Coarse-Grid Selection Using Simulated Annealing

TL;DR

This paper introduces a simulated annealing-based algorithm for selecting coarse grids in algebraic multigrid methods, improving upon previous greedy algorithms and effectively utilizing grid structure information.

Contribution

A novel coarsening algorithm using simulated annealing is proposed, offering better solutions for coarse grid selection in algebraic multigrid methods.

Findings

Improved coarse grid selection results over greedy algorithms.

Effective exploitation of grid structure information.

Demonstrated applicability on structured and unstructured meshes.

Abstract

Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying problem and its discretization, algebraic multigrid aims to be less intrusive, requiring less knowledge about the origin of the linear system. A key step in algebraic multigrid is the choice of the coarse/fine partitioning, aiming to balance the convergence of the iteration with its cost. In work by MacLachlan and Saad, a constrained combinatorial optimization problem is used to define the ``best'' coarse grid within the setting of a two-level reduction-based algebraic multigrid method and is shown to be NP-complete. Here, we develop a new coarsening algorithm based on simulated annealing to approximate solutions to this problem, which yields improved results over the greedy algorithm developed previously. We present numerical results for test problems on both structured and unstructured meshes, demonstrating the ability to exploit knowledge about the underlying grid structure if it is available.