  Uni- and bivariate data smoothing with spline functions is a well established
method in nonparametric regression analysis. The extension to multivariate data
is straightforward, but suffers from exponentially increasing memory and
computational complexity. Therefore, we consider a matrix-free implementation
of a geometric multigrid preconditioned conjugate gradient method for the
regularized least squares problem resulting from tensor product B-spline
smoothing with multivariate and scattered data. The algorithm requires a
moderate amount of memory and is therefore applicable also for high-dimensional
data. Moreover, for arbitrary but fixed dimension, we achieve grid independent
convergence which is fundamental to achieve algorithmic scalability.
