Computationally efficient barycentric interpolation of large grain boundary octonion point sets

Area of Science:

• Computational Geometry
• Data Science
• Materials Science

Background:

• Barycentric interpolation is a fundamental technique for estimating values within a simplex.
• Hyperspherical point sets present unique challenges for interpolation due to their complex geometry.
• Existing methods often struggle with computational complexity for large datasets.

Purpose of the Study:

• To develop an efficient barycentric interpolation method for large grain boundary octonion point sets on hyperspheres.
• To reduce computational complexity while preserving information integrity.
• To provide a versatile method applicable to diverse fields like materials science, robotics, and neural networks.

Main Methods:

• Dimensionality reduction using Singular Value Decomposition (SVD) transformations and linear projections to remove degenerate dimensions.
• Determination of intersecting facets through nearest neighbor (NN) searches with a maximum threshold.
• Integration of barycentric interpolation with facet intersection and dimensionality reduction techniques.

Main Results:

• A novel method significantly reduces computational complexity for hyperspherical interpolation.
• The approach effectively handles large grain boundary octonion point sets.
• Demonstrated applicability through a case study on the 7-sphere with 1-sphere and 2-sphere visualizations.

Conclusions:

• The proposed method offers an efficient and information-preserving solution for barycentric interpolation on hyperspheres.
• The combination of SVD, linear projections, and NN searches provides a robust framework.
• This technique is valuable for advanced applications requiring interpolation of high-dimensional, hyperspherical data.