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New Preconditioning Technique to Avoid Convergence Deterioration Due to the Zero-Tree Gauge Condition in Magnetostatic Analysis Takeshi Mifune1 , Yasuhito Takahashi2 , and Takeshi Iwashita3 Department of Electrical Engineering, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan Academic Center for Computing and Media Studies, Kyoto University, Kyoto 606-8501, Japan Magnetostatic edge-element formulation leads to a singular linear system of equations. Although a reduced system of equations can be derived by imposing the zero-tree gauge condition, the gauging causes undesirable deterioration in the convergence property of conventional iterative solvers. In this paper, we develop a new preconditioning technique that overcomes the slow convergence caused by zero-tree gauging, by utilizing the folded preconditioning proposed in our previous paper. Moreover, we present a novel theorem that clarifies the coincidence of the spectra of the preconditioned matrices when folded preconditioning is used. Index Terms—Edge-element, iterative solver, Krylov subspace method, tree-cotree decomposition.
I. INTRODUCTION
INITE-ELEMENT (FE) discretization of electromagnetic problems frequently leads to the requirement of solving a large linear system of equations with a sparse coefficient matrix [1]. In a large-scale problem, the derived system may include more than ten millions of unknowns and equations. Iterative methods, such as the Krylov subspace method [2], [3], are feasible for solving such large problems, whereas traditional direct solvers require vast amounts of storage and arithmetic operations. Since the FE formulation tends to give an ill-conditioned coefficient matrix that causes slow convergence of the iterative solvers, the preconditioning technique [2] is widely used to improve the condition number of the coefficient matrix. Several FE formulations, e.g., the edge-element formulation for magnetostatic problems, lead to singular systems of equations, which involve redundant unknowns [1], [4]. In general, we can eliminate the redundant unknowns from a singular system, thereby deriving a reduced system of equations. In the magnetostatic case, the reduced system can be obtained by imposing a zero-tree gauge condition [5], [6]. Due to a smaller number of unknowns, zero-tree gauging lowers the computational costs per step and memory requirements of iterative solvers. However, it also causes a significant deterioration in the convergence of iterative solvers [7], [8]. Consequently, zero-tree gauging does not contribute to a reduction in the total computation time, at least not when we adopt a conventional iterative solver such as the incomplete Cholesky preconditioned conjugate gradient (IC-CG) method. Reference [9] provides an explanation for the slow convergence due to zero-tree gauging, from the viewpoint of the spectral property of the reduced coefficient matrix. The aim of this paper is to develop a new preconditioner to resolve the slow convergence that zero-tree gauging causes. The
F
Manuscript received August 21, 2009; revised December 08, 2009; accepted February 17, 2010. First published March 22, 2010; current version published June 23, 2010. Corresponding author: T. Mifune (e-mail:
[email protected]. kyoto-u.ac.jp). Digital Object Identifier 10.1109/TMAG.2010.2045125
new preconditioner is based on a technique, called folded preconditioning, proposed in our recent work [10]. In general, for an arbitrary preconditioner for an original singular matrix, a corresponding folded preconditioner can be derived for the reduced matrix. A theorem presented in [10] states that, for any Krylov subspace method with any preconditioner for the original singular system, we are able to design a mathematically equivalent Krylov subspace method with the corresponding folded preconditioner for the reduced system. Although both Krylov subspace methods have the same convergence property, we do not encounter the slow convergence in the latter due to the elimination of redundant unknowns. The initial sections of this paper (Sections II and III) give a brief introduction to folded preconditioning, and, in particular, describe the relationship between the original and the folded preconditioner in a more comprehensible form than in the previous paper [10]. Moreover, we present a novel theorem with respect to the effect of folded preconditioning on the spectrum of the preconditioned matrix. The theorem clarifies the coincidence of the spectra of the preconditioned matrices in the singular and reduced systems of equations, excluding the zero eigenvalues. It should be noted that this theorem holds with respect to any singular and reduced linear systems and is not limited only to magnetostatic applications. The major issue in the later sections (Sections IV–VI) is to design a new computation procedure that enables us to apply folded preconditioning to magnetostatic applications with zerotree gauging. In practical use of folded preconditioners, the matrix-vector multiplication with respect to the auxiliary matrices [10] needs to be implemented efficiently. As pointed out in [10], the auxiliary matrices are sparse and given explicitly in a fullwave FE formulation, and consequently, we developed an efficient folded preconditioner utilizing the sparsity of the auxiliary matrices. On the contrary, the auxiliary matrices in a magnetostatic edge-element formulation are not sparse. In Section IV, we devise a new procedure, that may be utilized, even if the auxiliary matrices are not sparse, to compute the multiplication by the auxiliary matrix. Section V describes a specific usage of the new computation procedure in magnetostatic applications. Finally, Section VI presents a sample magnetostatic analysis that demonstrates the performance of the newly developed preconditioning.
