  Multigrid methods have proven to be an invaluable tool to efficiently solve
large sparse linear systems arising in the discretization of partial
differential equations (PDEs). Algebraic multigrid methods and in particular
adaptive algebraic multigrid approaches have shown that multigrid efficiency
can be obtained without having to resort to properties of the PDE. Yet the
required setup of these methods poses a not negligible overhead cost. Methods
from machine learning have attracted attention to streamline processes based on
statistical models being trained on the available data. Interpreting
algebraically smooth error as an instance of a Gaussian process, we develop a
new, data driven approach to construct adaptive algebraic multigrid methods.
Based on Gaussian a priori distributions, Kriging interpolation minimizes the
mean squared error of the a posteriori distribution, given the data on the
coarse grid. Going one step further, we exploit the quantification of
uncertainty in the Gaussian process model in order to construct efficient
variable splittings. Using a semivariogram fit of a suitable covariance model
we demonstrate that our approach yields efficient methods using a single
algebraically smooth vector.
