**Authors:** Chao Li, Raj Mittra

**Source:** FERMAT, Volume 20, Communication 4, Mar-Apr., 2017

**Abstract:** It has been demonstrated in the literature on Characteristic Basis Function Method (CBFM) that higher compression rate can be achieved by using larger blocks while carrying out domain decomposition in the context of CBFM [1]. However, the increased degrees of freedom (DOFs) in large domains make the generation of the Characteristic Basis Functions (CBFs) very time- and memory-consuming for the following reasons: (a) Impedance matrices for each domain need to be calculated; (b) Singular Value Decomposition (SVD) must be carried out to remove the redundancy between the CBFs. To mitigate the above problem, Multilevel CBFM (MLCBFM) has been proposed. Also, similar to MLCBFM, a hybrid approach namely the hybrid CBFM/ACA/UV method has been developed recently to address the same issue [2]. By using the UV technique, the computational time and memory complexity required by the
matrix filling process can be decreased from O(B N2RWG ) to O(B NRWG log NRWG ), where B is the number of blocks and NRWG is the average number of RWGs in each block. Compared to MLCBFM, the hybrid CBFM/ACA/UV has the advantage that it does not require the generation of either the CBFs or the reduced matrix at the lower level (two-level domain decomposition policy is normally adopted for MLCBFM).
In this work, a randomized singular value decomposition (rSVD) approach [3] is introduced in order to further accelerate the generation of the CBFs. It has been demonstrated that rSVD is capable of decomposing a rank-deficient matrix with dimension exceeding 300,000 in less than 10 seconds [4]. Using a random projection method, the decomposition can be implemented in a lower-dimensional space, and therefore the singular vectors (CBFs) can be derived in a highly efficient way.
Illustrative examples will be included in the presentation to demonstrate the efficacy of the proposed approach.

**Index Terms:** MLCBFM, rSVD, CBF Generation

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