Distributed-memory Hierarchical Interpolative Factorization

TL;DR

This paper introduces a distributed-memory implementation of the hierarchical interpolative factorization (HIF) for efficiently solving elliptic PDEs, achieving near-linear complexity and demonstrating scalability on large parallel systems.

Contribution

The paper presents DHIF, a parallel, distributed-memory version of HIF that maintains low communication overhead and scalable performance for large-scale elliptic PDE problems.

Findings

Achieves $O(Nrac{ ext{log}N}{P})$ complexity for construction

Achieves $O(rac{N}{P})$ complexity for applying the factorization

Demonstrates efficiency and scalability on up to 8192 processes

Abstract

The hierarchical interpolative factorization (HIF) offers an efficient way for solving or preconditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operators, the HIF achieves quasi-linear complexity for factorizing the discrete positive definite elliptic operator and linear complexity for solving the associated linear system. In this paper, the distributed-memory HIF (DHIF) is introduced as a parallel and distributed-memory implementation of the HIF. The DHIF organizes the processes in a hierarchical structure and keep the communication as local as possible. The computation complexity is $O\left(\frac{N\log N}{P}\right)$ and $O\left(\frac{N}{P}\right)$ for constructing and applying the DHIF, respectively, where $N$ is the size of the problem and $P$ is the number of processes. The communication complexity is $O\left(\sqrt{P}\log^3 P\right)\alpha + O\left(\frac{N^{2/3}}{\sqrt{P}}\right)\beta$ where $\alpha$ is the latency and $\beta$ is the inverse bandwidth. Extensive numerical examples are performed on the NERSC Edison system with up to 8192 processes. The numerical results agree with the complexity analysis and demonstrate the efficiency and scalability of the DHIF.