Error bound of the multilevel adaptive cross approximation (MLACA)
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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.
CitationChen, X., Gu, C., Heldring, A., Li, Z., Cao, QS. Error bound of the multilevel adaptive cross approximation (MLACA). "IEEE transactions on antennas and propagation", Gener 2016, vol. 64, núm. 1, p. 374-378.