A data driven heuristic for rapid convergence of Scheduled Relaxation Jacobi schemes

TL;DR

This paper introduces a data-driven heuristic for selecting relaxation schemes in the SRJ method, enabling rapid convergence of elliptic PDE solvers across various discretizations by training an algorithm on convergence data.

Contribution

It develops a family of SRJ schemes applicable to different discretizations and trains an algorithm to automatically select the best scheme during the solve process.

Findings

Heuristic provides good convergence across diverse problems.

Automatic scheme selection accelerates elliptic PDE solutions.

Method is effective for 1D Poisson equation and potentially beyond.

Abstract

The Scheduled Relaxation Jacobi (SRJ) method is a viable candidate as a high performance linear solver for elliptic partial differential equations (PDEs). The method greatly improves the convergence of the standard Jacobi iteration by applying a sequence of $M$ well-chosen overrelaxation and underrelaxation factors in each cycle of the algorithm to effectively attenuate the solution error. In previous work, optimal SRJ schemes (sets of relaxation factors) have been derived to accelerate convergence for specific discretizations of elliptic PDEs. In this work, we develop a family of SRJ schemes which can be applied to solve elliptic PDEs regardless of the specific discretization employed. To achieve favorable convergence, we train an algorithm to select which scheme in this family to apply at each cycle of the linear solve process, based on convergence data collected from applying these schemes to the one-dimensional Poisson equation. The automatic selection heuristic that is developed based on this limited data is found to provide good convergence for a wide range of problems.