Implicit Voronoi Mesh for High-Dimensional Neighbor Identification
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Conventional Voronoi domain decomposition methods face challenges in high-dimensional spaces due to the curse of dimensionality, leading to exponential increases in computational requirements and inefficiencies in identifying neighbor cells, which hampers accuracy and stability in numerical simulations and global optimization tasks.
Innovation Solution
A computing system performs hyperplane sampling and iterative line-hyperplane trimming to identify neighbor cells in an implicitly defined Voronoi mesh, allowing for efficient neighbor identification without characterizing every Voronoi vertex, thus overcoming the curse of dimensionality and enabling improved accuracy and stability in simulations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional Voronoi domain decomposition techniques (such as successive hyperplane trimming) are used to construct Voronoi cells, then the domain can be decomposed into cells with desired properties (positive Jacobian, convex elements, planar facets), but the computational requirements increase exponentially as the number of dimensions increases (curse of dimensionality)
Solution Approach 1:
The patent segments the domain decomposition process into two distinct phases: (1) an offline phase that generates the Voronoi mesh and stores it in an implicit data structure, and (2) an online phase that efficiently queries neighbor relationships. This segmentation allows the computationally intensive mesh generation to be performed once, while subsequent neighbor identification operations are dramatically accelerated through the use of implicit data structures and spatial hashing, thereby resolving the curse of dimensionality for high-dimensional neighbor queries.
Solution Approach 2:
The patent performs preliminary action by pre-computing and storing the Voronoi mesh structure in an implicit data structure during an offline phase. This includes pre-calculating Voronoi vertices, edges, and faces, and organizing them in a hash-based spatial data structure. This preliminary preparation enables rapid neighbor identification during online operations without requiring exponential computational resources, even in high-dimensional spaces.
2Loss of information
If all Voronoi vertices are characterized to identify neighbor cells, then complete neighbor information can be obtained, but the computational complexity and time required increase significantly in high-dimensional spaces
Solution Approach 1:
The patent extracts only the essential information needed for neighbor identification from the complete Voronoi mesh. Instead of characterizing all Voronoi vertices, the method uses implicit data structures that store and provide access to Voronoi vertices, edges, and faces on-demand. The spatial hashing mechanism extracts and utilizes only the relevant geometric relationships needed for neighbor cell identification, thereby obtaining complete neighbor information without the computational burden of processing the entire mesh explicitly.
Solution Approach 2:
The patent creates an implicit copy of the Voronoi mesh structure in a compressed, hash-based format. Rather than storing and processing the complete explicit mesh representation, the method maintains an implicit data structure that reproduces the necessary geometric relationships through hashing and spatial indexing. This implicit copying enables efficient neighbor queries by reconstructing only the necessary mesh information during online operations.
3Productivity
If conventional methods are used in high-dimensional domains, then domain decomposition can be performed, but the accuracy and stability of numerical simulations and global optimization tasks deteriorate due to computational inefficiencies
Solution Approach 1:
The patent replaces the traditional mechanical approach of explicitly constructing and storing complete Voronoi mesh representations with an implicit, algorithm-based system. Instead of using explicit geometric constructions that become intractable in high dimensions, the method employs implicit data structures, spatial hashing, and on-demand computation to substitute the mechanical mesh construction process. This substitution maintains simulation accuracy while achieving the computational efficiency needed for high-dimensional problems.
Data Source
AI summary
Described herein are technologies pertaining to generating a Voronoi mesh for an enclosed domain of arbitrarily many dimensions. An implicit Voronoi mesh is received that comprises locations of seeds in the domain. Neighbors of a seed are identified by extending a segment from the seed to the boundary of the domain and performing successive hyperplane trimming of the segment to identify a point on the facet of the Voronoi cell of the seed. A vertex of the cell can be identified by successively extending and trimming one or more additional spokes to identify points on a facet of the cell that are constrained to lie in a (d−n) space, where d is a number of dimensions of the domain and n is a number of iterations. A numerical simulation is performed based on the Voronoi mesh to generate simulation results in applications including surrogate modeling, global optimization, calibration, dimension reduction.