Evolving Generalizable Multigrid-Based Helmholtz Preconditioners with Grammar-Guided Genetic Programming

TL;DR

This paper presents a novel grammar-guided genetic programming approach to evolve multigrid preconditioners for Helmholtz equations, achieving superior performance over traditional methods on large-scale, ill-conditioned problems.

Contribution

It introduces a grammar-based genetic programming framework for designing multigrid preconditioners that generalize across problem difficulties and outperform existing methods.

Findings

Evolved preconditioners outperform human-designed methods.

Preconditioners effectively handle large, ill-conditioned Helmholtz problems.

Method scales to systems with over a million unknowns.

Abstract

Solving the indefinite Helmholtz equation is not only crucial for the understanding of many physical phenomena but also represents an outstandingly-difficult benchmark problem for the successful application of numerical methods. Here we introduce a new approach for evolving efficient preconditioned iterative solvers for Helmholtz problems with multi-objective grammar-guided genetic programming. Our approach is based on a novel context-free grammar, which enables the construction of multigrid preconditioners that employ a tailored sequence of operations on each discretization level. To find solvers that generalize well over the given domain, we propose a custom method of successive problem difficulty adaption, in which we evaluate a preconditioner's efficiency on increasingly ill-conditioned problem instances. We demonstrate our approach's effectiveness by evolving multigrid-based preconditioners for a two-dimensional indefinite Helmholtz problem that outperform several human-designed methods for different wavenumbers up to systems of linear equations with more than a million unknowns.