## Abstract

A physics-based rank-revealing multilevel algorithm to more efficiently compute low-rank (LR) approximations of the method of moments matrix blocks ${\mathbf{Z}}^{\textrm {os}}$ is presented. Using surface subsets of volumetric nonuniform spherical grids (proxy grids), an LR approximation ${\mathbf{Z}}^{\textrm {os}}\approx {\mathbf{AB}}^{\dagger }$ is obtained in two stages: 1) an upward pass and a downward pass on a multilevel tree, via truncated singular value decomposition-based analyses of interactions between source subclusters and their attendant proxy grids, rapidly estimates the rank and yields a domain of ${\mathbf{Z}}^{\textrm {os}}$ , in the form of a matrix ${\mathbf{B}}$ with orthonormal columns and 2) a fast matrix-matrix multiplication step yields ${\mathbf{A}}$. The algorithm reduces the ${\mathcal{ O}}(N^{3})$ computational costs of revealing the rank to ${\mathcal{ O}}(N^{2})$ or ${\mathcal{ O}}(N^{3/2})$ operations and that of finding an LR approximation to ${\mathcal{ O}}(N^{2})$ or ${\mathcal{ O}}(N^{3/2}\log N)$ operations by adopting enclosing or belt-like proxy grids for electromagnetically large basis/testing function distributions that are either densely packed in a volume or quasi-planar, respectively.

Original language | English |
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Article number | 8408813 |

Pages (from-to) | 5359-5369 |

Number of pages | 11 |

Journal | IEEE Transactions on Antennas and Propagation |

Volume | 66 |

Issue number | 10 |

DOIs | |

State | Published - 1 Oct 2018 |

## Keywords

- Algorithms
- fast solvers
- integral equations
- moment methods