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Fast Direct Solution of Integral Equations With Modified HODLR Structure for Analyzing Electromagnetic Scattering Problems

A modified hierarchically off-diagonal low-rank (HODLR) fast direct solver is presented to analyze the scattering by electrically large and complex perfect electric conducting objects. The overall idea of HODLR solver is that the impedance matrix can be decomposed into the multiplication of several diagonal block matrices and the inverse is obtained easily with Sherman–Morrison–Woodbury formula. In this paper, a novel modified matrix compression method is utilized for the low-rank approximation of the off-diagonal submatrices. The proposed method only compresses the far-group subblocks judged by extended admissibility condition. The low-rank representations of the off-diagonal submatrices are then reconstructed and recompressed with the help of adaptive tolerance strategy. Consequently, the computation time and storage requirements will reduce significantly compared with the conventional solver. Several numerical results are presented to demonstrate the effectiveness and accuracy of the proposed method.