O(N ) Iterative and O(NlogN ) Direct Volume Integral Equation Solvers for Large-Scale Electrodynamic Analysis Saad Omar and Dan Jiao * Abstract — State-of-the-art volume integral equa- tion (VIE) solvers for solving electrically large prob- lems are iterative solvers with the complexity of each matrix-vector multiplication being O(NlogN), where N is matrix size. In this work, we reduce this com- plexity to strictly O(N) irrespective of electrical size. Furthermore, we develop a fast inversion based di- rect VIE solver of O(NlogN) complexity, which is also independent of electrical size. Numerical ex- periments have demonstrated the complexity, accu- racy, and efficiency of the proposed new VIE solvers. Very large-scale VIE system matrices involving mil- lions of unknowns have been directly solved in fast CPU time and modest memory consumption on a single core running at 3 GHz. 1 Introduction Driven by the design of advanced engineering sys- tems, there is a continued need for reducing the complexity of computational electromagnetic (EM) methods. Among these methods, the volume inte- gral equation (VIE) based methods have a great flexibility in modeling both complicated geometry and inhomogeneous materials in open-region set- tings. Existing fast VIE solvers for solving large- scale problems are, in general, iterative solvers which rely on efficient matrix-vector multiplica- tions. The complexity of one matrix-vector mul- tiplication in a VIE solver has been significantly reduced from O(N 2 ) to O(NlogN ) for analyzing electrodynamic problems. In this work, we represent the dense system ma- trix resulting from the volume integral equation based analysis of general electromagnetic problems by an H 2 -matrix [1] with controlled accuracy. We develop an efficient algorithm to generate such an H 2 -matrix representation with the rank of each ad- missible block minimized based on prescribed accu- racy. The resultant rank hence scales linearly with the electrical size of the admissible block instead of cubically in a general 3-D problem. With the rank’s growth with electrical size taken into account, we * School of Electrical and Computer Engineering, Pur- due University, West Lafayette, IN 47907, USA, e-mail: djiao@purdue.edu, tel.: +1 765 4945240, fax: +1 765 4943371. This work was supported by a grant from NSF under award No. 0802178, and No. 1065318, a grant from SRC (Task 1292.073), and a grant from Office of Naval Re- search under award N00014-10-1-0482. developed linear-complexity H 2 -matrix-based stor- age and matrix-vector multiplication, and thereby an O(N ) iterative VIE solver regardless of electrical size. Moreover, we developed an O(NlogN ) matrix inversion, and hence a fast O(NlogN ) direct VIE solver for large-scale electrodynamic analysis. 2 Minimal-Rank Representation of the VIE Operator and the Resultant Rank’s Growth with Electrical Size Unlike static problems, the rank of an electrody- namic IE kernel increases with electrical size for achieving a prescribed accuracy, which results in a higher computational complexity if no advanced algorithms are conceived to effectively manage the rank’s growth with electrical size. The true indica- tor of this rank’s growth is singular value decom- position (SVD), since its resultant representation constitutes a minimal rank representation of a ma- trix for any prescribed accuracy. The SVD does not separate sources from observers in approximating Green’s function, and it finds a minimal rank rep- resentation of the IE kernel as a whole. The SVD is computationally O(N 3 ), and hence not practically feasible for studying the rank of electrically large IE operators. In view of the pivotal importance of this subject, a theoretical study has been carried out on the rank’s growth with electrical size in in- tegral equations [2]. The significance of this study lies in the fact that it derives a closed-form analyt- ical expression of the rank of the coupling Green’s function, which has the same scaling as that de- picted by SVD-based rank revealing. The findings on the rank-study are summarized as follows: 1) The rank (k) of the off-diagonal block, irrespec- tive of the electrical size, is far less than the size of the block, thus the off-diagonal block has a low rank representation, i.e. k ≪ N . 2) For static and one-dimensional configurations of sources and observers, the rank required by a pre- scribed accuracy remains constant irrespective of the problem size. 3) For 2- and 3-D configurations, the rank varies as square root of logarithm and linearly with the electrical size, respectively.   ,((( 593