THE solution to electromagnetic wave interaction with material

Transcription

1 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 9, SEPTEMBER Sparse Inverse Preconditioning of Multilevel Fast Multipole Algorithm for Hybrid Integral Equations in Electromagnetics Jeonghwa Lee, Jun Zhang, and Cai-Cheng Lu, Senior Member Abstract In computational electromagnetics, the multilevel fast multipole algorithm (MLFMA) is used to reduce the computational complexity of the matrix vector product operations. In iteratively solving the dense linear systems arising from discretized hybrid integral equations, the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the Krylov iterations. We show that a good quality SAI preconditioner can be constructed by using the near part matrix numerically generated in the MLFMA. The main purpose of this study is to show that this class of the SAI preconditioners are effective with the MLFMA and can reduce the number of Krylov iterations substantially. Our experimental results indicate that the SAI preconditioned MLFMA maintains the computational complexity of the MLFMA, but converges a lot faster, thus effectively reduces the overall simulation time. Index Terms Electromagnetic scattering, Krylov methods, multilevel fast multipole algorithm (MLFMA), sparse approximate inverse (SAI) preconditioning. I. INTRODUCTION THE solution to electromagnetic wave interaction with material coated objects has applications in radar cross section prediction for coated targets, printed circuit, and microstrip antenna analysis [16], [31], [34]. In three-dimensional (3-D) electromagnetic wave scattering problems, an arbitrarily shaped dielectric and conducting object can be modeled in the form of a hybrid volume and surface integral equation [20]. The surface integral equation is formed for the conducting surface and the volume integral equation (VIE) is formed for the dielectric. By making use of the method of moments (MoM) [20], [23], [33], Manuscript received December 11, 2002; revised July 28, The work of J. Lee was supported in part by the U.S. National Science Foundation (NSF) under Grant CCR The work of J. Zhang was supported in part by the U.S. National Science Foundation (NSF) under Grants CCR , CCR , and ACR , in part by the U.S. Department of Energy Office of Science under Grant DE-FG02-02ER45961, in part by the Japanese Research Organization for Information Science & Technology (RIST), and in part by the University of Kentucky Research Committee. The work of C.-C. Lu was supported in part by the U.S. National Science Foundation under Grant ECS and in part by the U.S. Office of Naval Research under Grant N J. Lee and J. Zhang are with the Laboratory for High Performance Scientific Computing and Computer Simulation, Department of Computer Science, University of Kentucky, Lexington, KY , USA ( leejh@engr.uky.edu; jzhang@cs.uky.edu). C.-C. Lu is with the Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY USA ( cclu@engr.uky.edu). Digital Object Identifier /TAP the hybrid integral equation is discretized into a complex valued linear system of the form where the coefficient matrix is large and dense, for electrically large targets in electromagnetic scattering. For some scattering structures the matrix can be poorly conditioned. The linear system (1) can be solved by some iterative methods, of which the Krylov (subspace) methods are considered to be the most effective ones currently available [2], [26]. The biconjugate gradient (BiCG) method is one of the many Krylov methods [17]. The Krylov methods such as the BiCG require the computation of some matrix vector product operations at each iteration, which accounts for the major computational cost of this class of methods. In electromagnetic simulations, the fast multipole method (FMM) is commonly used to reduce the computational complexity of a matrix vector product operation from to, where is the number of unknowns [10], [24]. With the multilevel fast multipole algorithm (MLFMA) the computational complexity is further reduced to [8], [19], [20], [29], [30]. In order to accelerate the convergence rate of a Krylov method, a preconditioner can be used to transform (1) into an equivalent form where is a nonsingular matrix of order. The purpose of this transformation is to make the product matrix better conditioned than the original matrix. The matrix is referred to as a preconditioner for the matrix, or for the linear system (1). For the preconditioning procedure to be cost effective, the construction of the matrix and the application operation with the matrix have to be inexpensive. Most existing preconditioners have been originally developed for solving sparse matrices [21], [26]. Recently, the development and practice of sophisticated preconditioning techniques in iteratively solving dense matrices have been a subject of growing interest [5] [7]. This is partly because of the increasing use of the wavelet techniques and the FMMs in engineering and scientific applications. In the FMM and the MLFMA, the diagonal and block diagonal preconditioners, constructed from the block diagonal submatrix, are popular [19], [29], [30]. In similar situations, several preconditioning (1) (2) X/04$ IEEE

