IC(𝒑) Preconditioning with Acceleration Factor for
High-Frequency Electromagnetic Field Analysis
Koki MASUI

Masao Ogino

Takahiro KATAGIRI

Fumihiko INO

Osaka UniversityJapan

Daido UniversityJapan

Nagoya UniversityJapan

Osaka UniversityJapanmasui@ist.osakau.ac.jp

m-ogino@daidoit.ac.jp

katagiri@cc.nagoyau.ac.jp

ino@ist.osakau.ac.jp

1. INTRODUCTION
Iterative methods suffer from oscillating residual norm histories
and slow convergence rates in solving the complex symmetriclinear equations derived from the edge finite element analysis ofhigh-frequency electromagnetic fields [1]. Various iterative
methods and preconditioners have been developed. Applying thepreconditioning method such as Incomplete Cholesky (IC)preconditioning with acceleration factor (AF) is a common
strategy to improve convergence in iterative methods forsymmetric linear equations. Although IC(𝑝) [2], where 𝑝 is thefill-in level, can improve the convergence, there are few examples
and an efficient implementation method has not been establishedin the past. Moreover, there is no example of applications forcomplex number problems such as electromagnetic field analysis.
Therefore, we implement and optimize IC( 𝑝 ), and show theresults when applied to electromagnetic field problems.

preconditioning matrix in IC(0), IC(1), and IC(2) are 1,129,211,2,094,069, 4,680,923, respectively.
All calculations were performed using an Intel CPU Core i99900 (3.60 GHz) processor with 32 GB memory, the gcc 10.2.0
compiler, and the “-O3 -mfma” optimization flag.
Figure 2 shows the iteration counts and calculation time as theAF of IC preconditioning was varied from 1.03 to 1.15 inincrements of 0.01. As can be seen in the figure, increasing thefill-in level improved the convergence regardless of theacceleration factor, indicating that is effective in electromagnetic
field analysis problems. Our proposed method succeeded inreducing the computation time by 51% compared to theconventional method (IC(0)) in the optimal parameters.

2. PRECONDITIONING
While IC(0) without fill-in generates the preconditioning matrix
those non-zero element positions are the same as the coefficientmatrix, IC(𝑝) generates the preconditioning matrix with additional
non-zero element positions depending on the value of 𝑝.
The additional non-zero element positions are determined by𝐿𝑒𝑣𝑒𝑙𝑖𝑗 = min{𝐿𝑒𝑣𝑒𝑙𝑖𝑗 ,min1≤𝑘≤min⁡(𝑖,𝑗){𝐿𝑒𝑣𝑒𝑙𝑖𝑘 + 𝐿𝑒𝑣𝑒𝑙𝑘𝑗 + 1}}.⁡⁡⁡⁡(1)⁡

If we implement as per (1), the computational complexity will beO( 𝑛3 ), where 𝑛 is the size of the matrix. Therefore, we
implemented an efficient method that focuses on the position of
non-zero elements in the coefficient matrix.
Figure 1 shows the method we used in this study. We computeadditional non-zero element positions from the non-zero element
positions contained in the column. If you want to compute IC(2),you should apply to the matrix computed by IC(1).

Figure 2. AF vs. iteration counts and calculation time.

4. CONCLUTION
In this study, we proposed an efficient IC(𝑝) preconditioning
method for complex symmetric matrices that appear in highfrequency electromagnetic field analysis and implemented an
iterative method with different fill-in levels to evaluate theperformance. As a result, our method succeeded in reducing boththe number of iterations and the computation time anddemonstrating its effectiveness in electromagnetic field analysis
problems.
In future work, we aim to realize further acceleration by
examining high-performance computing technology using Intel
AVX instructions.

REFERENCES
Figure 1. An efficient calculation method of preconditioning.

3. NUMERICAL EXPERIMENTS
As a high frequency electromagnetic field problem, let us
consider a wave equation having an electric field derived fromMaxwell’s equation. We constructed a medium-scale problem
with 134,573 degrees of freedom by ADVENTURE_Magnetic [3].
The number of nonzero elements in the coefficient matrix was2,123,849. The number of non-zero elements in the[1] Amane Takei, Shinobu Yoshimura, Hiroshi Kanayama. 2008.
Large-Scale Parallel Finite Element Analyses of HighFrequency Electromagnetic Field in Commuter Trains.
Comput. Modeling Engrg. Sci., 31 (Jul. 2008), pp.13-24.
[2] Meijerink J. A. and van der Vorst H. A. 1977. An Iterative
Solution Method for Linear Systems of Which theCoefficient Matrix is a Symmetric M-Matrix. Mathematics ofComputation, 31, 137 (Jan 1977), pp. 148-162.
[3] ADVENTURE Project, https://adventure.sys.t.u-tokyo.ac.jp/.