27,343 research outputs found
Composite Cyclotomic Fourier Transforms with Reduced Complexities
Discrete Fourier transforms~(DFTs) over finite fields have widespread
applications in digital communication and storage systems. Hence, reducing the
computational complexities of DFTs is of great significance. Recently proposed
cyclotomic fast Fourier transforms (CFFTs) are promising due to their low
multiplicative complexities. Unfortunately, there are two issues with CFFTs:
(1) they rely on efficient short cyclic convolution algorithms, which has not
been investigated thoroughly yet, and (2) they have very high additive
complexities when directly implemented. In this paper, we address both issues.
One of the main contributions of this paper is efficient bilinear 11-point
cyclic convolution algorithms, which allow us to construct CFFTs over
GF. The other main contribution of this paper is that we propose
composite cyclotomic Fourier transforms (CCFTs). In comparison to previously
proposed fast Fourier transforms, our CCFTs achieve lower overall complexities
for moderate to long lengths, and the improvement significantly increases as
the length grows. Our 2047-point and 4095-point CCFTs are also first efficient
DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also
advantageous for hardware implementations due to their regular and modular
structure.Comment: submitted to IEEE trans on Signal Processin
Signal Flow Graph Approach to Efficient DST I-IV Algorithms
In this paper, fast and efficient discrete sine transformation (DST)
algorithms are presented based on the factorization of sparse, scaled
orthogonal, rotation, rotation-reflection, and butterfly matrices. These
algorithms are completely recursive and solely based on DST I-IV. The presented
algorithms have low arithmetic cost compared to the known fast DST algorithms.
Furthermore, the language of signal flow graph representation of digital
structures is used to describe these efficient and recursive DST algorithms
having points signal flow graph for DST-I and points signal flow
graphs for DST II-IV
Accelerated identification of equilibrium structures of multicomponent inorganic crystals using machine learning potentials
The discovery of new multicomponent inorganic compounds can provide direct
solutions to many scientific and engineering challenges, yet the vast size of
the uncharted material space dwarfs current synthesis throughput. While the
computational crystal structure prediction is expected to mitigate this
frustration, the NP-hardness and steep costs of density functional theory (DFT)
calculations prohibit material exploration at scale. Herein, we introduce
SPINNER, a highly efficient and reliable structure-prediction framework based
on exhaustive random searches and evolutionary algorithms, which is completely
free from empiricism. Empowered by accurate neural network potentials, the
program can navigate the configuration space faster than DFT by more than
10-fold. In blind tests on 60 ternary compositions diversely selected
from the experimental database, SPINNER successfully identifies experimental
(or theoretically more stable) phases for ~80% of materials within 5000
generations, entailing up to half a million structure evaluations for each
composition. When benchmarked against previous data mining or DFT-based
evolutionary predictions, SPINNER identifies more stable phases in the majority
of cases. By developing a reliable and fast structure-prediction framework,
this work opens the door to large-scale, unbounded computational exploration of
undiscovered inorganic crystals.Comment: 3 figure
JDFTx: software for joint density-functional theory
Density-functional theory (DFT) has revolutionized computational prediction
of atomic-scale properties from first principles in physics, chemistry and
materials science. Continuing development of new methods is necessary for
accurate predictions of new classes of materials and properties, and for
connecting to nano- and mesoscale properties using coarse-grained theories.
JDFTx is a fully-featured open-source electronic DFT software designed
specifically to facilitate rapid development of new theories, models and
algorithms. Using an algebraic formulation as an abstraction layer, compact
C++11 code automatically performs well on diverse hardware including GPUs. This
code hosts the development of joint density-functional theory (JDFT) that
combines electronic DFT with classical DFT and continuum models of liquids for
first-principles calculations of solvated and electrochemical systems. In
addition, the modular nature of the code makes it easy to extend and interface
with, facilitating the development of multi-scale toolkits that connect to ab
initio calculations, e.g. photo-excited carrier dynamics combining electron and
phonon calculations with electromagnetic simulations.Comment: 9 pages, 3 figures, 2 code listing
De Novo Assembly of Nucleotide Sequences in a Compressed Feature Space
Sequencing technologies allow for an in-depth analysis
of biological species but the size of the generated datasets
introduce a number of analytical challenges. Recently, we
demonstrated the application of numerical sequence representations
and data transformations for the alignment of short
reads to a reference genome. Here, we expand out approach
for de novo assembly of short reads. Our results demonstrate
that highly compressed data can encapsulate the signal suffi-
ciently to accurately assemble reads to big contigs or complete
genomes
Quantum Fourier transform revisited
The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view
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