FAST EXPANSION INTO HARMONICS ON THE BALL

Area of Science:

• Mathematics
• Numerical Analysis
• Scientific Computing

Background:

• Representing 3D functions in Cartesian voxel grids is common in scientific applications.
• Ball harmonics provide an alternative, efficient basis for representing functions within a unit ball.
• Transforming between these representations is computationally intensive with naive methods.

Purpose of the Study:

• To devise fast and provably accurate algorithms for transforming 3D functions between Cartesian voxel and ball harmonic representations.
• To significantly reduce the computational complexity compared to existing direct methods.

Main Methods:

• Developed novel algorithms for transforming N x N x N Cartesian voxel data to ball harmonic expansions.
• Algorithms achieve provable accuracy (epsilon) with improved time complexity.
• Leveraged properties of the Dirichlet Laplacian eigenbasis on the unit ball.

Main Results:

• Achieved relative L1-L-infinity accuracy of epsilon in time complexity O(N^3(log N)^2 + N^3|log epsilon|^2).
• This is a substantial improvement over the naive O(N^6) complexity.
• Demonstrated the effectiveness of the algorithms through numerical examples.

Conclusions:

• The developed algorithms offer a computationally efficient solution for 3D function representation transformations.
• These methods have potential applications in various scientific fields requiring 3D data analysis and manipulation.
• The provable accuracy and improved speed make these algorithms valuable for complex simulations and modeling.