Application of the preconditioned conjugate-gradient algorithm to the edge FEM for electromagnetic boundary-value problems

Abstract

In this letter, an effective symmetric successive overrelaxation (SSOR) preconditioning scheme is applied to the conjugate‐gradient method for solving a large system of linear equations resulting from the use of the edge‐based finite‐element method (FEM). With SSOR as the preconditioner as well as its efficient implementation in the conjugate‐gradient (CG) algorithm, the PCG method converges five times as fast as the CG method. This result demonstrates that SSOR is a good preconditioner for the CG iterative method when the FEM is applied to solve large‐scale electromagnetic problems. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 235–238, 2000.

Robust GMRES recursive method for fast finite element analysis of 3D electromagnetic problems

“…This demonstrates the validation of our numerical FEM method and the SSOR-GMRESR͑m͒ algorithm implementation. To compare the relative speed-up of the SSOR-GMRESR͑m͒ and other popularly preconditioned CG iterative methods, Figure 2 displays the convergence characteristics of residual error versus number of iterations for SSOR-GMRESR͑m͒, incomplete factorization (IF) [17], factorized sparse inverse (FSAI) [10], SSOR [12] preconditioned, and conventional CG methods for the waveguide partially filled with dielectric when normalized wave number k 0 b ϭ 1.88.. As shown in Figure 2, the SSOR-GMRESR͑m͒ method achieves an iteration number convergence improvement 726.4 times less than the conventional CG method for the residual errors to reach Ϫ70 dB when the number of inner iterations for GM-RESR is chosen to be 15. The computation is carried out in Pentium IV 1.7 GHz PC and the CPU time is 175 s for SSOR-GMRESR (15), 835 s for IF-CG, 1462 s for SSOR-CG, 1767 s for FSAI-CG and 7396 s for the conventional CG algorithms, respectively.…”

“…Therefore, the key point is to choose a suitable approximation to A Ϫ1 r k . In the above implementation, as can be seen from line 5 of the algorithm, the SSOR preconditioning technique [12] for m steps of GMRES iteration is added to get a better approximation to A Ϫ1 r k . Meanwhile, a truncation strategy in line 7 is introduced to restrict memory requirements, where we only update from the last j outer iterations.…”

“…Although the approximate inverse technique is one of widely used preconditioning schemes, its performance heavily relies on finding an optimal sparsity pattern and the reduction in the solution time usually does not compensate for the huge setup time in the construction phase. Like diagonal or block diagonal matrix preconditioner, the symmetric successive overrelaxation (SSOR) preconditioner can directly be derived from the coefficient matrix without additional cost and can lead to significant convergence improvement for sparse linear systems [12]. The incomplete factorizations of the coefficient matrix are also a widely used class of preconditioners [13] for fast FEM analysis.…”

Robust GMRES recursive method for fast finite element analysis of 3D electromagnetic problems

“…This demonstrates the validation of our numerical FEM method and the SSOR-GMRESR͑m͒ algorithm implementation. To compare the relative speed-up of the SSOR-GMRESR͑m͒ and other popularly preconditioned CG iterative methods, Figure 2 displays the convergence characteristics of residual error versus number of iterations for SSOR-GMRESR͑m͒, incomplete factorization (IF) [17], factorized sparse inverse (FSAI) [10], SSOR [12] preconditioned, and conventional CG methods for the waveguide partially filled with dielectric when normalized wave number k 0 b ϭ 1.88.. As shown in Figure 2, the SSOR-GMRESR͑m͒ method achieves an iteration number convergence improvement 726.4 times less than the conventional CG method for the residual errors to reach Ϫ70 dB when the number of inner iterations for GM-RESR is chosen to be 15. The computation is carried out in Pentium IV 1.7 GHz PC and the CPU time is 175 s for SSOR-GMRESR (15), 835 s for IF-CG, 1462 s for SSOR-CG, 1767 s for FSAI-CG and 7396 s for the conventional CG algorithms, respectively.…”

