一种并行分层分块自适应交叉逼近算法
A Parallel Hierarchical Blocked Adaptive Cross Approximation Algorithm
摘要：
本文提出了一种分层低秩分解算法，假设任何矩阵元素都可以在$O(1)$时间内计算。该算法采用分块自适应交叉近似（BACA）算法计算子矩阵的秩揭示分解，然后通过截断奇异值分解（H-BACA）进行分层合并操作。该算法显着提高了基线ACA算法的收敛性，并实现了降低计算复杂性相比，如秩揭示QR分解的充分分解。数值结果表明了该算法的有效性、准确性和并行效率。
英文摘要：
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.
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