A Fast Geometric Multigrid Method for Curved Surfaces

TL;DR

This paper presents a geometric multigrid method tailored for solving linear systems on curved surfaces, leveraging point clouds for efficient hierarchy construction and extending applicability to various surface representations.

Contribution

Introduces Gravo MG, a multigrid solver that uses point clouds for hierarchy levels, enabling fast setup and applicability to diverse surface types.

Findings

Fast hierarchy construction using point clouds

Sparse prolongation matrices with three entries per row

Fast convergence of the multigrid solver

Abstract

We introduce a geometric multigrid method for solving linear systems arising from variational problems on surfaces in geometry processing, Gravo MG. Our scheme uses point clouds as a reduced representation of the levels of the multigrid hierarchy to achieve a fast hierarchy construction and to extend the applicability of the method from triangle meshes to other surface representations like point clouds, nonmanifold meshes, and polygonal meshes. To build the prolongation operators, we associate each point of the hierarchy to a triangle constructed from points in the next coarser level. We obtain well-shaped candidate triangles by computing graph Voronoi diagrams centered around the coarse points and determining neighboring Voronoi cells. Our selection of triangles ensures that the connections of each point to points at adjacent coarser and finer levels are balanced in the tangential directions. As a result, we obtain sparse prolongation matrices with three entries per row and fast convergence of the solver.