## 1. Introduction

[2] Efficient and accurate solutions of electromagnetic (EM) scattering and radiation problems have attained a lot of interest for decades. Typical applications include radar cross section (RCS) estimation, antenna analysis and design, electromagnetic compatibility (EMC), electromagnetic interference (EMI), radiation hazards (EMR), remote sensing, etc. Among many full-wave numerical methods, the algorithms developed based on the method of moments (MoM) [*Peterson et al.*, 1998] have been widely used due to its high fidelity and superior capability to handle arbitrarily shaped targets. A typical MoM solution procedure begins with properly meshing the target of interest and selecting basis functions to model the equivalent electric and magnetic currents. After modeling a target with a set of *N* expansion functions and performing the traditional Galerkin testing for the integral equation, an *N* × *N* dense impedance matrix is generated with the memory requirement of *O*(*N*^{2}). The resultant matrix system can be solved by direct or iterative solvers. The computational complexity of MoM is *O*(*N*^{3}) for a conventional direct solver in terms of CPU time, such as LU, and *O*(*N*^{2}) for an iterative algorithm, such as CG or GMRES. The RWG [*Rao et al.*, 1982] basis functions are the typical basis functions selected for discretizing the integral equations. To achieve accurate solutions, the average size of each element is generally on the order of 1/10 wavelength (*λ*). Consequently, the size of the associated MoM matrix grows very rapidly as the object size becomes larger with respect to *λ*; this challenges the MoM for a variety of applications. To make it worse, there are so-called multiscale applications. In these cases, targets are over-meshed to conduct wide-band calculations, or partly over-meshed to capture the tiny geometrical structures. The discretization size is virtually independent of *λ* and *N* subsequently can be very large even for electrically small target sizes.

[3] In the MoM, both the CPU time and memory space are a great burden for a moderate *N*, even on modern computers. To mitigate this technical difficulty, MoM matrix equations typically utilize iterative solvers along with techniques to accelerate the matrix vector multiplication (MVM). These accelerations are performed either by adopting sets of directional basis and testing functions which radiate narrow beams (giving rise to quasi-sparse impedance matrices) or by approximating MVM through the physical or mathematical properties of the MoM matrix. Examples of the former case include the impedance matrix localization (IML) [*Canning*, 1995], complex multipole beam approach (CMBA) [*Boag and Mittra*, 1994], and wavelet expansion [*Steinberg and Leviatan*, 1993]. The latter includes the fast multipole method (FMM) [*Coifman et al.*, 1993], its multilevel version [*Chew et al.*, 2001; *Velamparambil and Chew*, 2005; *Pan and Sheng*, 2006, 2008; *Ergul and Gurel*, 2009; *Taboada et al.*, 2010], adaptive integral method (AIM) [*Bleszynski et al.*, 1996], precorrected fast Fourier transform (pFFT) method [*Phillips and White*, 1997], multilevel matrix decomposition algorithm (MLMDA) [*Michielssen and Boag*, 1996; *Rius et al.*, 2008], IES^{3} [*Kapur and Long*, 1998], QR-based or SVD-based methods [*Tsang and Li*, 2004; *Breuer et al.*, 2003; *Gope and Jandhyala*, 2005; *Seo and Lee*, 2004; *Burkholder and Lee*, 2004], and adaptive cross-approximation (ACA) method [*Kurz et al.*, 2002; *Zhao et al.*, 2005; *Shaeffer*, 2008]. In some of these formulations the memory requirements and CPU time are reduced from *O*(*N*^{2}) to *O*(*N*^{1.5}) for single level implementations and *O*(*N* log ^{α}*N*)(1 ≤ *α* ≤ 2) for multilevel ones. The pFFT, FMM and its multilevel version are based on analytic property of the Green's function, while the MLMDA, IES^{3}, ACA and the other QR-based methods are based on the rank deficiency among the coupling matrices between well-separated mesh partitions. The approximating methods mentioned above differ in the implementation and performance despite similarity in essence. Among these methods, MLFMA seems to be the most appealing one because of its fidelity, efficiency and generality. Although MLFMA has well documented success in solving large-scale MoM-based problems, its applications on scenarios involving over-meshing still present challenges. To substantiate this claim, it is well known that the FMM and MLFMA suffer from sub-wavelength breakdown when targets are over-meshed [*Chew et al.*, 2001]. This results in expensive operations associated with near-field coupling submatrices which could consist of millions of entries. Computing and storing these are often impractical.

[4] One solution for this difficulty is to combine the traditional MLFMA with its low-frequency versions developed through analytic approaches [*Hu et al.*, 2001; *Darve and Have*, 2004; *Cheng et al.*, 2006; *Jiang and Chew*, 2004; *Daniela and Bunger*, 2009; *Vikram et al.*, 2009]. The efficiency issues associated with the aforementioned approaches are mentioned in Section 5.4. Another possibility is to adopt algebraic techniques such as ACA, QR or SVD-based methods, to approximate the near-field interactions in the MLFMA. However, the QR or SVD-based methods require all entries of the near-field matrix being computed. This prevents them from efficiently treating multiscale problems because evaluating and storing the near-field matrix would be too expensive. Furthermore, the complexity of the QR or SVD-based algorithms is of *O*(*N*^{3}), where *N* is the dimension of the objective matrix. Recently, *Rodriguez et al.* [2008] efficiently approximated near-field interactions of the FMM according to the corresponding data sparse representation of far-field interactions, which was obtained by applying the SVD to the aggregation matrix. But the error arising from this approximation is hard to analyze. On the other hand, the ACA can avoid the large time/memory requirement to compute all the matrix entries, while its error control scheme is still an ongoing area of research. In this paper, the interpolative decomposition (ID) [*Liberty et al.*, 2007] is combined with the conventional MLFMA for multiscale problems. Particularly, the ID is employed here to efficiently approximate near-field interactions (NFIs) with respect to MLFMA (MLFMA-NFIs). Furthermore, a specific mechanism for the ID approximation is developed to avoid the expensive operations on evaluation all of the MLFMA-NFI matrix elements.

[5] The rest of the paper is organized as follows. Section 2 begins with a brief outline of the conventional MLFMA and then discusses the main framework of the proposed ID-MLFMA. Section 3 gives the basic idea of the ID algorithm and its applications on matrix approximations. Some details about computations at the ID levels are discussed in Section 4, including the employment of the artificial sphere. Section 5 presents some illustrative numerical results, and finally, a summary and some conclusions are given in Section 6.