Coarse-grid selection using simulated annealing

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 (2007), a constrained combinatorial optimization problem is used to define the bestcoarse 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.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

Multilevel techniques are efficient approaches for solving large linear systems. Algebraic multigrid aims to require less knowledge about the origin of the linear system compared to geometric multigrid. This paper introduces a new coarsening algorithm based on simulated annealing to approximate solutions to the coarse/fine partitioning problem, showing improved results over the previous greedy algorithm.