Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

TL;DR

This paper introduces a scalable preconditioner for large 3D elliptic PDE systems using hierarchical low-rank approximations and cyclic reduction, demonstrating efficiency and robustness in distributed computing environments.

Contribution

It develops a novel hierarchical low-rank cyclic reduction preconditioner with log-linear complexity, scalable performance, and tunable parameters for optimized memory and computational efficiency.

Findings

Achieves good scalability in distributed environments.

Handles various types of elliptic PDE systems.

Allows control over iteration count through approximation accuracy.

Abstract

We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner.