Evolving Algebraic Multigrid Methods Using Grammar-Guided Genetic Programming

TL;DR

This paper introduces a novel approach to algebraic multigrid methods by using grammar-guided genetic programming to generate flexible cycle structures, leading to improved efficiency and performance over traditional cycles.

Contribution

It presents a new method for generating arbitrary-shaped multigrid cycles using grammar rules and optimizing them with genetic programming, enhancing AMG method flexibility and performance.

Findings

Flexible cycles outperform standard V-, F-, and W-cycles in efficiency.

Optimized cycles improve solver performance for 3D anisotropic problems.

Enhanced AMG preconditioner benefits multiphysics simulations.

Abstract

Multigrid methods despite being known to be asymptotically optimal algorithms, depend on the careful selection of their individual components for efficiency. Also, they are mostly restricted to standard cycle types like V-, F-, and W-cycles. We use grammar rules to generate arbitrary-shaped cycles, wherein the smoothers and their relaxation weights are chosen independently at each step within the cycle. We call this a flexible multigrid cycle. These flexible cycles are used in Algebraic Multigrid (AMG) methods with the help of grammar rules and optimized using genetic programming. The flexible AMG methods are implemented in the software library of hypre, and the programs are optimized separately for two cases: a standalone AMG solver for a 3D anisotropic problem and an AMG preconditioner with conjugate gradient for a multiphysics code. We observe that the optimized flexible cycles provide higher efficiency and better performance than the standard cycle types.