Stabilization of the Fast Multipole Method for Low Frequencies Using Multiple-Precision Arithmetic

Barışcan Karaosmanoğlu*1 and Özgür Ergül1

1Dept. Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey

bariscan@metu.edu.tr

Abstract We stabilize a conventional implementation of the fast multipole method (FMM) for low frequencies using multiple-precision arithmetic (MPA). We show that using MPA is a direct remedy for low-frequency breakdowns of the standard diagonalization, which is prone to numerical errors at short distances with respect to wavelength. By increasing the precision, rounding errors are suppressed until a desired level of accuracy is obtained with plane-wave expansions. As opposed to other approaches in the literature, using MPA does not require reimplementations of solvers, and it directly extends the applicability of FMM and similar methods to low-frequency problems, as well as multi-scale problems, that require globally or locally dense discretizations for accurate analysis.

1. Introduction It is well known that, despite their great success at higher frequencies, the fast multipole method (FMM) [1] and its multilevel version, namely, the multilevel fast multipole algorithm (MLFMA) [2], suffer from breakdowns that inhibit their efficient implementations for low-frequency problems [3]. For such a problem involving dense discretization with respect to the operating wavelength, the standard diagonalization based on plane-wave expansions of spherical waves cannot be used for most of the interactions. As these short-distance interactions must be calculated directly and stored in memory, even the complexities of FMM and MLFMA can be higher than desired. In order to handle these low-frequency problems, as well as multi scale problems that involve large objects with locally dense discretizations, it is common to modify FMM and MLFMA such that they become efficient without sacrificing the accuracy. One approach is employing multipoles without converting them into plane waves at short distances [4-9]. Based on the addition theorem, multipole expansion of the Green’s function is rapidly convergent for subwavelength interactions. Another popular approach is changing/deforming the angular integration line and adding evanescent waves into account to translate local radiations along short distances [10-12]. Even though both approaches have been very successful to implement low-frequency and broadband solvers, they require fundamental changes in the existing FMM and MLFMA codes. In fact, most of the low-frequency implementations need to be written from scratch, and their combinations with the conventional forms for broadband solutions require further efforts. From a numerical point of view, low-frequency problems encountered in FMM and MLFMA, and more specifically in the standard diagonalization of the Green’s function are due to rounding errors. Spherical Bessel functions that normally balance spherical Hankel functions (provided that the frequency is not zero [4,9]) are replaced with complex exponentials having nearly isotropic patterns at low frequencies. This leads to adding and subtracting large terms to derive translation operators, which are supposed to convert radiation patterns with tiny fluctuations into incoming fields. Furthermore, these fields are then tested by receiving patterns with almost no resolution, again due to the smooth behavior of complex exponentials with small arguments. Considering all these, it does not seem unusual that the standard diagonalization with single-precision and double-precision arithmetic systems become incapable of providing a desired level of accuracy for subwavelength interactions. Along this direction, we suggest using higher precision arithmetic systems to stabilize the diagonalization and convert standard implementations of FMM and MLFMA into broadband solvers. In fact, as shown in this paper, multiple-precision arithmetic (MPA) [13-15], which allows for different precisions for different computations, provides a straightforward remedy for low-frequency breakdowns. In this work, we present the stabilization of an FMM implementation (based on the standard diagonalization) using MPA. We show that, by increasing the precision, we are able to calculate ‘far-zone’ interactions accurately for both the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE). We also compute the electric field scattered from a small sphere to demonstrate the accuracy of the resulting FMM-MPA implementation.

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MPA is easy to apply to the conventional implementations of FMM and MLFMA, while using the advantages of these low-complexity methods.

2. Accurate Computations of Matrix Elements Using FMM-MPA As a test problem, we consider a perfectly conducting sphere of radius 0.3 m at 4 MHz. Hence, the radius of the sphere corresponds to approximately 1/250 of the wavelength. The surface of the sphere is discretized with triangles, on which Rao-Wilton-Glisson functions are defined as basis and testing functions. We use triangles of different sizes, i.e., 10 cm, 5 cm, and 3 cm, leading to matrix equations involving 510, 1476, and 4080 unknowns, respectively. Table 1. Maximum Relative Errors When FMM (With the Double-Precision Arithmetic and MPA) Is Used to Compute the Matrix Elements of the Sphere Problem Discretized With 510 Unknowns

Table 2. Relative Errors in Overall Matrix-Vector Multiplications When FMM (With the Double-Precision Arithmetic and MPA) Is Used For the Sphere Problem Discretized With 510 Unknowns

