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

Karaosmanoglu, Bariscan
Ergül, Özgür Salih
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.


Low-frequency multilevel fast multipole algorithm using an approximate diagonalization of the Green's function
Ergül, Özgür Salih (2014-08-23)
We present an approximate diagonalization of the Green's function to implement a stable multilevel fast multipole algorithm (MLFMA) for low-frequency problems. The diagonalization is based on scaled spherical functions, leading to stable computations of translation operators at all distances and for all frequencies. Similar to the conventional diagonalization, shift operators are expressed in terms of complex exponentials, while radiated and incoming fields are expanded in terms of scaled plane waves. Even ...
Error Control of MLFMA within a Multiple-Precision Arithmetic Framework
Kalfa, Mert; ERTÜRK, VAKUR BEHÇET; Ergül, Özgür Salih (2018-07-13)
We present a new error control scheme that provides the truncation numbers as well as the required digits of machine precision for the multilevel fast multipole algorithm (MLFMA). The proposed method is valid for all frequencies, whereas the previous studies on error control are valid only for high-frequency problems. When combined with a multiple-precision arithmetic framework, the proposed method can be used to solve low-frequency problems that would otherwise experience overflow issues. Numerical results...
Efficient and Accurate Electromagnetic Optimizations Based on Approximate Forms of the Multilevel Fast Multipole Algorithm
Onol, Can; Karaosmanoglu, Bariscan; Ergül, Özgür Salih (2016-01-01)
We present electromagnetic optimizations by heuristic algorithms supported by approximate forms of the multilevel fast multipole algorithm (MLFMA). Optimizations of complex structures, such as antennas, are performed by considering each trial as an electromagnetic problem that can be analyzed via MLFMA and its approximate forms. A dynamic accuracy control is utilized in order to increase the efficiency of optimizations. Specifically, in the proposed scheme, the accuracy is used as a parameter of the optimiz...
Rigorous Solutions of Large-Scale Scattering Problems Discretized with Hundreds of Millions of Unknowns
Guerel, L.; Ergül, Özgür Salih (2009-09-18)
We present fast and accurate solutions of large-scale scattering problems using a parallel implementation of the multilevel fast multipole algorithm (MLFMA). By employing a hierarchical partitioning strategy, MLFMA can be parallelized efficiently on distributed-memory architectures. This way, it becomes possible to solve very large problems discretized with hundreds of millions of unknowns. Effectiveness of the developed simulation environment is demonstrated on various scattering problems involving canonic...
Two-Step Lagrange Interpolation Method for the Multilevel Fast Multipole Algorithm
Ergül, Özgür Salih; Gurel, L. (Institute of Electrical and Electronics Engineers (IEEE), 2009)
We present a two-step Lagrange interpolation method for the efficient solution of large-scale electromagnetics problems with the multilevel fast multipole algorithm (MLFMA). Local interpolations are required during aggregation and disaggregation stages of MLFMA in order to match the different sampling rates for the radiated and incoming fields in consecutive levels. The conventional one-step method is decomposed into two one-dimensional interpolations, applied successively. As it provides a significant acce...
Citation Formats
B. Karaosmanoglu and Ö. S. Ergül, “Stabilization of the Fast Multipole Method for Low Frequencies Using Multiple-Precision Arithmetic,” 2014, Accessed: 00, 2020. [Online]. Available: