A Multigrid Preconditioner for Tensor Product Spline Smoothing
Abstract
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.
Cite
@article{arxiv.1901.00654,
title = {A Multigrid Preconditioner for Tensor Product Spline Smoothing},
author = {Martin Siebenborn and Julian Wagner},
journal= {arXiv preprint arXiv:1901.00654},
year = {2024}
}