Minimal-rank ℋ²-matrix-based iterative and direct volume integral equation solvers for large-scale scattering analysis

Accuracy directly controlled fast direct solutions of general H2-matrices and its application to electrically large electromagnetic analysis

“…At each level, the factorization proceeds from the left to the right across all the clusters. This is very different from the algorithm of [9], [11], [14], [15], where a recursive procedure is performed. To help better understand the proposed algorithm, we will use the H 2 -matrix shown in Fig.…”

“…For cluster i, we take all the fill-in blocks associated with its admissible blocks, and use them to update cluster basis V i . These fill-in blocks are shown in (15), which have been generated and stored in the corresponding admissible blocks during the elimination of previous clusters. Let them be F i,j k (k = 1, 2, ..., O(C sp )), where the first subscript denotes the row cluster i, and the second being the column cluster index.…”

“…Different from [9], [11], [15], in the proposed new direct solvers, the accuracy is directly controlled. When generating an H 2 -matrix to represent the original dense matrix, the accuracy is controlled by H 2 , which is the same as in [9], [11], [15]. The accuracy of the proposed direct solution is controlled by f ill−in .…”

“…We then simulate a dielectric slab with r = 2.54 at 300 MHz, which is also simulated in [14], [15]. The thickness of the slab is fixed to be 0.1λ 0 .…”

“…Later in [8], [9], fast matrix inversion and LU factorization algorithms are developed for H 2 -matrices. They have shown a complexity of O(N ) for solving constant-rank cases like electrically small and moderate problems for both surface and volume IEs (VIEs) [10], [11], [12], [13]; and a complexity of O(N logN ) for solving electrically large VIEs [14], [15].…”