# A Parallel Hierarchical Blocked Adaptive Cross Approximation Algorithm

## TL;DR

This paper introduces a hierarchical blocked adaptive cross approximation algorithm that enhances low-rank matrix decompositions with improved convergence and efficiency, suitable for parallel computing environments.

## Contribution

It proposes a novel hierarchical BACA algorithm that combines adaptive cross approximation with hierarchical merging, improving convergence and computational efficiency over traditional methods.

## Key findings

- Significantly improved convergence over baseline ACA
- Reduced computational complexity compared to full decompositions
- Demonstrated efficiency and parallel scalability in numerical tests

## 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.