2 2278 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 9, SEPTEMBER 2004 strategies are proposed for solving dense linear systems arising from the boundary integral equations [7], [32], or from the electric-field integral equations with the MLFMA implementation [36]. The preconditioners based on the incomplete LU (ILU) factorizations have been tested by a few authors with mixed success [3], [18], [27]. Our recent experience with a dynamically dropped ILU preconditioner (ILUT [25]) has been very successful. A few classes of difficult electromagnetic scattering problems have been solved by the ILUT preconditioner, but not by the block diagonal preconditioner [18]. A recent trend in preconditioning dense matrices from electromagnetics is to use a sparse approximate inverse (SAI) of the matrix. In most cases, a sparse matrix is first extracted from the matrix, then a sparse matrix is computed to approximate the matrix, the inverse of the matrix. The matrix is then used as a preconditioner for the matrix. The SAI preconditioning of dense matrices seems to be first considered in [15]. More recent studies can be found in [1], [4], [5], [7], [32], [36]. However, the construction of an SAI preconditioner for the MLFMA (or the FMM) has been less extensively studied or practiced [11], [22], [36]. In the MLFMA, the global coefficient matrix is not explicitly available. Thus the strategy of extracting a sparse matrix by dropping small magnitude entries from as a basis for constructing an SAI preconditioner is infeasible. Based on our successful experience in building an ILUT preconditioner for the MLFMA [18], we propose to construct an SAI preconditioner from the readily available near part submatrix of the coefficient matrix in the MLFMA. We will show how to preprocess the near part matrix and how to postprocess the computed SAI matrix in order to build an effective SAI preconditioner for the MLFMA. In our experimental tests, we use the BiCG method as the iterative solver, coupled with different preconditioning strategies to solve a few study cases of representative electromagnetic scattering problems. We mainly compare the SAI preconditioner with our recently developed ILUT preconditioner in terms of efficiency and effectiveness. It is well known that the convergence behavior of the iterative solvers are strongly related to the spectral properties of the coefficient matrix and the preconditioned matrices [12], [26]. We will demonstrate the good spectral properties of the SAI preconditioned matrix based on our numerical results. This paper is organized as follows. Section II gives a concise introduction to the hybrid integral equation approach in electromagnetic scattering and the MLFMA. In Section III, we outline the details to construct the SAI preconditioner from the near part matrix. Numerical experiments with a few electromagnetic scattering problems are used to show the efficiency of the SAI preconditioner in Section IV. Some concluding remarks are given in Section V. The hybrid integral equation approach combines the VIE and the surface integral equation (SIE) to model the scattering and radiation by mixed dielectric and conducting structures [20], [28]. For example, when a radome is applied to an antenna, the combined system consists of both dielectrics and conductors. Hence, the hybrid surface-vie is ideal for this problem [8]. The VIE is applied to the material region and the surface integral equation is enforced over the conducting surface. The integral equations can be formally written as follows: where stands for the excitation field produced by an instant radar, the subscript tan stands for taking the tangent component from the vector it applies to, and,,isan integral operator that maps the source to electric field and it is defined as Here, is the 3-D scalar Green s function for the background media, and. It should be pointed out that is related to in the above integral equations by. This results in a very general model as all the volume and surface regions are modeled properly. The advantage of this approach is that in the coated object scattering problems, the coating material can be inhomogeneous, and in the printed circuit and microstrip antenna simulation problems the substrate can be of finite size. The simplicity of the Green s function in both the VIE and the SIE has an important impact on the implementation of the fast solvers. However, the additional cost here is the increase in the number of unknowns since the volume that is occupied by the dielectric material is meshed. This results in larger memory requirement and longer CPU time for solving the corresponding matrix equation. But this deficiency can be overcome by applying fast integral equation solvers such as the MLFMA [8]. Using the method of moments (MoM), the hybrid integral equations are discretized into a matrix equation of the form where and stand for the vectors of the expansion coefficients for the surface current and the volume function, respectively [8], [20], and the matrix elements can be generally written as (3) II. HYBRID INTEGRAL EQUATION AND THE MLFMA The MLFMA for electromagnetic wave scattering problems is essentially an extension of that of scalar Helmholtz equation to vector problems[10]. The MLFMA is a state-of-the-art technique for solving the computational electromagnetic problems [8]. The material function if is a surface patch, and if is a volume cell. It can be seen that the coefficient matrix arising from discretized hybrid integral equations

