Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithms

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.

Figures (11)

• Figure 1: A high-level overview of the proposed automated AMG design workflow: given an input problem, the approach uses grammars, evolutionary algorithms, and high-performance solvers to generate flexible AMG cycles. Nodes in the resulting cycles are color-coded to indicate different smoothing operations.
• Figure 1: Evolution of AMG individuals with respect to cost per iteration and convergence rate, progressing from an initial random population to the final generation (clockwise from top left: initial population, generation 1, generation 10, generation 100). The dots represent the individuals, and are colored separately, representing each of the five independent runs of the G3P algorithm.
• Figure 1: Initial conditions of the zrad3D problem. A source is applied to the left side of the block as it is pictured above. As the problem evolves, radiation flows through the block, and the matter and radiation fields come into equilibrium with each other.
• Figure 2: An overview of the software setup for the automated AMG design.
• Figure 2: AMG cycling structure for the selected GP solvers, excluding the depiction of weighting parameters $\omega_i$, $\omega_0$, $\omega$, and $\alpha$.