O ( N ) Nested Skeletonization Scheme for the Analysis of Multiscale Structures Using the Method of Moments

We present a scheme to compress the method of moments (MoM) matrix, which is linear in complexity for low to intermediate frequency problems in electromagnetics. The method is fully kernel independent and easy to implement using existing codes. The O(N ) complexity for both memory and time (setup and matrix–vector product) is achieved thanks to the application of a recursive skeletonization to the H2 matrix structure of the MoM matrix that uses the nested nature of the far interactions. The interpolative decomposition is applied in a novel manner in order to compress the “far-field signature” of the groups of basis functions. Moreover the scheme is fully characterized and it proves itself well suited for the analysis of multiscale structures.

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

[2]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[3]  Jian-Ming Jin,et al.  Fast and Efficient Algorithms in Computational Electromagnetics , 2001 .

[4]  M.A. Khayat,et al.  An improved transformation and optimized sampling scheme for the numerical evaluation of singular and near-singular potentials , 2008, 2007 IEEE Antennas and Propagation Society International Symposium.

[5]  A. Heldring,et al.  Multilevel Adaptive Cross Approximation (MLACA) , 2009, IEEE Transactions on Antennas and Propagation.

[6]  Wolfgang Hackbusch,et al.  Construction and Arithmetics of H-Matrices , 2003, Computing.

[7]  F. Vipiana,et al.  Numerical Evaluation via Singularity Cancellation Schemes of Near-Singular Integrals Involving the Gradient of Helmholtz-Type Potentials , 2013, IEEE Transactions on Antennas and Propagation.

[8]  Jiming Song,et al.  Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects , 1997 .

[9]  Wolfgang Hackbusch,et al.  A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices , 1999, Computing.

[10]  Dan Jiao,et al.  An �-Matrix-Based Integral-Equation Solver of Reduced Complexity and Controlled Accuracy for Solving Electrodynamic Problems , 2009 .

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

[12]  Per-Gunnar Martinsson,et al.  A direct solver with O(N) complexity for integral equations on one-dimensional domains , 2011, 1105.5372.

[13]  Per-Gunnar Martinsson,et al.  An Accelerated Kernel-Independent Fast Multipole Method in One Dimension , 2007, SIAM J. Sci. Comput..

[14]  W. Chai,et al.  Linear-Complexity Direct and Iterative Integral Equation Solvers Accelerated by a New Rank-Minimized ${\cal H}^{2}$-Representation for Large-Scale 3-D Interconnect Extraction , 2013, IEEE Transactions on Microwave Theory and Techniques.

[15]  Jin-Fa Lee,et al.  A fast direct matrix solver for surface integral equation methods for electromagnetic wave scattering from non-penetrable targets , 2012 .

[16]  X. Sheng,et al.  Preconditioning Technique in the Interpolative Decomposition Multilevel Fast Multipole Algorithm , 2013, IEEE Transactions on Antennas and Propagation.

[17]  Giuseppe Vecchi,et al.  Wideband Fast Kernel-Independent Modeling of Large Multiscale Structures Via Nested Equivalent Source Approximation , 2015, IEEE Transactions on Antennas and Propagation.

[18]  Dan Jiao,et al.  A Linear Complexity Direct Volume Integral Equation Solver for Full-Wave 3-D Circuit Extraction in Inhomogeneous Materials , 2015, IEEE Transactions on Microwave Theory and Techniques.

[19]  M. Vouvakis,et al.  The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems , 2005, IEEE Transactions on Electromagnetic Compatibility.

[20]  Francesca Vipiana,et al.  EFIE Modeling of High-Definition Multiscale Structures , 2010, IEEE Transactions on Antennas and Propagation.

[21]  V. Rokhlin.  Rapid solution of integral equations of classical potential theory , 1985 .

[22]  Francesca Vipiana,et al.  Nested Equivalent Source Approximation for the Modeling of Multiscale Structures , 2014, IEEE Transactions on Antennas and Propagation.

[23]  G. Vecchi,et al.  A Multiresolution System of Rao–Wilton–Glisson Functions , 2007, IEEE Transactions on Antennas and Propagation.

[24]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

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

[26]  Leslie Greengard,et al.  A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization , 2012, SIAM J. Sci. Comput..

[27]  D. Wilton,et al.  Electromagnetic scattering by surfaces of arbitrary shape , 1980 .

[28]  Xin-Qing Sheng,et al.  HIERARCHICAL INTERPOLATIVE DECOMPOSITION MULTILEVEL FAST MULTIPOLE ALGORITHM FOR DYNAMIC ELECTROMAGNETIC SIMULATIONS , 2013 .

[29]  Lexing Ying,et al.  Fast Directional Multilevel Algorithms for Oscillatory Kernels , 2007, SIAM J. Sci. Comput..

[30]  Jiming Song,et al.  Multilevel fast‐multipole algorithm for solving combined field integral equations of electromagnetic scattering , 1995 .

[31]  Per-Gunnar Martinsson,et al.  On the Compression of Low Rank Matrices , 2005, SIAM J. Sci. Comput..

[32]  Mario Bebendorf,et al.  Approximation of boundary element matrices , 2000, Numerische Mathematik.

[33]  Per-Gunnar Martinsson,et al.  Fast direct solvers for integral equations in complex three-dimensional domains , 2009, Acta Numerica.