local interactions computation), likeZ 11 andZ 22 .…”

“…A way to solve this issue is to split each surface into sub-regions: the resulting self-impedance matrix can be then expressed into diagonal block matrices (which contain the self-coupling terms) and off-diagonal block-block matrices (which take into account the coupling between each sub-regions of the surface) which can be compressible when the regions are spaced some distance apart [12]. Besides, this can be done in a recursive scheme (each subregion can be split into many subregions to exhibit smaller off-diagonal blocks to compress) with a Multi-Level ACA [13], or also with the formalism of H-matrix theory [16,17].…”

“…Thus, in order to accelerate the matrix-vector products involved in the coupling steps, a recent algebraic method called Adaptive Cross Approximation (ACA) is applied to compress the coupling matrices. This method, developed in 2000 by Bebendorf [7,8], was then applied to electromagnetics [9][10][11][12][13] and can be seen as a truncated and partially pivoted Gaussian elimination [7].…”

Fast direct solution of characteristic basis function method using ACA-based LU decomposition

“…In the MoM, the integral equations are transformed into a dense linear matrix systems, which can be solved by direct methods and iterative methods. Compared with iterative methods [1,2,3], direct methods [4,5,6] are more suitable for the monostatic scattering problems, because the LU decomposition and inverse of the MoM impedance matrix can be reused for the different plane wave excitations. However, the conventional direct methods pose a burden on computer memory and CPU time.…”

“…Thus, after the LU decomposition to the reduced matrix, the off-diagonal submatrices of the lower and upper triangular matrices also have rank-deficient property, which can be easily compressed by the rank-deficient compression techniques [3,14,15]. Based on this fact, we apply the adaptive cross approximation (ACA)-based LU decomposition [3,6] to accelerate direct solution of the reduced matrix equation in the CBFM. It only requires knowledge of a few selected rows and columns of the original submatrices.…”