We present an iterative inner-outer scheme for the efficient solution of large-scale electromagnetics problems involving perfectly-conducting objects formulated with surface integral equations. Problems are solved by employing the multilevel fast multipole algorithm (MLFMA) on parallel computer systems. In order to construct a robust preconditioner, we develop an approximate MLFMA (AMLFMA) by systematically increasing the efficiency of the ordinary MLFMA. Using a flexible outer solver, iterative MLFMA solutions are accelerated via an inner iterative solver, employing AMLFMA and serving as a preconditioner to the outer solver. The resulting implementation is tested on various electromagnetics problems involving both open and closed conductors. We show that the processing time decreases significantly using the proposed method, compared to the solutions obtained with conventional preconditioners in the literature.

** MLFMA Truncation Numbers**
For a cluster of using the excess bandwidth formula [10] for the worst-case scenario and the one-box-buffer scheme [11], i.e.,

T_{l}≈ 1.73 *ka*_{l}+ 2.16(d_{o})^{2/3}(*ka*_{l})^{1/3},

where d

_{o} is the desired digits of accuracy. We note that

T_{min}=T_{1}=⌊2.72+2.51d_{o}^{2/3}⌋+1,

for the lowest level (l = 1) with a = 0.25λ where ⌊.⌋ represents the floor operation. Table 1 presents a list of truncation numbers for different box sizes and when the number of accurate digits varies from d_{o} = 1 to d_{o} = 5.
T_{l}≈ 1.73 *ka*_{l}+ 2.16(d_{o})^{2/3}(*ka*_{l)}^{1/3},

Truncation numbers (T_{l}) determined by the excess bandwidth formula for the worst-case scenario and the one-box-buffer scheme.
**Strategies for Building a Less-Accurate MLFMA**
We propose AMLFMA, which is based on systematically reducing the truncation numbers, i.e.,

T_{l}^{r} = T_{min} + α_{f} (T_{l} − T_{min}),

where T

_{min} is the minimum truncation number defined in (4) and T

_{l} is the ordinary truncation number for level

l. α

_{f} represents an approximation factor in the range from 0.0 to 1.0. As α

_{f} is increased from 0.0 to 1.0, AMLFMA becomes more accurate but less efficient, and it corresponds to the ordinary MLFMA when α

_{f} = 1.0.

Errors in the matrix-vector multiplications performed by
AMLFMA with α

_{f} in the 0.0−0.8 range and by IMLFMA (omitting the highest level) for a sphere problem discretized with 132,003 unknowns. The reference data is obtained by using an ordinary MLFMA with three digits of accuracy.

Errors in the matrix-vector multiplications performed by AMLFMA with α_{f} in the 0.0−0.8 range and by IMLFMA (omitting the highest level) for a patch problem discretized with 137,792 unknowns. The reference data is obtained by using an ordinary MLFMA with three digits of accuracy.

Processing time (seconds) for a single matrix-vector
multiplication.

**Iterative Preconditioning Based on AMLFMA**

Metallic objects modeled with open surfaces.

Electromagnetics problems involving open metallic objects.

Processing time (seconds) and the number of iterations for the solution of electromagnetics problems involving open metallic objects.

Metallic objects modeled with closed surfaces.

Electromagnetics problems involving closed metallic objects.

Processing time (seconds) and the number of iterations for the solution of electromagnetics problems involving closed metallic objects.

**For more information: **
- Please see the full article published in the
*Progress In Electromagnetics Research* in 2010.

**How to cite this paper:**

- Ö. Ergül, T. Malas and L. Gürel "Solutions of large-scale electromagnetics problems using an iterative inner-outer scheme with ordinary and approximate multilevel fast multipole algorithms,"
*Prog. Electromagn. Res., PIER 85, * vol. 106, pp. 203-223, 2010.