  Patch-based relaxation refers to a family of methods for solving linear
systems which partitions the matrix into smaller pieces often corresponding to
groups of adjacent degrees of freedom residing within patches of the
computational domain. The two most common families of patch-based methods are
block-Jacobi and Schwarz methods, where the former typically corresponds to
non-overlapping domains and the later implies some overlap. We focus on cases
where each patch consists of the degrees of freedom within a finite element
method mesh cell. Patch methods often capture complex local physics much more
effectively than simpler point-smoothers such as Jacobi; however, forming,
inverting, and applying each patch can be prohibitively expensive in terms of
both storage and computation time. To this end, we propose several approaches
for performing analysis on these patches and constructing a reduced
representation. The compression techniques rely on either matrix norm
comparisons or unsupervised learning via a clustering approach. We illustrate
how it is frequently possible to retain/factor less than 5% of all patches and
still develop a method that converges with the same number of iterations or
slightly more than when all patches are stored/factored.