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II. SINGULAR SYSTEM OF EQUATIONS
Now, we consider the reduced system of equations:
Throughout this paper, we consider solving a singular system of linear equations (1) , where is a singular for the unknown vector ), and . The right-hand matrix (i.e., side vector must be in the range of , to ensure the existence of a solution. . We can reorder Let be an integer that satisfies leading principal the rows and columns of such that the has the same rank as , i.e., submatrix
(2) with matrices
. Since and
, there exist that satisfy
(3) The corresponding reordering with respect to the unknown and right-hand-side vectors in (1) leads to
(9) that satisfies Once a particular solution we can obtain a solution of (1) by substituting and , respectively. This is easily verified by
is given, and 0 into
(10) To obtain a solution of (1), therefore, we can apply an iterative solver to either (1) or (9). Based on the computational cost per step and the memory usage, it appears to be more attractive to solve (9) rather than (1). However, in some practical applications, the performance of conventional iterative solvers for (9) is very poor because of the slow convergence [7]–[9], [11]. The folded preconditioning technique has been proposed as a solution to this problem. III. FOLDED PRECONDITIONING A. Preconditioning
(4) and
Preconditioning is a widely used technique to improve the convergence of iterative solvers. Let be a preconditioning matrix (preconditioner) for (1). The (left-) preconditioned system is given by (11)
(5) where , , , and . and the rightmost equality Vector must be in the range of in (5) must be satisfied, to ensure that is in the range of . Let
and . If we choose an approwhere priate , the convergence property of iterative solvers for (11) may be better than that of iterative solvers for (1) [2], [3]. B. Definition of Folded Preconditioning for (1), the corresponding Given a specific preconditioner for (9) is defined by folded preconditioner (12)
(6) where denotes the identity matrix. Using (6), the coefficient matrix and right-hand side vector can be rewritten as
(7)
and to denote the inverse matrices of (In [10], we used the respective preconditioners, for convenience of explanation.) in (12) may, in rare circumAlthough matrix stances, be singular, use of the superscript “ 1” as the MoorePenrose generalized inverse can deal with all situations in the theorem and discussions presented below. C. Spectra of Preconditioned Matrices
and
(8)
The convergence property of iterative solvers depends greatly on the spectrum of the preconditioned coefficient matrix [2]. Here, we present a new theorem with respect to the spectra of
MIFUNE et al.: NEW PRECONDITIONING TECHNIQUE TO AVOID CONVERGENCE DETERIORATION
the preconditioned matrices in singular and reduced systems of equations. Theorem 1: The Jordan canonical form of coincides with that of , excluding those Jordan blocks corresponding to their zero eigenvalues. , i.e., Proof: Let be the Jordan canonical form of (13) are the generalized where the column vectors of . The diagonal entries of give the eigenvectors of eigenvalues of . and by Here, we denote the Null-spaces of and , respectively. Since from linearly independent vectors that sat(6), there exist isfy . Since from (7), it follows corresponding to the zero that all are eigenvectors of eigenvalues. We can select such that all are in the columns of . Multiplying both sides of (13) by and using (7) and (12), we obtain (14) from the definitions of and , we Since from are able to obtain the Jordan canonical form of from and the corresponding rows and (14), by dropping columns from . This theorem implies that the convergence of iterative solvers is similar to that of iterative solvers for (9) preconditioned by for (1) preconditioned by . Since any similar transformation keeps the spectrum of the matrix, right- or two-sided preconditioning also leads to the coincidence of the spectra of the preconditioned coefficient matrices. D. Equivalency of Preconditioned Krylov Subspace Methods Our previous paper [10] has revealed that, for any preconditioned Krylov subspace solver for (1), we can derive an equivalent solver in the form of the preconditioned Krylov subspace solver for (9), by using folded preconditioning. This is more easily confirmed using the notation introduced in this paper. In general, the preconditioned Krylov subspace method for (1) generates an approximate solution in the th step, such that
(16)
and another approximate solution,
It is worth noting that
