An HSS Matrix-Inspired Butterfly-Based Direct Solver for Analyzing Scattering From Two-Dimensional Objects

A butterfly-based fast direct integral equation solver for analyzing high-frequency scattering from two-dimensional objects is presented. The solver leverages a randomized butterfly scheme to compress blocks corresponding to near- and far-field interactions in the discretized forward and inverse electric field integral operators. The observed memory requirements and computational cost of the proposed solver scale as <inline-formula><tex-math notation="LaTeX"> $O(N{\log ^2}N)$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$O({N^{1.5}}\log N)$ </tex-math></inline-formula>, respectively. The solver is applied to the analysis of scattering from electrically large objects spanning over 10 000 wavelengths and modeled in terms of five million unknowns.

[1]  Per-Gunnar Martinsson,et al.  A fast direct solver for scattering problems involving elongated structures , 2007, J. Comput. Phys..

[2]  Weng Cho Chew,et al.  Scattering from elongated objects: Direct solution in O(N log2 N) operations , 1996 .

[3]  V. Rokhlin,et al.  A fast direct solver for boundary integral equations in two dimensions , 2003 .

[4]  Jun Hu,et al.  On MLMDA/Butterfly Compressibility of Inverse Integral Operators , 2013, IEEE Antennas and Wireless Propagation Letters.

[5]  Leslie Greengard,et al.  A Fast Direct Solver for High Frequency Scattering from a Large Cavity in Two Dimensions , 2014, SIAM J. Sci. Comput..

[6]  J. Shaeffer,et al.  Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1 M Unknowns , 2008, IEEE Transactions on Antennas and Propagation.

[7]  A. Boag,et al.  A multilevel fast direct solver for EM scattering from quasi-planar objects , 2009, 2009 International Conference on Electromagnetics in Advanced Applications.

[8]  Amir Boag,et al.  Multilevel evaluation of electromagnetic fields for the rapid solution of scattering problems , 1994 .

[9]  Per-Gunnar Martinsson,et al.  An O(N) Direct Solver for Integral Equations on the Plane , 2013, 1303.5466.

[10]  Mark Tygert,et al.  Fast algorithms for spherical harmonic expansions, III , 2009, J. Comput. Phys..

[11]  W. Hackbusch,et al.  Introduction to Hierarchical Matrices with Applications , 2003 .

[12]  E. Michielssen,et al.  A multilevel matrix decomposition algorithm for analyzing scattering from large structures , 1996 .

[13]  Yaniv Brick,et al.  Fast Direct Solver for Essentially Convex Scatterers Using Multilevel Non-Uniform Grids , 2014, IEEE Transactions on Antennas and Propagation.

[14]  Mario Bebendorf,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Hierarchical Lu Decomposition Based Preconditioners for Bem Hierarchical Lu Decomposition Based Preconditioners for Bem , 2022 .