The crystallographic fast Fourier transform. Recursive symmetry reduction

Area of Science:

• Crystallography
• Computational Science
• Materials Science

Background:

• Fast Fourier Transform (FFT) is crucial for analyzing crystallographic data.
• Existing FFT algorithms can be computationally intensive, requiring significant memory and time.
• Exploiting crystallographic symmetry can potentially optimize FFT computations.

Purpose of the Study:

• To develop maximally efficient algorithms for computing the crystallographic fast Fourier transform (FFT).
• To reduce computation time and memory usage in crystallographic data analysis.
• To apply these algorithms across all 230 crystallographic space groups.

Main Methods:

• Recursive reduction of the crystallographic FFT problem to smaller transforms on grids without special points.
• Leveraging the symmetry operators inherent in each of the 230 space groups.
• Utilizing previously established maximally efficient FFT algorithms for general grids.

Main Results:

• Achieved reduction in computation time and memory usage by a factor equal to the number of symmetry operators.
• Developed a unified approach applicable to all 230 crystallographic space groups.
• Demonstrated the effectiveness of recursive problem reduction for optimizing FFT.

Conclusions:

• The presented algorithms offer significant computational advantages for crystallographic FFT.
• The approach effectively integrates crystallographic symmetry into FFT computation.
• This work provides a pathway for more efficient analysis of crystallographic datasets.