3 LEE et al.: SPARSE INVERSE PRECONDITIONING OF MLFMA FOR HYBRID EQUATIONS 2279 Fig. 1. Sparse data structure of a dense matrix A from electromagnetic scattering. (a) Block diagonal part A and (b) block diagonal and block near-diagonal part B. TABLE I INFORMATION ABOUT THE MATRICES USED IN THE EXPERIMENTS (ALL LENGTH UNITS ARE IN, THE WAVELENGTH IN FREE-SPACE) is nonsymmetric. Once the matrix equation (3) is solved by numerical matrix equation solvers, the expansion coefficients and can be used to calculate the scattered field and the radar cross section. In antenna analysis problems the coefficients can be used to retrieve the antenna s input impedance and calculate the antenna s radiation pattern. In the following, we use to denote the coefficient matrix in (3),, and for simplicity. The basis function is an elementary current that has local support. To solve the matrix equation by an iterative method, matrix-vector multiplications are needed at each iteration. Physically, a matrix-vector multiplication corresponds to one cycle of interactions between the basis functions. Using the addition theorem [8], [10], the matrix-vector product can be written as where,, and are sparse matrices. In fact, the dense matrix can be structurally divided into three parts,,, and. is the block diagonal part of, is the block near-diagonal part of, and is the far part of. Here the terms near and far refer to the distance between two groups of elements. We denote by for simplicity. Fig. 1 shows the sparse data structure of a partitioned dense matrix from one of our examples, the P1B case in (4) Table I. Fig. 1(a) shows, Fig. 1(b) shows, and the far part of is scattered in the rest of the area of the Fig. 1(b). The side bands shown in Fig. 1(a) are not used in computing the block diagonal preconditioner. They indicate the interactions between group-to-group in the FMM. In the FMM, the matrix elements are calculated differently depending on the distance between a testing function and a basis function. For near-neighbor matrix elements (those in and ), the calculation remains the same as in the MoM procedure. However, those elements in are not explicitly computed and stored. Hence they are not numerically available in the FMM. It can be shown that with optimum grouping, the number of nonzero elements in the sparse matrices in (4) are all on the order of, and hence the operation count to perform is [10]. If the above process is implemented in multilevel, the total cost (in memory and CPU time) can be reduced further to be proportional to for one matrix vector multiplication. Due to this reduced computational complexity, the memory and CPU time savings can be orders of magnitude compared with the direct dense matrix multiplication methods of if is very large [19], [22]. As the level of the MLFMA decreases, we find that the number of nonzeros in the near part matrix increases significantly. The accuracy of the computed solution is strongly related to the number of levels of the MLFMA [18]. That is,

4 2280 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 9, SEPTEMBER 2004 as the number of the MLFMA levels decreases, the computed solution is close to the exact solution, but the near part matrix becomes denser. It is well-known that the MLFMA is an approximation method. The accuracy of the MLFMA solution also depends on the number of terms in the truncated infinite summations [10]. In Fig. 3(a) of the Section IV, we can see some slight difference between the exact solution (LUD) and the iterative solutions. This difference can be reduced if the level of the MLFMA is decreased. III. SAI PRECONDITIONERS The purpose of the preconditioning is to make the preconditioned matrix as close to the identity matrix as possible. To this end, we try to construct a matrix that approximates the matrix. It is difficult to make the matrix sparse, since in most cases the inverse of a matrix is dense even if is sparse. In practice, the SAI preconditioner is computed as a matrix which minimizes, subject to a certain sparsity pattern constraint. Here is the Frobenius norm of a matrix [26]. The key requirement is to keep sparse and to minimize the norm of the residual matrix. The Frobenius norm minimization problem can be decoupled into the sum of the squares of the 2-norms of the individual columns of the residual matrix as where and are the column vectors of the matrices and, respectively. Each of the subminimization problems in the right hand side of (5) can be solved independently. We begin with solving one equation (1) and end up with solving equations of (5). The difference between the solution vector in (1) and the solution vector in (5) is that the former is dense and the latter is sparse. The order of each subminimization problem in the right