Abstract

This paper unifies multigrid and multilevel (hierarchical) preconditioners, two widely-used approaches for solving computational photography and other computer graphics simulation problems. It provides detailed experimental comparisons of these techniques and their variants, including an analysis of relative computational costs and how these impact practical algorithm performance. We derive both theoretical convergence rates based on the condition numbers of the systems and their preconditioners, and empirical convergence rates drawn from real-world problems. We also develop new techniques for sparsifying higher connectivity problems, and compare our techniques to existing and newly developed variants such as algebraic and combinatorial multigrid. Our experimental results demonstrate that, except for highly irregular problems, adaptive hierarchical basis function preconditioners generally outperform alternative multigrid techniques, especially when computational complexity is taken into account.