Tuning Multigrid Methods with Robust Optimization

TL;DR

This paper introduces optimization algorithms to efficiently tune multigrid methods by solving minimax problems in local Fourier analysis, improving the prediction and performance of PDE solvers.

Contribution

It develops novel optimization techniques to solve minimax problems in local Fourier analysis, enabling efficient parameter tuning for multigrid methods.

Findings

Optimization algorithms effectively solve minimax problems in Fourier analysis.

The methods improve the accuracy of spectral radius and condition number estimates.

Examples demonstrate enhanced algorithm performance with the proposed approach.

Abstract

Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius of a stationary iteration, or the condition number of a preconditioned system, in terms of a symbol representation of the algorithm. In practice, this is a "minimax" problem, minimizing with respect to solver parameters the appropriate measure of work, which involves maximizing over the Fourier frequency. Often, several algorithmic parameters may be determined by local Fourier analysis in order to obtain efficient algorithms. Analytical solutions to minimax problems are rarely possible beyond simple problems; the status quo in local Fourier analysis involves grid sampling, which is prohibitively expensive in high dimensions. In this paper, we propose and explore optimization algorithms to solve these problems efficiently. Several examples, with known and unknown analytical solutions, are presented to show the effectiveness of these approaches.