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

2025-08-08

AI Summary

Traditional adaptive mesh refinement (AMR) methods—e.g., octrees—enforce isotropic refinement, leading to redundant subdivision along non-critical directions when resolving anisotropic features, thereby incurring excessive storage and computational overhead. To address this, we propose *omnitree*, a general-purpose multidimensional adaptive grid structure supporting anisotropic refinement. Omnitree introduces, for the first time in discrete tree-based representations, dynamic dimension selection and axis-aligned anisotropic splitting: each node independently refines along an optimal subset of dimensions, breaking octree’s fixed 2^d splitting constraint. This yields significantly shallower trees and fewer nodes. Evaluated on 4,166 3D shapes, omnitree achieves 1.5× faster average convergence versus baselines while reducing memory consumption at equivalent accuracy; its advantages amplify in higher-dimensional settings. Omnitree establishes a more compact, efficient, and scalable discretization paradigm for high-dimensional adaptive modeling.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Ensemble Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Octrees enforce inefficient isotropic refinement for anisotropic problems
• Omnitrees enable anisotropic refinement by focusing on important dimensions
• Omnitrees improve convergence and storage efficiency in high-dimensional problems

Innovation

• Omnitrees enable anisotropic mesh refinement
• Omnitrees optimize dimension-specific refinement efficiency
• Omnitrees enhance convergence and reduce storage