Compressed Fast Multipole Representations for Homogeneous 3-D Kernels

Authors

  • R. J. Adams Electrical & Computer Engineering, University of Kentucky, Lexington, KY, USA
  • J. C. Young Electrical & Computer Engineering, University of Kentucky, Lexington, KY, USA https://orcid.org/0000-0002-5559-5764
  • S. D. Gedney Electrical Engineering, University of Colorado Denver, Denver, CO, USA

DOI:

https://doi.org/10.13052/2024.ACES.J.390201

Keywords:

fast multipole method, integral equation

Abstract

For homogeneous kernels, the memory requirements associated with H2 representations of integral equation matrices can be reduced by incorporating translational invariance. Starting with a non-translationally invariant H2 representation, this can be accomplished using a left/right iterative algorithm. In this paper, it is shown that a similar algorithm can also be used to compress an existing fast multipole method (FMM). It is observed that the iterative compression converges faster when used to compress an FMM than when it is applied to an H2 representation. Resulting savings in floating-point operations are indicated, and extensions of the reported method are discussed.

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Author Biographies

R. J. Adams, Electrical & Computer Engineering, University of Kentucky, Lexington, KY, USA

Robert J. Adams received the B.S. degree from Michigan Technological University, Houghton, MI, USA, in 1993, and the M.S. and Ph.D. degrees in electrical engineering from Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, VA, USA, in 1995 and 1998, respectively.

From 1999 to 2000, he was a Research Assistant Professor with Virginia Tech. Dr. Adams joined the University of Kentucky in 2001, where he is currently a Professor with the Department of Electrical and Computer Engineering.

Dr. Adams has made novel contributions to mesh and frequency stable integral equation formulations of electromagnetic problems, constraint-based methods for high-order MOM discretizations, spectral splitting methods for implementing shadowing effects in integral equations at high frequencies, and sparse direct solution methods for low-to-moderate frequency electromagnetics applications. Dr. Adams’ group developed the first O(N) sparse direct solver for 3-D H2

and FMM representations of static and low-frequency EM problems. Dr. Adams is a senior member of the IEEE.

J. C. Young, Electrical & Computer Engineering, University of Kentucky, Lexington, KY, USA

John C. Young received the B.E.E. degree in electrical engineering from Auburn University in 1997, the M.S. degree in electrical engineering from Clemson University in 2000, and the Ph.D. degree in electrical engineering also from Clemson University in 2002.

From January 2003 to April 2003, he served as a post-doctoral researcher at Clemson University, and from 2003 to 2005, he served as a post-doctoral researcher at Tokyo Institute of Technology, Tokyo, Japan. From 2005 to 2008, he worked at Japan Radio Co. Since 2008, he has been with the Department of Electrical and Computer Engineering at the University of Kentucky, Lexington, KY, where he is currently an associate professor.

Dr. Young’s research interests include integral equation methods, finite element methods, electromagnetic theory, waveguides, array antennas, and magnetic signature modeling of hysteretic materials. He is a member of IEEE, the Applied Electromagnetics Society (ACES), and URSI Commission B. He currently serves as an Associate Editor for the IEEE Transactions on Antennas and Propagation and on the Education Committee of the Antennas and Propagation Society. He also served (2020-2023) on the Board of Directors of ACES where he is currently Secretary.

S. D. Gedney, Electrical Engineering, University of Colorado Denver, Denver, CO, USA

Stephen D. Gedney received the B.Eng.-Honors degree from McGill University, Montreal, P.Q., in 1985, and the M.S. and Ph.D. degrees in Electrical Engineering from the University of Illinois, Urbana-Champaign, IL, in 1987 and 1991, respectively.

He is currently the Don and Karen White Professor of the Department of Electrical Engineering at the University of Colorado Denver (CUD). Previously he was a Professor of Electrical Engineering at the University of Kentucky from 1991-2014. He worked for the U.S. Army Corps of Engineers, Champaign, IL (1985-1987). He was a visiting Professor at the Jet Propulsion Laboratory, (1992-1993), HRL laboratories (1996-1997) and Alpha Omega Electromagnetics (2004-2005). He received the Tau Beta Pi Outstanding Teacher Award in 1995 and 2013. From 2002-2014, he was the Reese Terry Professor of Electrical and Computer Engineering at the University of Kentucky. He was titled as a Distinguished Professor of the University of Colorado in 2022. He is a past Associate Editor of the IEEE Transactions on Antennas and Propagation (1997-2004), a member of the IEEE Antennas and Propagation Society ADCOM (2000-2003), and served as the chair of the IEEE Antennas and Propagation Society Membership Committee (1995-2002). He is a Fellow of the IEEE and member of Tau Beta Pi.

References

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., vol. 73, no. 2, pp. 325-348, 1987.

W. Fong and E. Darve, “The black-box fast multipole method,” J. Comput. Phys., vol. 228, no. 23, pp. 8712-8725, Dec. 2009.

W. Hackbusch, Hierarchical Matrices: Algorithms and Analysis (Springer Series in Computational Mathematics). Berlin: Springer-Verlag, p. 511, 2015.

X. Xu and R. J. Adams, “Sparse matrix factorization using overlapped localizing LOGOS modes on a shifted grid,” IEEE T. Antenn. Propag., vol. 60, no. 3, pp. 1414-1424, 2012.

R. J. Adams, J. C. Young, and S. D. Gedney, “Compressing H2 matrices for translationally invariant kernels,” Applied Computational Electromagnetics Society (ACES) Journal, vol. 35, no. 11, pp. 1392-1393, Nov. 2020.

R. J. Adams, J. C. Young, and S. D. Gedney,“Efficiency improvements in compressing H2 matrices for translationally invariant kernels,” presented at the 2022 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Denver, CO, July 10-15, 2022.

R. J. Adams, J. C. Young, and S. D. Gedney, “Compressing a fast multipole method representation of an integral equation matrix,” presented at the 2023 International Applied Computational Electromagnetics Society Symposium, Monterey/Seaside, CA, 2023.

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Published

2024-02-29

How to Cite

[1]
R. J. Adams, J. C. Young, and S. D. Gedney, “Compressed Fast Multipole Representations for Homogeneous 3-D Kernels”, ACES Journal, vol. 39, no. 02, pp. 91–96, Feb. 2024.

Issue

2024: Vol 39 Iss 2

Section

ACES-Monterey 2023

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