Tables 1 and 2 present relative errors obtained while using FMM to compute matrix elements compared to direct calculations when the mesh size is 10 cm. In FMM, cubic boxes that are 1/500 of the wavelength are used, and non-touching boxes are consider ‘far-zone’ such that interactions between these boxes are computed via diagonalization. For both EFIE and MFIE, the truncation number is increased from 10 to 19 in order to improve the accuracy of the interactions. Table 1 lists the maximum error encountered among all interactions. Without MPA (and using the double-precision arithmetic), error values are very high and they increase drastically as the truncation number increases, as a typical behavior of low-frequency breakdowns. Using MPA, however, error values are significantly reduced for MFIE. In the case of EFIE, we observe that errors are lower with MPA, but they are not fully controllable. Our investigations show that this is due to the dyadic Green’s function that requires more harmonics compared to the scalar Green’s function for accurate computations. Hence, we use an alternative implementation of EFIE, where vector and scalar potential parts of EFIE are diagonalized separately. This way, we can also obtain small errors using EFIE (denoted by EFIE-2). In Table 2, we consider the mean errors in ‘far-zone’ interactions, corresponding to overall matrix-vector multiplications (MVMs), which further demonstrate significant improvements via MPA. The truncation numbers used on Tables 1 and 2 are obtained via optimizations. The required precisions, i.e., number of digits, are also obtained in accordance with these optimizations, where different testing-basis scenarios are considered in the diagonalization of the Green’s function. Hence, we are able to control the accuracy for a given box size and error threshold, which leads to a straightforward implementation of a broadband MLFMA using MPA.

3. Accurate Field Computations Using FMM-MPA

Fig. 1 presents the far-zone scattering results obtained for the sphere of radius 0.3 m at 4 MHz, when the problem is formulated with MFIE and discretized with different mesh sizes. It can be observed that, using FMM-MPA,

MFIE Maximum Error EFIE Maximum Error EFIE-2 Maximum Error Truncation Number FMM MPA-FMM FMM MPA-FMM MPA-FMM

10 1.13e+8 1.93e-2 5.26e+6 1.22e-1 6.83e-2 11 9.03e+11 4.12e-3 1.11e+9 1.28e-1 2.40e-2 13 8.28e+17 2.69e-3 1.77e+15 1.33e-1 3.29e-3 14 1.18e+20 2.57e-3 1.15e+19 1.33e-1 1.48e-2 19 3.20e+34 4.04e-4 1.06e+34 1.33e-1 1.57e-3

MFIE MVM Error EFIE MVM Error EFIE-2 MVM Error Truncation Number FMM MPA-FMM FMM MPA-FMM MPA-FMM

10 5.93e+7 1.70e-3 2.76e+6 5.69e-2 1.14e-2 11 1.60e+11 4.16e-4 1.04e+9 5.82e-2 4.89e-3 13 1.75e+17 2.29e-4 5.50e+14 5.90e-2 6.75e-4 14 4.50e+19 2.00e-4 2.32e+18 5.90e-2 2.12e-3 19 6.25e+33 3.14e-5 2.30e+33 5.89e-2 2.13e-4

we are able to obtain accurate electric field values. Specifically, field curves obtained with FMM-MPA perfectly agree with those obtained with MOM for all discretization. The discrepancies between the computational and analytical Mie-series solutions, which decrease as the discretization is refined, seems to be related to the triangular modeling of the surface of the sphere. RCS results obtained with FMM using the double-precision arithmetic are not shown in Fig. 1, since these solutions do not converge iteratively due to significantly inaccurate ‘far-zone’ interactions.

5. Conclusion

We use MPA to stabilize a conventional FMM implementation for low-frequency problems involving dense discretizations with respect to wavelength. We show that, by using MPA, matrix elements and scattered field values can be computed accurately at low frequencies using the standard diagonalization, without resorting to alternative expansion and diagonalization methods. MPA is easy to apply and it can be used to convert existing implementations of FMM and MLFMA to broadband solvers for low-frequency and multi-scale problems.

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Figure 1. Far-zone electric field (V/m) scattered from a perfectly conducting sphere of radius 0.3 m at 4 MHz calculated via MFIE and Mie-series solutions.

6. Acknowledgments

This work was supported by the Scientific and Technical Research Council of Turkey (TUBITAK) under the Research Grants 113E129 and 113E276, and by a BAGEP Grant from Bilim Akademisi (The Science Academy, Turkey).

7. References

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