Optimizing Geometric Multigrid Methods with Evolutionary Computation

TL;DR

This paper introduces a novel method that uses evolutionary computation to optimize geometric multigrid solvers, improving their efficiency for solving PDE discretizations by automatically generating and selecting effective solver configurations.

Contribution

The paper presents a new approach combining evolutionary algorithms, grammar-based solver representation, and model-based evaluation to optimize multigrid methods for PDEs, outperforming standard cycles.

Findings

Optimized multigrid solvers outperform V- and W-cycles in tests.

Automated solver generation is feasible with the ExaStencils framework.

Multi-objective optimization improves convergence and performance.

Abstract

For many linear and nonlinear systems that arise from the discretization of partial differential equations the construction of an efficient multigrid solver is a challenging task. Here we present a novel approach for the optimization of geometric multigrid methods that is based on evolutionary computation, a generic program optimization technique inspired by the principle of natural evolution. A multigrid solver is represented as a tree of mathematical expressions which we generate based on a tailored grammar. The quality of each solver is evaluated in terms of convergence and compute performance using automated local Fourier analysis (LFA) and roofline performance modeling, respectively. Based on these objectives a multi-objective optimization is performed using strongly typed genetic programming with a non-dominated sorting based selection. To evaluate the model-based prediction and to target concrete applications, scalable implementations of an evolved solver can be automatically generated with the ExaStencils framework. We demonstrate our approach by constructing multigrid solvers for the steady-state heat equation with constant and variable coefficients that consistently perform better than common V- and W-cycles.