Some new strategies for RCM ordering in solving electromagnetic scattering problems

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Abstract

We describe some ordering strategies for improving the incomplete Cholesky factorization used in the preconditioned conjugate gradient method applied to electromagnetic scattering problems. Some matrix ordering strategies, derived from the greedy partitioning algorithm for multilevel methods, are combined with RCM ordering. These ordering techniques are tested and compared with normal RCM ordering. Some tuning in selecting special nodes as first and last nodes in reverse Cuthill–McKee ordering is shown to apparently improve the quality of incomplete Cholesky factorization preconditioners.

Introduction

We are concerned with solving large linear systems that arise from edge element-based finite element discretization of high-frequency electromagnetic field simulations. The coefficient matrix of the linear equations which stem from the finite-element analysis of high-frequency electromagnetic field simulations such as scattering [1], [2], [3], [4], [5], [6], [7], [8], [9] is generally symmetric and indefinite. At present, the incomplete Cholesky factorization [10], [11], [12], [13], [14], [15], [16], [17] (IC) preconditioners applied with the preconditioned Conjugate Gradient (PCG) method are rather popular. Here RCM ordering methods preceding the incomplete Cholesky factorization are studied in the case of finite element (FEM) matrices arising from the discretization of the following electromagnetic scattering problem: ×(1μr×Esc)k02εrEsc=×(1μr×Einc)+k02Einc, with some absorbing boundary conditions, where Esc is the scattering field, Einc is the incident field, and μr and εr are relative permeability and permittivity, respectively.

The solution of Eq. (1) will result in a linear system Ax=b, where A=(aij)n×nCn×n is sparse complex symmetric (usually indefinite), x,bCn.

In the literature, the preconditioned conjugate gradient (PCG) method has been proven to be a competitive iterative method for the solution of symmetric linear systems arising from PDEs in many applications [15]. In order to perform a successful application of the PCG method, an effective preconditioner, such as the incomplete Cholesky factorization preconditioner, is needed first. It was illustrated in many studies that reordering of matrices plays a crucial effect in constructing a preconditioner with high quality.

In [18], Notay investigated the reverse Cuthill–McKee ordering and fill-in strategies for approximate factorization preconditioning. In [15], Clift and Tang described a kind of basis of a matrix ordering heuristic for improving the incomplete factorization used in preconditioned conjugate gradient techniques applied to anisotropic PDEs. And they presented a variation of RCM which is shown to generally improve the quality of incomplete factorization preconditioners. For convenience, we will use a general framework of RCM ordering, provided by Notay, which is close to the implementation form.

In order to test different strategies for RCM presented in this paper, we use a modified incomplete Cholesky factorization algorithm (MIC(p,τ)) [14] which allows us to decide the sparsity of incomplete factorization preconditioners by two fill-in control parameters: (1) p, the number of the largest number p of nonzero entries in each row; and (2) dropping tolerance.

The rest of the paper is organized as follows. Some new strategies used in RCM ordering are presented in Section 2. In Section 3, numerical experiments are carried out to illustrate the efficiency of these strategies. Finally, concluding remarks are given in Section 4.

Section snippets

Application strategies for RCM

Reverse Cuthill–McKee ordering is normally used as a generally effective matrix coefficient insensitive ordering which is quick to compute [15], [17].

In the literature, the RCM ordering algorithm has been developed with a focus on how to select a starting node in the Cuthill–McKee ordering [18], i.e. the last node in the Reverse Cuthill–McKee ordering. In this paper, we will focus on selecting the first node of the RCM ordering method and the last node of the RCM ordering method.

In [18], Y.

Numerical tests

In order to evaluate the performance of our reordering strategies, we examined matrices arising from electromagnetic scattering problems. We implemented numerical experiments based on ITSOL packages [20]. All numerical tests were performed on the Linux operating system. All codes are programmed in C language and run on a PC, with 2 GB memory and a 2.66 GHz Intel(R) Core(TM)2 Duo CPU. The maximal iteration number is taken as 1000. The stopping criteria is r(k)/r(0)<108. In all tests, the

Conclusions

We have presented and tested RCM ordering techniques with new strategies which consider the coefficients of matrices from electromagnetic problems by using a measure (see Eq. (3)) as that in [19]. These strategies associated with RCM ordering methods, based on Algorithm 2, result in an improved incomplete Cholesky factorization for PCG methods.

Numerical results indicate that the modified RCM technique with two strategies is generally better than normal RCM especially for difficult problems. We

Acknowledgments

We would like to thank the anonymous referees for their detailed comments which greatly improved this paper. We also thank Dr. Liang Li for his help in providing test problems. This research was supported by the NSFC (60973015, and 61170311), the Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), the Youth Sci. and Tech. Fund of UESTC and the Fundamental Research Funds for the Central Universities.

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