“…Therefore, the key point is to choose a suitable approximation to A Ϫ1 r k . In the above implementation, as can be seen from line 5 of the algorithm, the SSOR preconditioning technique [12] for m steps of GMRES iteration is added to get a better approximation to A Ϫ1 r k . Meanwhile, a truncation strategy in line 7 is introduced to restrict memory requirements, where we only update from the last j outer iterations.…”

“…Although the approximate inverse technique is one of widely used preconditioning schemes, its performance heavily relies on finding an optimal sparsity pattern and the reduction in the solution time usually does not compensate for the huge setup time in the construction phase. Like diagonal or block diagonal matrix preconditioner, the symmetric successive overrelaxation (SSOR) preconditioner can directly be derived from the coefficient matrix without additional cost and can lead to significant convergence improvement for sparse linear systems [12]. The incomplete factorizations of the coefficient matrix are also a widely used class of preconditioners [13] for fast FEM analysis.…”

Adaptive sampling cubic‐spline interpolation method for efficient calculation of monostatic RCS

“…One method that can alleviate this difficulty is to interpolate the required data by analytic model-based function, among which the rational function interpolation proves to be the most versatile, effective, and apposite one. In this article, we investigate a new rational function interpolation method based on Stoer-Bulirsch algorithm [1][2][3], combined with the method of moments (MoM) to accelerate the reflective and perspective parameter computation of FSS [4][5][6].…”

Adaptive sampling bicubic spline interpolation method for fast calculation of monostatic RCS

“…The fast multipole method (FMM) and multilevel fast multipole method (MLFMM) [2,3] are able to alleviate this difficulty by accelerating the operation of matrix-vector product, and preconditioning techniques [4][5][6][7][8] can speed up convergence of iterative solvers by improving spectral properties of the EFIE matrix. However, it is still time-and memory-consuming for calculation of monostatic RCS since it requires repeated solution of EFIE at each incident direction and frequency.…”

Robust GMRES recursive method for fast finite element analysis of 3D electromagnetic problems

“…This demonstrates the validation of our numerical FEM method and the SSOR-GMRESR͑m͒ algorithm implementation. To compare the relative speed-up of the SSOR-GMRESR͑m͒ and other popularly preconditioned CG iterative methods, Figure 2 displays the convergence characteristics of residual error versus number of iterations for SSOR-GMRESR͑m͒, incomplete factorization (IF) [17], factorized sparse inverse (FSAI) [10], SSOR [12] preconditioned, and conventional CG methods for the waveguide partially filled with dielectric when normalized wave number k 0 b ϭ 1.88.. As shown in Figure 2, the SSOR-GMRESR͑m͒ method achieves an iteration number convergence improvement 726.4 times less than the conventional CG method for the residual errors to reach Ϫ70 dB when the number of inner iterations for GM-RESR is chosen to be 15. The computation is carried out in Pentium IV 1.7 GHz PC and the CPU time is 175 s for SSOR-GMRESR (15), 835 s for IF-CG, 1462 s for SSOR-CG, 1767 s for FSAI-CG and 7396 s for the conventional CG algorithms, respectively.…”

“…Therefore, the key point is to choose a suitable approximation to A Ϫ1 r k . In the above implementation, as can be seen from line 5 of the algorithm, the SSOR preconditioning technique [12] for m steps of GMRES iteration is added to get a better approximation to A Ϫ1 r k . Meanwhile, a truncation strategy in line 7 is introduced to restrict memory requirements, where we only update from the last j outer iterations.…”

“…Although the approximate inverse technique is one of widely used preconditioning schemes, its performance heavily relies on finding an optimal sparsity pattern and the reduction in the solution time usually does not compensate for the huge setup time in the construction phase. Like diagonal or block diagonal matrix preconditioner, the symmetric successive overrelaxation (SSOR) preconditioner can directly be derived from the coefficient matrix without additional cost and can lead to significant convergence improvement for sparse linear systems [12]. The incomplete factorizations of the coefficient matrix are also a widely used class of preconditioners [13] for fast FEM analysis.…”

Adaptive sampling cubic‐spline interpolation method for efficient calculation of monostatic RCS