, satisfy the following relation with respect to the residual vector: (18) Considering the above equality, an algorithm that computes can be regarded as equivalent to one that com. putes Substituting (7) and (8) into (15) and (16), multiplying both , and using (12), we have sides of (15) by (19) (20) denotes the upper elements of that where satisfy . Equations (19) and (20) indicate that can be designed in the form of a an algorithm to compute preconditioned Krylov subspace solver for the reduced system as the preconditioning matrix. (9) with IV. NEW COMPUTATION PROCEDURE FOR NON-SPARSE AUXILIARY MATRICES The algorithm for an iterative solver with preconditioning computes the matrix-vector multiplication with respect to the inverse of the preconditioner [2], [3]. For the folded preconditioner derived from the original preconditioner , we need an , where efficient procedure to compute is a given -dimensional vector. Since the original preconditioning procedure should provide , we only need procedures to comthe multiplication by and . pute the multiplications by the auxiliary matrices, and are sufficiently sparse, such as in the problem When dealt with in [10], it is straightforward to implement an efficient multiplication. However, this is not the case in magnetostatic analysis, as described in the next section. We therefore, propose a new computation procedure that may be efficient for non-sparse auxiliary matrices. For this new procedure, we need to find two nonsingular matrices, and (21) with
(15)
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, and must satisfy
,
, . Here, matrices
and
(22) where and residual vector space
denote the initial approximate solution and , respectively. The order- Krylov subis given as follows:
and (23) Equation (22) leads to
(17)
(24)
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that is (25) (From the assumption that is nonsingular, gular.) From (23), similarly, we have
is also nonsin-
(26) and Fig. 1. Example of FE mesh and tree-cotree decomposition.
(27) Let denote the transpose of the reduced incidence matrix [12]. Matrix is written in the following form:
Moreover, we can verify the following relationships: (28)
(31) and (29) where and are arbitrary - and -dimensional vectors, respectively. Equations (28) and (29) give the new procedures to compute the multiplications by the auxiliary matrices. The left-hand sides of (28) and (29) can be efficiently computed if, for example, we find sparse triangular matrices and .
and . Each row of with and is associated with an edge in the co-tree and tree, respectively. The nonzero entries of represent the relationship between the edges and nodes in the FE mesh: (32) otherwise with (33)
V. FOLDED PRECONDITIONING FOR MAGNETOSTATIC EDGE-ELEMENT ANALYSIS This section focuses on an application of the computation procedure, discussed in the previous section, to the magnetostatic problem. More specifically, we aim to find sparse triangular matrices and that satisfy (22) and (23), in the magnetostatic edge-element formulation. The edge-element formulation for the magnetostatic problem coefficient matrix [4]: leads to the following
to node . where edge is directed from node , not . Note (The number of nodes in a FE mesh is that the column associated with the root node is not included in .) For instance, the matrix for Fig. 1 is given by
(34)
(30) where , , and denote the edge-element basis functions [1], magnetic reluctivity that is element-wise constant, and the whole analyzed region, respectively. We do not deal with the higher order basis functions [1] in this paper, and hence is identical to the number of edges in the FE mesh. , the It is known that given by (30) is singular and that nullity of , is equal to the number of edges in the spanning tree of the FE mesh [4]. Hence, is equal to the number of edges in the co-tree. Fig. 1 shows an example of the FE mesh and treecotree decomposition, in which solid and dashed edges are in the tree and co-tree [12], respectively. We can derive the reduced system (9) by eliminating unknowns which are associated with edges in the tree [4], [9]. for simplicity, and consequently Hereafter, we enforce a reordering must be applied so that and in (4) are associated with edges in the co-tree and tree, respectively.
In general, the matrix satisfies , and [4]. Since is nonsingular when have
, , we (35)
and (36) in (34) is given in an upper-triangular form. In Moreover, becomes a triangular matrix by, general, we can ensure that for example, the following reordering with respect to the nodes and edges in the tree.
