Generating optimized Fourier interpolation routines for density function theory using SPIRAL

© 2015 IEEE.Upsampling of a multi-dimensional data-set is an operation with wide application in image processing and quantum mechanical calculations using density functional theory. For small up sampling factors as seen in the quantum chemistry code ONETEP, a time-shift based implementation that shifts samples by a fraction of the original grid spacing to fill in the intermediate values using a frequency domain Fourier property can be a good choice. Readily available highly optimized multidimensional FFT implementations are leveraged at the expense of extra passes through the entire working set. In this paper we present an optimized variant of the time-shift based up sampling. Since ONETEP handles threading, we address the memory hierarchy and SIMD vectorization, and focus on problem dimensions relevant for ONETEP. We present a formalization of this operation within the SPIRAL framework and demonstrate auto-generated and auto-tuned interpolation libraries. We compare the performance of our generated code against the previous best implementations using highly optimized FFT libraries (FFTW and MKL). We demonstrate speed-ups in isolation averaging 3x and within ONETEP of up to 15%

Efficient Retrieval of Similar Time Sequences Using DFT

We propose an improvement of the known DFT-based indexing technique for fast retrieval of similar time sequences. We use the last few Fourier coefficients in the distance computation without storing them in the index since every coefficient at the end is the complex conjugate of a coefficient at the beginning and as strong as its counterpart. We show analytically that this observation can accelerate the search time of the index by more than a factor of two. This result was confirmed by our experiments, which were carried out on real stock prices and synthetic data

Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs

This paper presents a systematic methodology based on the algebraic theory of signal processing to classify and derive fast algorithms for linear transforms. Instead of manipulating the entries of transform matrices, our approach derives the algorithms by stepwise decomposition of the associated signal models, or polynomial algebras. This decomposition is based on two generic methods or algebraic principles that generalize the well-known Cooley-Tukey FFT and make the algorithms' derivations concise and transparent. Application to the 16 discrete cosine and sine transforms yields a large class of fast algorithms, many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar