Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms

2026-03-18

AI Summary

This work proposes a novel approach to automating the design of Algebraic Multigrid (AMG) methods by introducing context-free grammar-based genetic programming. Traditional AMG performance heavily relies on manually crafted cycle structures and smoothing strategies, which struggle to efficiently explore complex, nonstandard configuration spaces. The proposed method overcomes these limitations by automatically generating AMG algorithms with flexible hierarchical structures—including level-specific smoothing sequences and nonrecursive cycles—thereby transcending the constraints of human-designed search spaces. Implemented and evaluated within the hypre library, the automatically discovered nonstandard cycles consistently outperform classical schemes both as standalone solvers and as preconditioners. These results demonstrate the effectiveness and superiority of grammar-guided evolutionary strategies for the automated design of high-performance AMG algorithms.

Technology Category

• Multiagent Systems: Distributed Problem Solving
• Machine Learning: Evolutionary Learning
• Search and Optimization: Evolutionary Computation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• algebraic multigrid
• flexible cycling
• evolutionary algorithms
• grammar-based design
• non-standard cycles

Innovation

• evolutionary algorithms
• genetic programming
• algebraic multigrid
• flexible cycling
• grammar-based design