The Beauty of Anisotropic Mesh Refinement: Omnitrees for Efficient Dyadic Discretizations

TL;DR

This paper introduces omnitrees, an anisotropic mesh refinement data structure that improves efficiency and convergence in adaptive mesh refinement, especially for highly anisotropic and high-dimensional problems.

Contribution

Omnitrees generalize octrees to allow dimension-specific refinement, reducing tree depth and increasing convergence rates in anisotropic adaptive mesh refinement.

Findings

Omnitrees increase convergence rate by up to 1.5x in 3D shape representation.

Omnitrees require less storage for the same error bounds compared to octrees.

Advantages of omnitrees are expected to be greater in higher-dimensional applications.

Abstract

Structured adaptive mesh refinement (AMR), commonly implemented via quadtrees and octrees, underpins a wide range of applications including databases, computer graphics, physics simulations, and machine learning. However, octrees enforce isotropic refinement in regions of interest, which can be especially inefficient for problems that are intrinsically anisotropic--much resolution is spent where little information is gained. This paper presents omnitrees as an anisotropic generalization of octrees and related data structures. Omnitrees allow to refine only the locally most important dimensions, providing tree structures that are less deep than bintrees and less wide than octrees. As a result, the convergence of the AMR schemes can be increased by up to a factor of the dimensionality d for very anisotropic problems, quickly offsetting their modest increase in storage overhead. We validate this finding on the problem of binary shape representation across 4,166 three-dimensional objects: Omnitrees increase the mean convergence rate by 1.5x, require less storage to achieve equivalent error bounds, and maximize the information density of the stored function faster than octrees. These advantages are projected to be even stronger for higher-dimensional problems. We provide a first validation by introducing a time-dependent rotation to create four-dimensional representations, and discuss the properties of their 4-d octree and omnitree approximations. Overall, omnitree discretizations can make existing AMR approaches more efficient, and open up new possibilities for high-dimensional applications.