hand side of (5) is substantially smaller than that of (1). The most important step in constructing an SAI preconditioner is to choose the sparsity pattern constraint. Both static and dynamic sparsity pattern selection strategies have been developed [9], [13]. Compared with the static sparsity pattern strategy, the dynamic sparsity pattern strategy can be more accurate within the given nonzero density of the matrix,but it is also more expensive to compute [35]. In the MLFMA implementation, the global matrix is not numerically available, but the near part matrix is. We thus construct the SAI preconditioner with respect to the matrix. We will use a static sparsity pattern strategy for constructing.wefirst construct a sparsified matrix from the near part matrix. Here the matrix is obtained from by removing some small magnitude entries of with respect to a threshold parameter. The computational procedure is given in Algorithm 3.1, in which,, and are three user provided threshold drop tolerance parameters. (5) ALGORITHM 3.1 Computing an SAI matrix M from the near part matrix BN 1) Obtain a sparsified matrix BN ~ from BN with respect to 1 2) Further sparsify BN ~ with respect to 2 to obtain a sparse matrix CN ~ 3) Compute an SAI matrix CN of the matrix BN ~ based on the sparsity pattern of CN ~ 4) Further sparsify the matrix CN with respect to 3 to obtain the SAI preconditioner M Here, we detail the dropping strategies used in Algorithm 3.1. Let,,, be entries of a matrix in question. If, we set in the preprocessing phase to sparsify the matrix to obtain the matrix in Step 1. Note that the diagonal entries are not dropped regardless of their values. If for any in, we set in the preprocessing phase. In Step 2, a similar procedure is employed to obtain a sparsity pattern matrix with a larger value of. After the initial SAI matrix is computed in Step 3, it is further sparsified by using the parameter to obtain the final SAI matrix in Step 4. To compute the SAI in Step 3, the minimization problem (5) is equivalent to minimizing the individual functions with the sparsity pattern of being chosen as the th column of the matrix. Note that for notation convenience, we keep using the matrices and, but they are corresponding to the matrices and in Algorithm 3.1, respectively. We assume that entries of at certain locations are allowed to be nonzero, the rest of the entries are forced to be zero. Denote the nonzero entries of by and the columns of corresponding to by. Since is sparse, the submatrix has many rows that are identically zero. After removing the zero rows, we have a reduced matrix with rows. The individual minimization problem (6) is reduced to a least squares problem of order We note that the matrix is usually a very small rectangular matrix. It has full rank if is nonsingular. There are a variety of methods available to solve the least squares problem (7). One approach [13] is to solve (7) using a QR factorization as where is a nonsingular upper triangular matrix. is an unitary matrix, such that. The least squares problem (7) is solved by first computing and then obtaining the solution as. In this way, can be computed for each, independently. This yields an approximate inverse matrix, which minimizes for the given sparsity pattern. (6) (7) (8)

5 LEE et al.: SPARSE INVERSE PRECONDITIONING OF MLFMA FOR HYBRID EQUATIONS 2281 Fig. 2. Sparsity patterns for each preconditioned matrix in the P1B case. (a) Matrix ~ B : sparsified from B ( = 0:03); (b) matrix M: SAI( = 0:04, =0:05); (c) matrix LU : ILUT(10, 30). Fig. 2 shows the sparsity patterns of the preconditioned matrices in the P1B case in Section IV. The sparsified matrix is 96% sparse with respect to the matrix but maintains mostly the nonzero pattern of the matrix as in Fig. 2(a). The SAI matrix [Fig. 2(b)] is more than 98% sparse with respect to the matrix. Fig. 2(c) shows that the ILUT matrix is denser than the SAI matrix. This sparsity pattern guarantees that the sparsified matrix is sufficient to construct the SAI preconditioner. It should be mentioned that the ILU preconditioners can be implemented in a block-by-block manner. This allows for the use of preconditioners that may need storage space equal to many times of the size of the available RAM in a computer [14]. In contrast, the storage cost of the SAI preconditioner implemented in this paper is comparable to that of the near part matrix, the storage and floating point operation complexity of the MLFMA is maintained. IV. NUMERICAL TESTS AND ANALYSIS A number of numerical examples are presented to demonstrate the efficiency of the SAI preconditioner for accelerating the BiCG iterations. We examine its convergence behavior based on the number of preconditioned iterations and some theoretical facts (such as the eigenvalue clustering and the condition numbers). We compare the SAI preconditioner with the ILUT preconditioner, the block diagonal preconditioner, and without a preconditioner. The numerical results and analysis of the ILU preconditioner with a dual dropping strategy (ILUT) are given in [18]. Since the ILUT has been shown to be efficient for solving the dense complex linear systems from