Error bound of the multilevel adaptive cross approximation (MLACA)

Abstract

An error bound of the multilevel adaptive cross approximation (MLACA 1, which is a multilevel version of the adaptive cross approximation-singular value decomposition (ACA-SVD), is rigorously derived. For compressing an off-diagonal submatrix of the method of moments MAD impedance matrix with a binary tree, the L-level MIACA includes L + 1 steps, and each step includes 2(L) ACA-SVD decompositions. If the relative Frobenius norm error of the ACA-SVD used in the MLACA is smaller than epsilon, the rigorous proof in this communication shows that the relative Frobenius norm error of the L-Ievel MLACA is smaller than (1 + epsilon)(L+1) - 1. In practical applications, the error bound of the MLACA can be approximated as epsilon(L + 1), because epsilon is always << 1. The error upper bound can he used to control the accuracy of the MLACA. To ensure an error of the L-level MLACA smaller than epsilon for different L, the ACA-SVD threshold can be set to (1 + epsilon)1/L+1 - 1, which approximately equals epsilon/(L + 1) for practical applications.