Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms

TL;DR

This paper introduces an evolutionary algorithm-based approach to automatically design efficient algebraic multigrid methods with flexible cycling patterns, improving solver performance for PDEs.

Contribution

It presents a novel method using genetic programming guided by grammars to generate non-standard multigrid cycles, enhancing automation and performance.

Findings

GP-generated cycles improve multigrid efficiency

Enhanced performance as solver and preconditioner

Demonstrated on hypre library benchmarks

Abstract

Although multigrid is asymptotically optimal for solving many important partial differential equations, its efficiency relies heavily on the careful selection of the individual algorithmic components. In contrast to recent approaches that can optimize certain multigrid components using deep learning techniques, we adopt a complementary strategy, employing evolutionary algorithms to construct efficient multigrid cycles from proven algorithmic building blocks. Here, we will present its application to generate efficient algebraic multigrid methods with so-called \emph{flexible cycling}, that is, level-specific smoothing sequences and non-recursive cycling patterns. The search space with such non-standard cycles is intractable to navigate manually, and is generated using genetic programming (GP) guided by context-free grammars. Numerical experiments with the linear algebra library, \emph{hypre}, demonstrate the potential of these non-standard GP cycles to improve multigrid performance both as a solver and a preconditioner.