Nested Pseudo Skeleton Approximation Algorithm for Generating ^-Matrix Representations of Electrically Large Surface Integral Equations

• Published in: IEEE journal on multiscale and multiphysics computational techniques Vol. 9; pp. 393 - 402
• Main Authors: Yang, Chang, Jiao, Dan
• Format: Journal Article
• Language: English
• Published: IEEE 2024
• Subjects: <named-content xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" content-type="math" xlink:type="simple"> <inline-formula> <tex-math notation="LaTeX"> {\mathcal H}</tex-math> </inline-formula> </named-content>-matrix; <named-content xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" content-type="math" xlink:type="simple"> <inline-formula> <tex-math notation="LaTeX"> {\mathcal H}^{2}</tex-math> </inline-formula> </named-content>-matrix; Accuracy; Approximation algorithms; Closed-form solutions; Clustering algorithms; Couplings; electric field integral equations; electrically large analysis; Integral equations; Matrix decomposition; Method of moments; nested representation; Observers; pseudo-skeleton approximation; Surface integral equations; Time complexity
• ISSN: 2379-8815; 2379-8815
• DOI: 10.1109/JMMCT.2024.3487779

In this paper, we develop a kernel-independent and purely algebraic method, Nested Pseudo-Skeleton Approximation (NPSA) algorithm, to generate a low-rank <inline-formula><tex-math notation="LaTeX">{\mathcal H}^{2}</tex-math></inline-formula>-matrix representation of electrically large surface integral equations (SIEs). The algorithm only uses <inline-formula><tex-math notation="LaTeX">O(NlogN)</tex-math></inline-formula> entries of the original dense SIE matrix of size <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> to generate the <inline-formula><tex-math notation="LaTeX">{\mathcal H}^{2}</tex-math></inline-formula>-representation. It also provides a closed-form expression of the cluster bases and coupling matrices with respect to original matrix entries. The resultant <inline-formula><tex-math notation="LaTeX">{\mathcal H}^{2}</tex-math></inline-formula>-matrix is then directly solved for electrically large scattering analysis. Numerical experiments have demonstrated the accuracy and efficiency of the proposed algorithm. In addition to surface integral equations, the proposed algorithms can also be applied to solving other electrically large integral equations.