Browsing by Subject "Multiple-precision arithmetic"

Now showing 1 - 3 of 3

Open Access
Error analysis of MLFMA with closed-form expressions
(IEEE, 2021-04-06) Kalfa, Mert; Ertürk, Vakur B.; Ergül, Özgür
The current state-of-the-art error control of the multilevel fast multipole algorithm (MLFMA) is valid for any given error threshold at any frequency, but it requires a multiple-precision arithmetic framework to be implemented. In this work, we use asymptotic approximations and curve-fitting techniques to derive accurate closed-form expressions for the error control of MLFMA that can be implemented in common fixed-precision computers. Moreover, using the proposed closed-form expressions in conjunction with the state-of-the-art scheme, we report novel design curves for MLFMA that can be used to determine achievable error limits, as well as the minimum box sizes that can be solved with a given desired error threshold for a wide range of machine precision levels.
Open Access
Error control in MLFMA with multiple-precision arithmetic
(Institution of Engineering and Technology, 2018-04) Kalfa, Mert; Ergül, Ö.; Ertürk, Vakur B.
We present a new error control method that provides the truncation numbers as well as the required digits of machine precision for the translation operator of 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 (i.e., electrically large translation distances). When combined with a multiple-precision implementation of MLFMA, the proposed method can be used to solve low-frequency problems that are problematic with a fixed-precision implementation. Numerical results in the form of optimal truncation numbers and machine precisions for a variety of box sizes and desired relative error thresholds are presented and compared with the methods or numerical surveys available in the literature.
Open Access
Multiple-precision MLFMA for efficient and accurate solutions of broadband electromagnetic problems
(2020-08) Kalfa, Mert
The multilevel fast multipole algorithm (MLFMA) is a popular full-wave electromagnetic solver that enables the solution of electrically large problems with an extremely large number of unknowns. As with all computational electromagnetics solvers, active research is ongoing to extend the limitations of MLFMA for larger problems with ﬁner geometrical details. For electrically small structures MLFMA suﬀers from the low-frequency breakdown, while more eﬃcient schemes are required for electrically larger problems. We propose and demonstrate an elegant solution to the aforementioned problems by introducing a multiple-precision arithmetic (MPA) framework to the inherent hierarchical tree structure of MLFMA, dubbed the multiple-precision multilevel fast multipole algorithm (MP-MLFMA). With the introduction of MPMLFMA we show that a distinct machine precision can be assigned to each level of the tree structure of MLFMA, which enables accurate and eﬃcient solutions of problems with deep tree-structures over arbitrarily large frequency bandwidths. To determine the required machine precisions for a given tree structure, as well as the number of harmonics required for an accurate error control of the translation operator of MP-MLFMA, we introduce and validate a novel error control scheme with accurate design curves that is valid at all frequencies, for the ﬁrst time in the literature. Combined with the proposed error control scheme, we present the capabilities of MP-MLFMA over a wide range of broadband and deep tree-structure scattering problems. We also illustrate the true potential eﬃciency of MP-MLFMA, with a simple MPA framework implementation on a single-precision processor. With the hardware-deﬁned implementation, we show the super-linear speed-up potential of MP-MLFMA for low-precisions.