A multigrid preconditioner for tensor product spline smoothing

Abstract Penalized spline smoothing is a well-established, nonparametric regression method that efficient for one and two covariates. Its extension to more than covariates straightforward but suffers from exponentially increasing memory demands computational complexity, which brings the its numerical limit. with multiple requires solving large-scale, regularized least-squares problem where occurring matrices do not fit into storage of common computer systems. To overcome this restriction, we introduce matrix-free implementation conjugate gradient method. We further present simple diagonal as well advanced geometric multigrid preconditioner significantly speed up convergence All algorithms require negligible amount therefore allow penalized Moreover, arbitrary fixed covariate dimension, show grid independent fundamental achieve algorithmic scalability.