Sparse Hierarchical Solvers with Guaranteed Convergence

TL;DR

This paper introduces a new hierarchical solver for sparse linear systems from PDE discretizations that guarantees convergence and maintains linear complexity, improving robustness against high condition numbers.

Contribution

The paper presents a novel hierarchical solver that enhances robustness and convergence guarantees for sparse linear systems while preserving linear computational cost.

Findings

Achieves improved robustness with respect to condition number.

Maintains linear computational complexity and memory footprint.

Provides convergence guarantees for the hierarchical solver.

Abstract

Solving sparse linear systems from discretized PDEs is challenging. Direct solvers have in many cases quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and efficient. Approximate factorization preconditioners, such as incomplete LU factorization, provide cheap approximations to the system matrix. However, even a highly accurate preconditioner may have deteriorating performance when the condition number of the system matrix increases. By increasing the accuracy on low-frequency errors, we propose a novel hierarchical solver with improved robustness with respect to the condition number of the linear system. This solver retains the linear computational cost and memory footprint of the original algorithm.