A Parallel Hierarchical Blocked Adaptive Cross Approximation Algorithm

Abstract: This paper presents a hierarchical low-rank decomposition algorithm assuming any matrix element can be computed in $O(1)$ time. The proposed algorithm computes rank-revealing decompositions of sub-matrices with a blocked adaptive cross approximation (BACA) algorithm, followed by a hierarchical merge operation via truncated singular value decompositions (H-BACA). The proposed algorithm significantly improves the convergence of the baseline ACA algorithm and achieves reduced computational complexity compared to the full decompositions such as rank-revealing QR decompositions. Numerical results demonstrate the efficiency, accuracy and parallel efficiency of the proposed algorithm.