Efficient matrix filling of multilevel simply sparse method via multilevel fast multipole algorithm

Authors


Abstract

[1] In this paper, an improved multilevel simply sparse method (MLSSM) is proposed for solving electromagnetic scattering problems that are formulated using the electric field integral equation approach. Previously, the matrix filling procedure of the conventional MLSSM is based on the adaptive cross approximation (ACA) method. Although the ACA is more efficient than direct filling, it requires a longer filling time for the far-field matrix than that of the multilevel fast multipole algorithm (MLFMA). Three problems with moderate electrical sizes are used to demonstrate that the far-field matrix filling memory of the ACA is also higher than that of the MLFMA. Hence, the MLFMA is utilized to reduce both the far-field matrix filling time and memory of the conventional MLSSM. Since the MLSSM recompresses the far-field interaction matrix of the MLFMA, the matrix-vector multiplication of the proposed method is more efficient than that of the MLFMA. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

1. Introduction

[2] Electromagnetic integral equations are often discretized using the method of moments (MoM) [Mahachoklertwattana et al., 2008; Ye and Jin, 2008; Kim et al., 2004; Rao et al., 1982], one of the most widespread and well established techniques for electromagnetic problems. However, the matrix associated with the resulting linear systems is usually large and dense for electrically large targets for electromagnetic scattering problems [Mahachoklertwattana et al., 2008]. It is basically impractical to solve electric field integral-equation (EFIE) matrix equations using direct methods because their memory requirement and computational complexity are O(N2) and O(N3), respectively, where N is the number of unknowns. Recently, a number of fast methods have been developed for accelerating the iterative solution of electromagnetic integral equations discretized using MoM. Most of them are based on the multilevel subdomain decomposition, requiring a computational complexity with the order of NlogN. One of the most popular techniques is the fast multipole method (FMM) [He et al., 2001; Ergül and Gürel, 2007, 2009; Chew et al., 2001; Pan et al., 2001; Malas and Gurel, 2009; Oo et al., 2002, 2004; Rodriguez et al., 2008], which has been widely used to solve very large electromagnetic problems due to its excellent computational efficiency.

[3] Another fast method is the multilevel simply sparse method (MLSSM) [Canning and Rogovin, 2002; Zhu et al., 2005; Cheng et al., 2008; Adams et al., 2006; Xu et al., 2010; Xu, 2009]. The single level of SSM is presented by Canning and Rogovin [2003], whereas the MLSSM is shown by Canning and Rogovin [2002], Zhu et al. [2005], Cheng et al. [2008], Adams et al. [2006], Xu et al. [2010], and Xu [2009]. This method can be implemented easily, with a critical advantage that it can be programmed on top of any existing computer codes. The adaptive cross approximation (ACA) [Zhao et al., 2005] was used to fill up the far-field matrix of the conventional MLSSM, which is more efficient than the direct filling. However, both the memory and computation complexities of the ACA are of the order of O(N4/3logN) for moderate-size problems that are formulated using the surface integral equation approach [Zhao et al., 2005], which are higher than that of the MLFMA.

[4] The MLSSM introduced by Canning and Rogovin [2002, 2003], Adams et al. [2006], Xu et al. [2010], and Xu [2009] is applied to the direct solution of the matrix equation, whereas the method introduced by Zhu et al. [2005] and Cheng et al. [2008] is applied to the iterative solution of the matrix equation. An improved MLSSM presented in this paper is an iterative solution for general 3-D electromagnetic problems. Numerical results show that when compared to the ACA, the MLFMA gives a much more efficient far-field matrix filling operation for the MLSSM. Hence, the MLFMA can be used to fill up the far-field matrix of MLSSM efficiently.

[5] This paper is organized as follows. Section 2 gives a brief introduction to the MLFMA and the electric field integral equations (EFIE) for electromagnetic scattering problems. Section 3 describes the improved MLSSM in detail. Section 4 gives some numerical examples to demonstrate the accuracy and computation efficiency of our approach.