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(i) Arbitrarily choose a spanning tree and a root node. Set . (ii) From the tree structure, choose a leaf node as node . In the tree structure, there is only a single edge connected to node . Label this edge as . (iii) Remove node and edge from the tree structure. , exit. Otherwise, and go back to (ii). (iv) If Here, a leaf node is a vertex incident to only one edge in the tree (i.e., a vertex of degree 1 [12] in the tree), other than the root node. In fact, the ordering used in Fig. 1 is set by the above algorithm. Obviously, the above reordering ensures that (37) . Hence becomes an upper-triangular with matrix. Consequently, we obtain suitable matrices and for the new computation procedure proposed in the previous section, into and , and into and . by substituting Equations (22) and (23) are satisfied because of (35) and (36), respectively. Both and are nonsingular, due to (37). Since and are sparse and triangular, the left-hand sides of (28) and (29) can be computed efficiently using backward has and forward substitution, respectively. In fact, matrix at most three nonzero entries in each row (column) associated with the co-tree, and at most two nonzero entries in each row (column) associated with the tree. Because all nonzero entries are 1 or 1, each of the forward and backward substitutions can additions/subtractions. Note that be executed by at most the diagonal entries of and are not explicitly needed in the computation if we apply a suitable scaling such that the diagonal entries become unity. obtained from It can be additionally confirmed that and (24) and (26) are identical to the principal parts of the fundamental circuit and cutset matrices [12], respectively, in the graph theory. These are not generally sparse matrices. VI. NUMERICAL RESULTS In this section, the performance of iterative solvers with the new preconditioner is examined in a magnetostatic test problem. Computations are executed on a Windows PC (Intel Core 2 X9770, 8 GB RAM). The Fortran 95/2003 code is compiled using the Intel Fortran compiler 11.0 with the optimization option “/O2.” Fig. 2 depicts the test problem (1/8 model) that is similar to the box shield problem adopted in [13]. The brick edge-elements [1] are used in the FE formulation. Dirichlet boundary conditions are imposed in the - and planes. In the tree-cotree splitting, all edges parallel to the -axis are chosen as elements of the tree. IC factorization is adopted as the original preconditioner for (1), with the acceleration parameter in the IC factorization set to 1.03. We use the IC-CG method for (1) and the CG method with the folded variant of IC preconditioning (FIC-CG) for (9). For further comparison, we also use the standard IC-CG method for (9), although its performance is very poor.
Fig. 2. Test model.
TABLE I PERFORMANCE OF THE ITERATIVE SOLVERS
The convergence criterion is 1e-8 for the relative residual norm with respect to (9), and not to (1). The same criterion is used in the solver for (1), for fair comparison [10]. This causes no disadvantage to the solver for (1). Table I compares the performance of the iterative solvers. The zero-tree gauging reduces the number of unknowns by approximately 2/3, and the number of nonzero entries in the upper triangular part of the coefficient matrix by nearly 4/9. This results in a considerable advantage to the FIC-CG solver for (9) in terms of the memory used. The memory requirement for matrix is relatively small as shown in Table I. It is also confirmed that the number of iterations of the FIC-CG solver for (9) is nearly equal to that of the IC-CG solver for (1). As discussed in [10], the convergence properties of both solvers are mathematically equivalent, although rounding errors may cause the number of iterations in each to differ slightly. In contrast, the convergence of the standard IC-CG solver for (9) is very poor. The new solver for (9) shows a considerable improvement in both memory usage and solution time, compared with the conventional one for (1). This implies good efficiency of the new computation procedure presented in Sections IV and V.
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The numerical discrepancy between the magnetic enerand , computed using the IC-CG solver for gies (1) and the FIC-CG solver for (9), is sufficiently small: .
The theoretical result in Section III and computation procedure proposed in Section IV can be applied in singular and reduced systems of equations in general, e.g., those that arise from the singularity decomposition technique [13] or the implicit error correction method [16], [17], among others.
VII. CONCLUSION In this paper, we developed a new preconditioning technique that overcomes the convergence deterioration due to zero-tree gauging in magnetostatic edge-element analysis. The new technique is based on the folded preconditioning proposed in [10]. Moreover, we presented a new theorem with respect to the spectra of preconditioned matrices. To apply the folded preconditioning to magnetostatic edge-element analysis, we devised a new computational procedure, that stems from the new preconditioning, for the matrix-vector multiplication with respect to the auxiliary matrices. Despite the fact that the auxiliary matrices are not sparse in the magnetostatic edge-element formulation, the new procedure gives an efficient implementation for the multiplication. The numerical test demonstrates the good efficiency of the proposed preconditioning. In the test analysis, IC factorization was adopted as the original preconditioner for the singular matrix. However, we could utilize any arbitrary preconditioner, e.g., the multigrid methods [4], [14], thereby developing the corresponding folded preconditioner for the reduced matrix. Compared with the conventional solver, the new solver achieves, approximately, a 17% and 24% improvement in the solution time and memory requirement, respectively, in the test analysis. It should be noted that the degree of the improvement hardly depends on the configuration of the analyzed model, such as the magnetic property of the shield, and that we can estimate the effect of the proposed method beforehand, by calculating the number of reduced unknowns and nonzero entries in the matrices. The computation procedure developed in this paper can also be applied to the eddy-current FE analysis using both vector and scalar potentials, in which the zero-tree gauge condition can be imposed in the nonconductive region [4]. We can apply the technique discussed in Section V in a straightforward manner for the nonconductive region, together with the one developed in [10] for the conductive region. The folded preconditioning technique can also be used with nonlinear problems, when each linear system of equations in the nonlinear iteration is singular. Reference [15] deals with a nonlinear analysis, in which the Newton-Raphson iteration is used. Additionally, we presented a theoretical result with respect to the spectra of preconditioned coefficient matrices in singular and reduced systems of equations. The theorem presented states that, if folded preconditioning is used, the spectrum of the preconditioned coefficient matrix in the singular system coincides with that in the reduced system, excluding the zero eigenvalues.
