Surface multigrid via intrinsic prolongation

Fig. 1. We introduce a novel geometric multigrid solver for curved surfaces. Our key ingredient is an intrinsic prolongation operator computed via parameterizing the high resolution shape via its coarsened counterpart, visualized using colored triangles. By recursively applying this self-parameterization, we obtain a hierarchy (from left to right) for our multigrid method (e.g., to solve heat geodesics [Crane et al. 2017], far left). ©model by Benoît Rogez under CC BY-NC.

This paper introduces a novel geometric multigrid solver for unstructured curved surfaces. Multigrid methods are highly efficient iterative methods for solving systems of linear equations. Despite the success in solving problems defined on structured domains, generalizing multigrid to unstructured curved domains remains a challenging problem. The critical missing ingredient is a prolongation operator to transfer functions across different multigrid levels. We propose a novel method for computing the prolongation for triangulated surfaces based on intrinsic geometry, enabling an efficient geometric multigrid solver for curved surfaces. Our surface multigrid solver achieves better convergence than existing multigrid methods. Compared to direct solvers, our solver is orders of magnitude faster. We evaluate our method on many geometry processing applications and a wide variety of complex shapes with and without boundaries. By simply replacing the direct solver, we upgrade existing algorithms to interactive frame rates, and shift the computational bottleneck away from solving linear systems.