An Adaptive HODBF Direct Solver for Fast Solutions of Surface Integral Equations in Electromagnetic Analysis

This article presents an adaptive hierarchically off-diagonal butterfly (A-HODBF) direct solver for fast solutions of surface integral equation (SIE) in electromagnetic analysis. The solver utilizes the hierarchically off-diagonal (HOD) structure in conjunction with the butterfly (BF) algorithm to compress the impedance matrix and intermediate factors during the inversion process. To maintain low computational complexity, a novel adaptive BF compression and reconstruction strategy is developed for off-diagonal blocks, avoiding oversampling during the filling process and reducing the dimension of test random matrices during the reconstruction process. This adaptive approach not only enhances computational efficiency but also ensures high accuracy. By employing the A-HODBF direct solver, we have successfully extended the BF algorithm from the strong admissible condition to the weak admissible condition while maintaining computational complexity at $O(N^{1.5}log N)$ and memory requirements at $O(N log^{2}N)$ for SIE in 3-D scattering problems. Numerical results demonstrate the accuracy and efficiency of the proposed solver, indicating significant improvements over existing methods.