Simplification Directional Fields

Abstract

In their recent paper, Liu et al. developed a new geometric multigrid solver that achieves better convergence than existing multigrid methods, and is orders of magnitude faster than conventional solvers. This was achieved by using a novel way to calculate the prolongation operator for scalar fields. In this work, we introduce similar multigrid solvers for directional fields, where we will use hierarchical simplification on directional fields to convergence to solutions quickly and accurately. Our main contributions are three new structure preserving prolongation operators that can transfer signals between the fine and the coarse multigrid levels for directional fields in discrete exterior algebra, face-based fields, and power fields. The prolongation operators can be defined as an interpolation from the fine to the coarse version of a collapsed 1-ring. We tested each of these operators with three different multigrid solvers on a most harmonic field problem for twelve different meshes with sizes ranging from 13K to 270K faces. Although the precomputation time for our operators is long, our prolongation operators in combination with our multigrid solvers are poised to outperform linear least squares by a significant margin when solving for larger meshes.