2. The Formulations of MLFMA

[6] The MoM formulation of electromagnetic scattering problems using Rao-Wilton-Glisson (RWG) basis functions for surface modeling is given by Rao et al. [1982]. The MLFMA implementation used here was based on Chew et al. [2001]. For the electric field integral equations (EFIE), the resulting linear system after taking the Galerkin's testing has the following form (an ejwt time convention is assumed and suppressed throughout the paper)

equation image

where

equation image

and

equation image

where Ei(r) is the incident excitation plane wave. G(r,r′) refers to the Green's function in free space and an is the column vector containing the unknown coefficients of the surface current expansion using RWG basis functions. As usual, r and r′ denote the observation and source points, respectively, whereas η and k denote the impendence and wave number in free space, respectively. N is the number of unknowns used to discretize the object. Using the fast multipole technique, we first divide the N basis functions into M groups, denoted by Gp (p = 1, 2,…, M). Now let rm be a field point in group Gp centered at rp, and rn be a source point in another group Gq centered at rq. With rmp = rmrp, rpq = rprq, rqn = rqrn, the matrix-vector product of level l can be written as follows.

equation image

[7] In (2), Fqnlequation image and Rmplequation image are the aggregation factor and disaggregation factor at level l, respectively, and αpql(k, equation image) is the translation factor. The forms of the three matrices in equation (2) are shown by Chew et al. [2001]. The integral in (2) is evaluated using the Gaussian quadrature with K = 2L2 points, where L is selected as L = kd + 2 ln (kd + π) in our calculation with d being the maximum diameter of a group size and Bpl being the neighboring groups of Gp including itself at level l. Once the far field patterns are known at level l, they can be interpolated and translated to the center of the farther box at level l-1.

[8] The operator Pll−1equation image is used to denote an interpolation operator that interpolates the discrete far field values from level l to level l-1 equation image. The matrix-vector product at level l-1 can be written as follows.

equation image

where r′ denotes the distance between the field point and the group that contains the field point, whereas r″ denotes the distance between the source point and the group which contains the source point. The above procedure has been described in more detail by Chew et al. [2001].

3. The Improved MLSSM

[9] The impedance matrix filled by MLSSM is carried out based on the same multilevel spatial decomposition of the underlying geometry. The structure of the MLSSM representation is given in a multilevel recursion manner [Canning and Rogovin, 2002; Zhu et al., 2005; Cheng et al., 2008; Adams et al., 2006; Xu et al., 2010; Xu, 2009]:

equation image

where Zl is the reduced order impedance matrix and consists of only far interactions at level l + 1, which will be compressed in the coarser levels recursively up to level-3. There is no level L + 1 near interactions at the finest level L. Thus, ZL is the impedance matrix Z. In (4), equation imagel is the sparse matrix containing all near-neighbor interactions at level l of the oct-tree excluding near neighbor interactions at any finer level. Ul and Vl are the new testing and basis function matrices, respectively, which are block diagonal unitary matrices that compress interaction between sources in non-touching groups at level l. From equations (2) and (4), it can be seen that they have many similarities. The matrices Ul, Vl and Zl-1 are similar to the disaggregation factor, the aggregation factor, and the translation factor of MLFMA, respectively. The difference between them is that the aggregation factor, the disaggregation factor and the translation factor of MLFMA are not unitary. In order to transform the expression of MLFMA into that of MLSSM, QR and SVD is used to orthonormalize the far-field three submatrices of (2).

3.1. The Single Level of the Improved MLSSM

[10] For simplicity, the procedure of single level of the improved MLSSM is first described in the following. The impedance matrix can be expressed as

equation image

[11] The following procedure is used to form the matrices U3, Z2 and V3H at level 3.

[12] 1. At finest level 3, the near interaction matrix ZN is computed directly.

[13] 2. First, using FMM to fill the far field sub-matrix.

[14] 3. Second, this step is used to form the new testing function matrix U3 which is the row transformation matrix at level-3. For a given observer group 3(i), loop over source group j, which are not near-neighbors of field group 3(i). Extract the corresponding sub-matrix A(3(i),Gq), GqFar(3(i)) of the impedance matrix Z. The sub-matrix A(3(1),Gq) represents the collection of non-near-neighbors of 3(1), which is shown in Figure 1

Figure 1.

Shaded block indicates the sub-matrix A(3(1),Gq), GqFar(3(1)).

[15] The sub-matrix A(3(i), Gq) is approximated by FMM as follows:

equation image

[16] Maintaining an admissible error ɛ, QR and SVD are used to compress the Rmp3equation image

equation image

that gives the final expression of Rmp3equation image

equation image

[17] QU′(3(i)) is the i-th diagonal block of U3. Implement the procedure as the above for the all observer groups, we can build U3. The residual matrix SVH is multiplied with the remaining matrices in equation (6) to form intermediate matrix C.

