A fast direct matrix solver for surface integral equation methods for electromagnetic wave scattering from non‐penetrable targets

Abstract

[1] The implementation details of a fast direct solver is described herein for solving dense matrix equations from the application of surface integral equation methods for electromagnetic field scatterings from non-penetrable targets. The proposed algorithm exploits the smoothness of the far field and computes a low rank decomposition of the off-diagonal coupling blocks of the matrices through a set of skeletonization processes. Moreover, an artificial surface (the Huygens' surface) is introduced for each clustering group to efficiently account for the couplings between well-separated groups. Furthermore, a recursive multilevel version of the algorithm is presented. Although asymptotically the algorithm would not alter the bleak outlook of the complexity of the worst case scenario, O(N 3 ) for required CPU time where N denotes the number of unknowns, for electrically large electromagnetic (EM) problems; through numerical examples, we found that the proposed multilevel direct solver can scale as good as O(N ) in CPU time for moderate-sized EM problems. Note that our conclusions are drawn based on a few sample examples that we have conducted and should not be taken as a true complexity analysis for general electrodynamic applications. However, for the fixed frequency (h-refinement) scenario, where the discretization size decreases, the computational complexities observed agree well with the theoretical predictions. Namely, the algorithm exhibits O(N ) and O(N

Accurate and Efficient Evaluation of Spatial Electromagnetic Responses of Large Scale Targets

“…There exists a complex matrix whose columns consist of a subset of the columns of and a complex matrix such that: • some subset of the columns of makes up the identity matrix; • no element of has an absolute value greater than 1; • ; • the least (that is the th greatest) singular value of is at least 1; • when and , , where is the -st greatest singular value of . Upon above statement, an approximation can be reached as (1) when the exact rank of is greater than , but the st greatest singular value of is small. Before conducting the decomposition as shown in (1), a threshold is often prescribed to control the error of the approximation.…”

“…But, in general, direct solvers only work well for small-or moderate-scale targets due to the high factorization cost. The factorization time ranges from to for dynamic electromagnetic problems [1], [2] although it can be reduced to as low as for static or quasi-static problems in the fast methods [1]- [6]. In contrast, the cost for solving each RHS iteratively is of the order [7]- [9], with the aid of fast methods, where is the number of iterations.…”

A Doubly Hierarchical MoM for High-Fidelity Modeling of Multiscale Structures

“…Inspired by [26] and [27], and differently from [11], in this paper we obtain a reciprocal algorithm rMLMCM, where the radiation matrix V t satisfies…”

ADAPTIVE AND PARALLEL SURFACE INTEGRAL EQUATION SOLVERS FOR VERY LARGE-SCALE ELECTROMAGNETIC MODELING AND SIMULATION (Invited Paper)

“…Fast and parallel algorithms such as the fast multipole method (FMM) [15][16][17][18], hybrid FMM and fast Fourier transform (FFT) [19], and parallel adaptive integral method [20,21] have been developed to accelerate the dense matrix-vector product (MVP). Other significant developments include the parallel higher-order method of moments [22] and fast direct solver of the SIE linear system [23][24][25]. With these advancements, the solutions with over several hundred millions and a billion unknowns have been possible [17,18,26].…”