One-Stage O(N \log N) Algorithm for Generating Nested Rank-Minimized Representation of Electrically Large Volume Integral Equations

In this paper, we develop a new one-stage <inline-formula><tex-math notation="LaTeX"> O(N \log N)</tex-math></inline-formula> algorithm to generate a rank-minimized <inline-formula><tex-math notation="LaTeX">\mathcal {H}^{2}</tex-math><...

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Published inIEEE journal on multiscale and multiphysics computational techniques Vol. 10; pp. 169 - 178
Main Authors Wang, Yifan, Jiao, Dan
Format Journal Article
LanguageEnglish
Published IEEE 2025
Subjects
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Accuracy
Approximation algorithms
Clustering algorithms
Complexity theory
Costs
electrically large analysis
fast solvers
Indexes
Integral equations
low-rank representation
matrix compression
nested representation
rank-minimized compression
Scattering
Three-dimensional displays
Training
Volume integral equations
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ISSN2379-8815
2379-8815
DOI10.1109/JMMCT.2025.3544143

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Abstract In this paper, we develop a new one-stage <inline-formula><tex-math notation="LaTeX"> O(N \log N)</tex-math></inline-formula> algorithm to generate a rank-minimized <inline-formula><tex-math notation="LaTeX">\mathcal {H}^{2}</tex-math></inline-formula>-representation of electrically large volume integral equations (VIEs), which significantly reduces the CPU run time of state-of-the-art algorithms for completing the same task. Unlike existing two-stage algorithms, this new algorithm requires only one stage to build nested cluster bases. The cluster basis is obtained directly from the interaction between a cluster and its admissible clusters composed of real or auxiliary ones that cover all interaction directions. Furthermore, the row and column pivots of the resultant low-rank representation are chosen from the source and observer points in an analytical way without the need for numerically finding them. This further speeds up the computation. Numerical experiments on a suite of electrically large 3D scattering problems have demonstrated the efficiency and accuracy of the proposed new algorithm.
AbstractList In this paper, we develop a new one-stage <inline-formula><tex-math notation="LaTeX"> O(N \log N)</tex-math></inline-formula> algorithm to generate a rank-minimized <inline-formula><tex-math notation="LaTeX">\mathcal {H}^{2}</tex-math></inline-formula>-representation of electrically large volume integral equations (VIEs), which significantly reduces the CPU run time of state-of-the-art algorithms for completing the same task. Unlike existing two-stage algorithms, this new algorithm requires only one stage to build nested cluster bases. The cluster basis is obtained directly from the interaction between a cluster and its admissible clusters composed of real or auxiliary ones that cover all interaction directions. Furthermore, the row and column pivots of the resultant low-rank representation are chosen from the source and observer points in an analytical way without the need for numerically finding them. This further speeds up the computation. Numerical experiments on a suite of electrically large 3D scattering problems have demonstrated the efficiency and accuracy of the proposed new algorithm.
Author Jiao, Dan
Wang, Yifan
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Snippet In this paper, we develop a new one-stage <inline-formula><tex-math notation="LaTeX"> O(N \log N)</tex-math></inline-formula> algorithm to generate a...
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Accuracy
Approximation algorithms
Clustering algorithms
Complexity theory
Costs
electrically large analysis
fast solvers
Indexes
Integral equations
low-rank representation
matrix compression
nested representation
rank-minimized compression
Scattering
Three-dimensional displays
Training
Volume integral equations
Title One-Stage O(N \log N) Algorithm for Generating Nested Rank-Minimized Representation of Electrically Large Volume Integral Equations
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