electromagnetic wave scattering problems, we mainly compare the performance of the SAI and the ILUT. We calculate the radar cross section (RCS) to demonstrate the performance of our preconditioned BiCG solver for different conducting geometries with and without coating. The geometries considered include plates, spheres, antenna arrays, and cavities (see Table I). The mesh sizes for all the test structures are about one tenth of a wavelength. Here are the explanations of the notations used in the tables with numerical data and in some figures. LUD: the exact solution computed from the exact LU factorization method. prec: the preconditioner used with the BiCG method. SAI (,, ): the SAI preconditioner with three parameters: : the drop tolerance to sparsify the matrix ;, : the drop tolerances to compute the preconditioner. NONE: no preconditioner. BLOCK: the block diagonal preconditioner [19], [29], [30]. ILUT (, ): the incomplete LU preconditioner with a dual dropping strategy [18], [25]: : the threshold drop tolerance (real) used in the ILUT; : the fill-in parameter (integer) used in the ILUT. level: the number of levels in the MLFMA. ratio: the sparsity ratio of the number of nonzeros of a preconditioner with respect to the number of nonzeros of the near part matrix. setup: the setup CPU time in seconds for constructing a preconditioner. : the number of the (preconditioned) BiCG iterations. : the CPU time in seconds for the iteration phase. : the CPU time in seconds for both the setup and the iteration phase. All cases are tested on one processor of an HP superdome supercluster at the University of Kentucky. The processor has 2 GB local memory and runs at 750 MHz. The code is written in Fortran 77 and is run in single precision. The iteration process is terminated when the 2-norm of the residual vector is reduced by, or the number of iterations exceeds Due to limited space, we only report a few significant numerical results in this paper. The test results are given in Tables II V. In order to get a feeling on the effects of the parameters on the performance of the SAI and the ILUT, we give in each table the results with a few sets of parameters. Based on our test results,

6 2282 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 9, SEPTEMBER 2004 TABLE II NUMERICAL DATA OF THE P1B CASE (n =1416) TABLE III NUMERICAL DATA OF THE S2B CASE (n = 32400) TABLE IV NUMERICAL DATA OF THE P3A CASE (n = ) we can conclude that the values of the user provided parameters ( and for the ILUT,,, and for the SAI) affect the performance of the two preconditioners. However, the ranges of the parameters may be determined after a few test runs, and some of them can be determined by the physical problem set-up. E.g., the parameter in the ILUT seems to be good.

7 LEE et al.: SPARSE INVERSE PRECONDITIONING OF MLFMA FOR HYBRID EQUATIONS 2283 TABLE V NUMERICAL DATA OF THE C1A CASE (n =20176) Fig. 3. P1B case: ILUT (10, 30), SAI(0.03, 0.04, 0.05). (a) Solution comparison and (b) convergence history. The parameter usually depends upon a particular problem and affects the performance of the ILUT once the parameter is fixed. The three parameters of the SAI can be chosen safely in the interval (0.01,0.1). Without better information, the setting can be tested first with. According to our numerical data, we find that the effectiveness of the SAI and ILUT preconditioners are affected by the shape of the models. E.g., the problems with the sphere and the cavity configurations need more iterations to converge. The preconditioners are most effective for the P1B case, in which the number of iterations has been reduced significantly by using the SAI and the ILUT, compared with the unpreconditioned case and with using the block diagonal preconditioner. For the S2B and C1A cases, the two preconditioners still reduce a large number of iterations, but the rate of reductions is not as large as that in the P1B and P3A cases. Fig. 3(a) shows that the solutions of the P1B case computed with four different iterative solution strategies are similar and approximate to the exact solution computed from the LU factorization. The solution graphs indicate the scattering pattern of the objects. Fig. 3(b) shows the convergence history of the residual 2-norm of the BiCG method with the SAI preconditioner, the ILU preconditioner, the block diagonal preconditioner, and without a preconditioner. The SAI and the ILUT are seen to converge sharply faster. Next, we use the P1B case to demonstrate the efficiency of the SAI preconditioning based on our numerical data and some known theoretical results. The convergence behavior of the Krylov methods depends on the distribution of the eigenvalues and on the condition number of the coefficient matrix. Eigenvalues tightly clustered around a single point (away from the origin) provide fast convergence. While widely spread eigenvalues, especially around the origin, cause the convergence to be very slow. This is because a low

