  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.
