Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms
This work addresses the problem of optimizing multigrid efficiency for solving partial differential equations, offering an automated alternative to deep learning approaches, though it is incremental as it builds on existing algorithmic building blocks.
The paper tackles the challenge of designing efficient algebraic multigrid methods by using evolutionary algorithms to automate the selection of level-specific smoothing sequences and non-recursive cycling patterns, which are intractable to optimize manually. Numerical experiments with the hypre library show that these non-standard cycles can improve multigrid performance as both a solver and a preconditioner.
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.