8 2284 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 9, SEPTEMBER 2004 TABLE VI EXTREME EIGENVALUES AND CONDITIONING INFORMATION OF THE MATRICES IN THE P1B CASE Fig. 4. Clustering of the eigenvalues in the P1B case. (a) Eigenvalues of A, (b) eigenvalues of (A ) A, (c) eigenvalues of (LU ) A, and (d) eigenvalues of MA. degree polynomial with the value 1 at the origin cannot be small at a large number of such points [12], [26]. According to [12], [26], we can only link the convergence rate of the GMRES (not the BiCG) with the condition number of the eigenvector matrix. For a given (diagonalizable) matrix, it can be decomposed as, where is the diagonal matrix of the eigenvalues and is the matrix of the eigenvectors of the matrix. The condition number of the eigenvector matrix measures the normality of the matrix. Although existing theoretical result is not for the BiCG, it may be used to explain the convergence of the Krylov methods (including the BiCG) heuristically. Table VI lists the eigenvalue and the condition number information related to the P1B case. The condition number of the eigenvector matrix of the original dense matrix is relatively large. The smallest magnitude eigenvalues of the original matrix is near the origin. The original dense matrix and the block diagonal preconditioned matrix make the spectrum spread around the origin, see Fig. 4(a) and (b). (We note that these subfigures are not drawn on the same scale.) This makes the preconditioned matrix more indefinite. These features cause slow convergence

9 LEE et al.: SPARSE INVERSE PRECONDITIONING OF MLFMA FOR HYBRID EQUATIONS 2285 Fig. 5. Total CPU time comparison of the SAI and the ILUT based on the sparsity ratio. (a) P3A case and (b) C1A case. of the BiCG method on the original dense matrix and on the block diagonal preconditioned matrix. However, the eigenvalue decomposition shows that the condition number of the eigenvector matrix of the block diagonal preconditioned matrix is smaller than that of the original matrix. The eigenvalues of the SAI preconditioned matrix [Fig. 4(d)] and the ILUT preconditioned matrix [Fig. 4(c)] are shifted to the right hand side of the origin and they are away from zero. They are also clustered around 1. The smallest eigenvalue in magnitude of the SAI preconditioned matrix has the real part of 0.11, see Table VI. In addition, the condition number of the eigenvector matrix is significantly decreased (the SAI preconditioned matrix is close to normal). The condition number of the SAI preconditioned matrix is relatively small. These features explain the good convergence behavior of the BiCG method on this SAI preconditioned matrix. The S2B case is one of the challenging problems because the SAI and ILUT preconditioners with several parameter choices do not perform well, compared with the BiCG without preconditioning. This is mainly due to the expensive setup time. However, Table III shows that the SAI preconditioner reduces the setup time dramatically if we choose the parameters well. The SAI also converges in a relatively small number of iterations. In particular, we point out the small sparsity ratios of the SAI preconditioner. They are around 0.04 for most of the parameter choices, which means that the SAI preconditioner needs only a few percentages of the storage required for storing the near part matrix. On the other hand, the storage cost of the ILUT preconditioner is usually comparable to that of the matrix. Lower sparsity ratio reduces not only the storage cost, but also the CPU time in each iteration. Fig. 5 is a graphical demonstration of the CPU time with respect to the sparsity ratio in the P3A and C1A cases. In the P3A case, we see that the total CPU time mainly depends on the iteration time [see Fig. 5(a)]. Fig. 5(b) shows that the total CPU time of the C1A case mainly depends on the setup time. The sphere and cavity models require large setup time (see Tables III and V). For each case, there is a certain range of the sparsity ratio which shows better results according to the numerical data. That is, for the P3A case, if the sparsity ratio is between 0.1 and 0.5 the SAI performance is good with respect to the total CPU time. For the C1A case, the sparsity ratio between 0.05 and 0.25 requires a small amount of CPU time. For the storage space, we can clearly see that the SAI preconditioner is much better than the ILUT preconditioner (see Fig. 5). We see that the sparsity ratio of the SAI in all cases is far smaller than 1. The SAI preconditioner with suitable parameter choices does not need a large amount of storage space. Thus, the memory and floating point operation complexity of the MLFMA in each iteration is maintained. All cases show that the BiCG method with the SAI preconditioner converges very fast with a small number of iterations, compared to other preconditioners. Since the number of iterations of the BiCG is decreased significantly using the SAI, the total simulation time is reduced substantially. V. CONCLUSION We described a preconditioning strategy based on the SAI of the near part matrix in the MLFMA. This SAI preconditioner is easy to construct, as the near part matrix is naturally available in the MLFMA implementation. We gave a detailed algorithm on sparsifying the intermediate matrices at each step in order to construct an SAI preconditioner that is cost effective.

