A survey of parallel solvers for the finite element method in computational electromagnetics <s> The Conjugate Gradient Method <s> A new parallelized ICCG scheme is applied to a finite edge element analysis of a 3-D eddy current problem with a moving conductor. In the new method, the global matrix is automatically reordered to the matrix form appropriate to parallel processing. The method requires no particular pre-processing step. It is shown that the newly proposed method has a better parallel efficiency than the Block ICCG method because of a higher preconditioning effect. <s> BIB001 </s> A survey of parallel solvers for the finite element method in computational electromagnetics <s> The Conjugate Gradient Method <s> Proposes a new black-box-type parallel processing method for the incomplete Cholesky conjugate gradient (ICCG) solver. The new method is based on a multicolor ordering concept and an automatic reordering process in the solver. Parallel performance is evaluated in the context of three-dimensional finite edge-element eddy-current analysis. The proposed method attains high parallelism with a small increase in CG iterations and achieves high parallel performance. <s> BIB002 </s> A survey of parallel solvers for the finite element method in computational electromagnetics <s> The Conjugate Gradient Method <s> The conjugate gradient squared (CGS) algorithm is a Krylov subspace algorithm that can be used to obtain fast solutions for linear systems (Ax=b) with complex nonsymmetric, very large, and very sparse coefficient matrices (A). By considering electromagnetic scattering problems as examples, a study of the performance and scalability of this algorithm on two MIMD machines is presented. A modified CGS (MCGS) algorithm, where the synchronization overhead is effectively reduced by a factor of two, is proposed in this paper. This is achieved by changing the computation sequence in the CGS algorithm. Both experimental and theoretical analyses are performed to investigate the impact of this modification on the overall execution time. From the theoretical and experimental analysis it is found that CGS is faster than MCGS for smaller number of processors and MCGS outperforms CGS as the number of processors increases. Based on this observation, a set of algorithms approach is proposed, where either CGS or MGS is selected depending on the values of the dimension of the A matrix (N) and number of processors (P). The set approach provides an algorithm that is more scalable than either the CGS or MCGS algorithms. The experiments performed on a 128-processor mesh Intel Paragon and on a 16-processor IBM SP2 with multistage network indicate that MCGS is approximately 20% faster than CGS. <s> BIB003
The Conjugate Gradient (CG) method is one of the most popular linear solvers for Finite Element (FE) analysis, since it is well adapted to solve sparse matrix systems. In order to improve the solving, preconditioning techniques have to be used together with the solving method. The basic idea consists in replacing the initial system A x = b by a new system where M is the preconditioning matrix. Two conditions are required for efficiency: -the conditioning number of the new matrix Ã has to be lower than the initial conditioning number, -and the matrix system M has to be to not expensive to build or to solve. Several preconditioning techniques exist: Incomplete Cholesky (ICP), SSOR, Diagonal (DP) preconditioning. The following example shows the efficiency of the preconditioning method. It is used for the modeling of radiofrequency hyperthermia  . The problem is computed with more than 130 000 degrees of freedom. From Table I , by comparing both DP and ICP, it is shown that better results are obtained with the ICP in terms of number of iterations. On the other hand, since the cost of one iteration is greater than when using the DP, the total CPU time of computation is larger. In this case, the best results are obtained with a SSOR preconditioning: both number of iterations and CPU time are divided by two. In BIB001 , a parallelized transient eddy current FE analysis is presented. Due to the presence of moving conductors, the right-hand side of the system of equations is changed at each time step. First-order bricktype edge elements are used for the FE discretization. Parallel solving is performed using the ICCG, leading to difficulties due to the forward-backward substitutions. Two parallel implementations are compared. First, on N p processors, the block ICCG (BICCG) method divides the global matrix into N p submatrices and performs the IC factorization for the local submatrix in each processor, with the entries between different processors being ignored in the factorization. Second, a renumbering process allows to reorder the matrix into a form similar to the dissection ordering case (PICCG-RP) ( fig. 1 ). Since no entries are neglected during the preconditioning step, a better convergence rate is observed with the PICCG-RP method, leading to a better parallel efficiency ( fig. 2 ). This reordering technique is extended in BIB002 for the same 3D eddy-current analyses. The key of the method is coloring of the unknowns. Several colors are assigned to the unknowns, such that the unknowns belonging to the same color must have no data dependency on each other. Due to the presence of edge elements, the assignment is algebraic: it is carried out by using only information obtained from a coefficient matrix. This method is actually a complete black-box solver, and its implementation appears to be easy. Parallel performances obtained on a distributed memory Fujitsu VPP-800 are also better than when using a classical BICCG ( fig. 3 ). It is also shown that the use of many colors improves the convergence rate, but it causes an increase in communication costs and a decline in parallelism. For the example of transient eddy currents analysis, made of more than 10 6 unknowns, best results are obtained with 60 colors. The Conjugate Gradient Squared (CGS) is also used for the modeling of 2D open-region frequencydomain scattering problems by the FEM in BIB003 . By modifying the computation sequence in the algorithm, the synchronization overhead is reduced by a factor of two. The basic idea of this modification is to merge the inner products present in the CGS algorithm so that they can be evaluated using a single global summing operation per iteration of the "while" loop. From both theoretical and experimental analyses it is found that this Modified CGS (MCGS) algorithm becomes slightly faster as the number of processors increases. A set of algorithms is then proposed, where either CGS or MCGS is selected depending on the number of unknowns and number of processors.