[18] 4. Third, this step is utilized to form the new basis function matrix V3 which is the column transformation matrix at level-3. For a given source group 3(j), loop over observer group i, which are not near-neighbors of source group 3(j). Concatenate all matrices C(ij) and obtain the matrix B(3(j))

equation image

[19] Maintaining the admissible error ɛ, use QR and SVD to compress the B(3(j))

equation image

[20] V(3(j)) is the j-th diagonal block of V3. Implement the procedure as the above for all source groups, we can build V3. The residual matrix US″forms the matrix Z2.

[21] The essential part of above procedure is to post-compress the far interaction of impedance matrix. A large portion of memory requirement are used in the improved MLSSM is to store the matrices U3, V3H, and Z2 at level 3. The matrices U3, V3H are very sparse, where U3 and V3H are both unitary and block-diagonal. The dimension of the matrix Z2 is very small. The forms of matrices U3, Z2 and V3H are shown in Figure 2 at level 3.

Figure 2.

Level 3 SSM matrices (a):U3, (b):Z2, (c):V3H.

3.2. The Multilevel of the Improved MLSSM

[22] The procedure of multilevel is a little more complicated than the single level improved MLSSM. For simplicity, suppose that the object is decomposed in 4-level oct-tree, the impedance matrix can be expressed as

equation image

where

equation image

[23] The procedures to form U4 and V4 are a little different from that of single level case. The main differences are in the procedures of forming the matrices A and B. In the multilevel procedure, it needs to concatenate not only all sub-matrices which is not the near interaction groups of 4(i) for matrix A(4(i)) at level-4, but also the corresponding sub-matrices which is not the near interaction groups of 4(i) for matrix A(4(i)) at level-3. Figure 3 shows the forms of the A(4(1),Gq).

Figure 3.

Shaded block indicates the sub-matrix A(4(1),Gq), Gq = ∉ Near(4(1)).

[24] The form of A(4(i), Gq) is expressed as

equation image

where Near(l(i)) denotes the near interaction groups of the i-th nonempty group for each observer group l(i) at level l. To form the matrix B, it also needs to concatenate corresponding matrices C(ij) at level-3.

[25] When U4, V4 and Z3 are obtained, the matrices U3 and V3 are obtained from the matrix Z3far, far interaction part of Z3 at level-3. Repeat steps 3 and 4 at level 3 to obtain U3 and V3. Using the MLSSM for finding the impedance matrix Z, a fast matrix-vector multiplication algorithm can be obtained as follows:

equation image

[26] The above matrix-vector multiplication algorithm is very similar to that for the MLFMA. Because of recompressing far interactions of MLFMA, the matrix-vector multiplication of MLSSM is more efficient than for MLFMA.

4. Numerical Results

[27] To validate and demonstrate the accuracy and efficiency of the improved MLSSM, some numerical results are presented in this section. In implementing the improved MLSSM, the restarted version of the GMRES algorithm [Saad and Schultz, 1986; Saad, 1996] is used as the iterative method. The restarting number of GMRES is set to 30, whereas the stop precision for the restarted GMRES is set to 10−3. The truncating tolerance of each of MLSSM and ACA is 10−3 relative to the largest singular value. All experiments were performed on a Core-2 6300 with 1.86 GHz CPU and 1.96GB RAM, computed using the single precision.

[28] First, we consider the scattering of a perfectly electrically conducting (PEC) sphere with radius 1λ at 150 MHz. The incident direction is θi = 0°, ϕi = 0° and the scattered angles vary from 0° to 180° in the azimuthal direction when the pitch angle is fixed at 0°. Figure 4 compares the analytical Mie solution with the computed bistatic RCS for a sphere, with excellent agreement between them. The result validates the accuracy of the improved MLSSM.

Figure 4.

Bistatic RCS for a sphere with 10998 unknowns.

