A Scalable Crystallographic FFT

Abstract

Computational X-ray crystallography is the most accurate method for determining the atomic structure of crystals. Some large scale problems of current interest, such as the determination of macromolecular configurations at atomic level, demand a reiterated computation of large three-dimensional discrete Fourier transforms (DFT). Although fast Fourier transforms (FFT) are widely available, significant improvements in operation count and storage requirements can be obtained by using instead FFT variants tailored to crystal structure calculations. These are called crystallographic FFTs. A crystallographic FFT uses a-priori knowledge of the specimen’s crystal symmetries to reduce the size of input and output data sets, and to eliminate redundant computations. Since in most cases of interest the modified FFT is still fairly large, parallel computation is regarded as an alternative to further speedup X-ray crystallography computations. Traditional divide-and-conquer multidimensional FFTs do not scale up well. In this paper we describe a parallel multidimensional crystallographic FFT for data arrays of prime edge-length that is endowed with good scalability properties. We perform a simple theoretical analysis and some computer experiments to validate our claim.