“…One method that can alleviate this difficulty is to interpolate the required data by analytic model-based function, among which the rational function interpolation proves to be the most versatile, effective, and apposite one. In this article, we investigate a new rational function interpolation method based on Stoer-Bulirsch algorithm [1][2][3], combined with the method of moments (MoM) to accelerate the reflective and perspective parameter computation of FSS [4][5][6].…”

Adaptive sampling bicubic spline interpolation method for fast calculation of monostatic RCS

“…The fast multipole method (FMM) and multilevel fast multipole method (MLFMM) [2,3] are able to alleviate this difficulty by accelerating the operation of matrix-vector product, and preconditioning techniques [4][5][6][7][8] can speed up convergence of iterative solvers by improving spectral properties of the EFIE matrix. However, it is still time-and memory-consuming for calculation of monostatic RCS since it requires repeated solution of EFIE at each incident direction and frequency.…”

Robust GMRES recursive method for fast finite element analysis of 3D electromagnetic problems

“…This demonstrates the validation of our numerical FEM method and the SSOR-GMRESR͑m͒ algorithm implementation. To compare the relative speed-up of the SSOR-GMRESR͑m͒ and other popularly preconditioned CG iterative methods, Figure 2 displays the convergence characteristics of residual error versus number of iterations for SSOR-GMRESR͑m͒, incomplete factorization (IF) [17], factorized sparse inverse (FSAI) [10], SSOR [12] preconditioned, and conventional CG methods for the waveguide partially filled with dielectric when normalized wave number k 0 b ϭ 1.88.. As shown in Figure 2, the SSOR-GMRESR͑m͒ method achieves an iteration number convergence improvement 726.4 times less than the conventional CG method for the residual errors to reach Ϫ70 dB when the number of inner iterations for GM-RESR is chosen to be 15. The computation is carried out in Pentium IV 1.7 GHz PC and the CPU time is 175 s for SSOR-GMRESR (15), 835 s for IF-CG, 1462 s for SSOR-CG, 1767 s for FSAI-CG and 7396 s for the conventional CG algorithms, respectively.…”

“…Therefore, the key point is to choose a suitable approximation to A Ϫ1 r k . In the above implementation, as can be seen from line 5 of the algorithm, the SSOR preconditioning technique [12] for m steps of GMRES iteration is added to get a better approximation to A Ϫ1 r k . Meanwhile, a truncation strategy in line 7 is introduced to restrict memory requirements, where we only update from the last j outer iterations.…”

“…Although the approximate inverse technique is one of widely used preconditioning schemes, its performance heavily relies on finding an optimal sparsity pattern and the reduction in the solution time usually does not compensate for the huge setup time in the construction phase. Like diagonal or block diagonal matrix preconditioner, the symmetric successive overrelaxation (SSOR) preconditioner can directly be derived from the coefficient matrix without additional cost and can lead to significant convergence improvement for sparse linear systems [12]. The incomplete factorizations of the coefficient matrix are also a widely used class of preconditioners [13] for fast FEM analysis.…”

Adaptive sampling cubic‐spline interpolation method for efficient calculation of monostatic RCS

“…One method that can alleviate this difficulty is to interpolate the required data by analytic model-based function, among which the rational function interpolation proves to be the most versatile, effective, and apposite one. In this article, we investigate a new rational function interpolation method based on Stoer-Bulirsch algorithm [1][2][3], combined with the method of moments (MoM) to accelerate the reflective and perspective parameter computation of FSS [4][5][6].…”

Adaptive sampling bicubic spline interpolation method for fast calculation of monostatic RCS

“…The fast multipole method (FMM) and multilevel fast multipole method (MLFMM) [2,3] are able to alleviate this difficulty by accelerating the operation of matrix-vector product, and preconditioning techniques [4][5][6][7][8] can speed up convergence of iterative solvers by improving spectral properties of the EFIE matrix. However, it is still time-and memory-consuming for calculation of monostatic RCS since it requires repeated solution of EFIE at each incident direction and frequency.…”