[29] Next, the far-field matrix-filling efficiency of the improved MLSSM is checked by computing the bistatic RCS of three scattering geometries. The far-field memories in Figures 5–7 indicate the final memories of the matrix-filling procedure of MLSSM, which are not the memories of the final MLSSM. Figures 5–7 show the far-field matrix filling time and memory of the MLSSM using the ACA and MLFMA as the size of the problems is increased. In the figures, “improved filling” refers to the far-field matrix filling of the MLSSM using the MLFMA, while “without improved filling” refers to the far-field matrix filling of the MLSSM using the ACA. A cylinder-plate geometry with the electrical size increasing from 1.6λ to 6.4λ is considered in the first example. The cylinder has a radius of 0.5 m and a height of 5 m, whereas the edge of the plate has a length of 8 m. The z-axis is used as the rotation axis. In the second example, a sphere geometry with the electrical size increasing from 1.3λ to 5.3λ is considered. The radius of the sphere is given by 2 m. For the last example, a split ring resonator (SRR) array with the electrical size increasing from 1λ to 4λ is studied. The dimension of the structure is the same as that presented by Parron et al. [2006]. It can be found that the filling time of the conventional MLSSM is 100 times longer than that of the improved MLSSM, whereas the filling memory of the conventional MLSSM is 10 times longer than that of the improved MLSSM.

Figure 5.

Far-field matrix filling time and memory for cylinder-plate geometry.

Figure 6.

Far-field matrix filling time and memory for sphere geometry.

Figure 7.

Far-field matrix filling time and memory for SRR array.

[30] Next, the complexity of the MLSSM is analyzed. Figures 8–10 show the MVP times and far-field memories (which indicate the final memories of the MLSSM) for the cylinder-plate, sphere, and SRR array, respectively, where MVP refers to a matrix-vector operation. It can be seen that the MVP time of the MLSSM is much shorter than that of the MLFMA, while the memory requirement of the MLSSM is very close to that of the MLFMA.

Figure 8.

One matrix-vector operation time and far-field memory for cylinder-plate geometry.

Figure 9.

One matrix-vector operation time and far-field memory for sphere geometry.

Figure 10.

One matrix-vector operation time and far-field memory for SRR array.

[31] The relative errors of the MLFMA and improved MLSSM for the three scattering geometries as mentioned above are analyzed in Table 1. The formulation of the relative error is given by

equation image

where M denotes the induced current computed using the MoM iterative solution, and T is the induced current computed using the MLFMA, the improved MLSSM, and the conventional MLSSM. The sphere and cylinder-plate are discretized with 10998 unknowns at 300 MHz and 11862 unknowns at 150 MHz, respectively. The SRR array has 11776 unknowns at 6 GHz.

Table 1. Relative Error for the Sphere, Cylinder-Plate, and SRR Array
ObjectsMLFMAImproved MLSSMConventional MLSSM
Sphere0.41%0.62%0.60%
Cylinder-plate0.13%0.19%0.21%
SRR array0.59%0.70%0.72%

[32] Figure 11 shows the computational efficiency of the improved MLSSM. The electrical size of the cylinder varies from 1.5λ to 11.7λ. The cylinder has a radius of 0.5 m and a height of 4 m. Its rotation axis is z-axis. The truncating tolerance of the Matrix decomposition algorithm - singular value decomposition (MDA-SVD) [Rius et al., 2008; Jiang et al., 2010] is 10−3 relative to the largest singular value. The results are computed using an Intel(R) Core(TM) 2 Quad CPU at 2.83 GHz and 8 GB of RAM. The MVP time of the MLSSM is compared with that of the MDA-SVD and MLFMA. It can be observed from Figure 11 that matrix vector multiplication is more efficient for the MLSSM than for the MDA-SVD and MLFMA. The recompression time of the far interactions of MLSSM for the cylinder example is also analyzed, which is shown in Figure 12.

Figure 11.

One matrix-vector operation time for cylinder geometry.

Figure 12.

The recompression time of the MLSSM for cylinder geometry.

[33] To summarize, the far-field matrix filling procedure of the improved MLSSM is much more efficient than that of the conventional MLSSM. Since the MLSSM recompresses the far interaction of the MLFMA, matrix-vector multiplication of the MLSSM is more efficient than for the MLFMA. It can be applied to monostatic RCS problems of complex objects.

5. Conclusion

[34] In this paper, an improved MLSSM is proposed for solving electromagnetic problems efficiently. The far-field matrix filling procedure of the MLSSM is previously based on the ACA. Since either the far-field matrix filling time or memory of the ACA is much more expensive than that of the MLFMA for moderate-size problems, the improved MLSSM utilizes the MLFMA instead of the ACA. Our numerical results show that the improved MLSSM is more efficient than the conventional MLSSM. The MLSSM recompresses the far-field interaction of the MLFMA, so matrix-vector multiplication of the MLSSM is more efficient than that of the MLFMA. It can be used to compute monostatic RCSs of complex objects efficiently.

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