10 2286 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 9, SEPTEMBER 2004 We conducted a few numerical tests to show that the SAI preconditioner is effective in solving some electromagnetic scattering problems. We performed some experimental analysis in order to understand the properties of the SAI preconditioned matrices. Our studies indicate that the eigenvalues of the SAI preconditioned matrix are tightly clustered around 1, which ensures the fast convergence of the preconditioned BiCG method. According to the results from our numerical experiments, we can see that the SAI preconditioner constructed from the near part matrix improves the computational efficiency in terms of both the memory requirements and the CPU time. The results show that the BiCG method with the SAI preconditioner is robust for solving 3-D model cases from electromagnetic scattering simulations. In all cases, the memory cost of the SAI preconditioner is much less than the cost of storing the near part matrix. With well chosen parameters, the computational (construction) cost of the SAI preconditioner is even smaller than the preconditioned iterative solution cost in some range of the sparsity ratio. REFERENCES [1] G. Alléon, M. Benzi, and L. Giraud, Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics, Numer. Algorithms, vol. 16, pp. 1 15, [2] O. Axelsson, Iterative Solution Methods. Cambridge, U.K.: Cambridge Univ. Press, [3] B. Carpentieri, I. S. Duff, and L. Giraud, Experiments with sparse preconditioning of dense problems from electromagnetic applications, CERFACS, Toulouse, France, Tech. Rep. TR/PA/00/04, [4], Sparse pattern selection strategies for robust Frobenius-norm minimization preconditioners in electromagnetism, Numer. Linear Algebra Appl., vol. 7, pp , [5] B. Carpentieri, I. S. Duff, and M. 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Technol. Lett., vol. 30, no. 1, pp , Jeonghwa Lee received the M.S. degree in mathematics from Chonnam National University, South Korea, in 1998 and the Ph.D. degree in computer science from the University of Kentucky, Lexington, in Her research interests include large scale parallel and scientific computing, iterative and preconditioning techniques for large scale matrix computation, and computational electromagnetics.

11 LEE et al.: SPARSE INVERSE PRECONDITIONING OF MLFMA FOR HYBRID EQUATIONS 2287 Jun Zhang received the Ph.D. degree from The George Washington University, Washington, DC, in He is an Associate Professor of computer science and Director of the Laboratory for High Performance Scientific Computing and Computer Simulation, University of Kentucky, Lexington. His research interests include large scale parallel and scientific computing, numerical simulation, iterative and preconditioning techniques for large scale matrix computation. Dr. Zhang is the recipient of a National Science Foundation CAREER Award and several other awards. His research work is currently funded by the U.S. National Science Foundation and the Department of Energy. He is an Associate Editor and on the editorial boards of three international journals in computer simulation and computational mathematics, and is on the program committees of a few international conferences. Cai-Cheng Lu (S 95 M 95 SM 98) received the Ph.D. degree from the University of Illinois at Urbana Champaign, in Previously, he was with Demaco, Inc. (now SAIC), where he worked on a number of new features for the Xpatch code. Currently, he is an Associate Professor in the Department of Electrical and Computer Engineering, University of Kentucky, Lexington. He is especially experienced in fast algorithms for computational electromagnetics and is one of the authors for a CEM code FISC. His research interests are in wave scattering, microwave circuit simulation, and antenna analysis. Dr. Lu is a recipient of the 2000 Young Investigator Award from the Office of Naval Research and a CAREER Award from the National Science Foundation.