ACKNOWLEDGMENT The authors would like to thank A. Kameari and Prof. K. Fujiwara for their comments and insight into the potential of the folded preconditioning technique in magnetostatic edge-element analyses. REFERENCES [1] J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. New York: Wiley, 2002. [2] A. Greenbaum, Iterative Methods for Solving Linear Systems. Philadelphia: SIAM, 1997. [3] H. A. van der Vorst, Iterative Krylov Methods for Large Linear Systems. Cambridge, U.K.: Cambridge University Press, 2003. [4] Y. Zhu and A. C. Cangellaris, Multigrid Finite Element Methods for Electromagnetic Field Modeling. Piscataway, NJ: IEEE Press, 2006. [5] R. Alvanese and G. Rubinacci, “Magnetostatic field computations in terms of two-component vector potentials,” Int. J. Numer. Methods Eng., vol. 29, pp. 515–531, 1990. [6] J. B. Manges and Z. Cendes, “A generalized tree-cotree gauge for magnetic field computation,” IEEE Trans. Magn., vol. 31, no. 3, pp. 1342–1347, May 1995. [7] N. A. Golias and T. D. Tsiboukis, “Magnetostatics with edge elements: A numerical investigation in the choice of the tree,” IEEE Trans. Magn., vol. 30, no. 5, pp. 2877–2880, Sep. 1994. [8] O. Biro, K. Preis, and K. R. Richter, “On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3D magnetostatic problems,” IEEE Trans. Magn., vol. 32, no. 3, pp. 651–654, May 1996. [9] H. Igarashi, “On the property of the curl-curl matrix in finite element analysis with edge elements,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3129–3132, Sep. 2001. [10] T. Mifune, Y. Takahashi, and T. Iwashita, “Folded preconditioner: A new class of preconditioners for Krylov subspace methods to solve redundancy-reduced linear systems of equations,” IEEE Trans. Magn., vol. 45, no. 5, pp. 2068–2075, May 2009. [11] K. Fujiwara, T. Nakata, and H. Ohashi, “Improvement of convergence characteristics of ICCG method for the A-' method using edge elements,” IEEE Trans. Magn., vol. 32, no. 3, pp. 804–807, May 1996. [12] W. Chen, Applied Graph Theory. Amsterdam, The Netherlands: North-Holland, 1971. [13] A. Kameari, “Improvement of ICCG convergence for thin elements in magnetic field analyses using the finite-element method,” IEEE Trans. Magn., vol. 44, no. 6, pp. 1178–1181, Jun. 2008. [14] W. Hackbusch, Multi-Grid Methods and Applications. New York: Springer-Verlag, 1985. [15] Y. Takahashi, T. Mifune, and T. Iwashita, “Novel block IC based preconditioning for edge finite-element electromagnetic field analyses,” in Proc. 17th Conf. Computation of Electromagnetic Fields (COMPUMAG2009), 2009, pp. 1026–1027. [16] T. Iwashita, T. Mifune, and M. Shimasaki, “Similarities between implicit correction multigrid method and A-phi formulation in electromagnetic field analysis,” IEEE Trans. Magn., vol. 44, no. 6, pp. 946–949, Jun. 2008. [17] T. Mifune, S. Moriguchi, T. Iwashita, and M. Shimasaki, “Convergence acceleration of iterative solvers for the finite element analysis using the implicit and explicit error correction methods,” IEEE Trans. Magn., vol. 45, no. 3, pp. 1104–1107, Mar. 2009.
