36,379 research outputs found
Deterministic Sparse FFT Algorithms
The discrete Fourier transform (DFT) is a well-known transform with many applications in various fields. By fast Fourier transform (FFT) algorithms, the DFT of a vector can be efficiently computed. Using these algorithms, one can reconstruct a complex vector x of length N from its discrete Fourier transform applying O(N log N) arithmetical operations. In order to improve the complexity of FFT algorithms, one needs additional a priori assumptions on the vector x. In this thesis, the focus is on vectors with small support or sparse vectors for which several new deterministic algorithms are proposed that have a lower complexity than regular FFT algorithms. We develop sublinear time algorithms for the reconstruction of complex vectors or matrices with small support from Fourier data as well as an algorithm for the reconstruction of real nonnegative vectors. The algorithms are analyzed and evaluated in numerical experiments. Furthermore, we generalize the algorithm for real nonnegative vectors with small support and propose an approach to the reconstruction of sparse vectors with real nonnegative entries
Efficient implementation of the Gutzwiller variational method
We present a self-consistent numerical approach to solve the Gutzwiller
variational problem for general multi-band models with arbitrary on-site
interaction. The proposed method generalizes and improves the procedure derived
by Deng et al., Phys. Rev. B. 79 075114 (2009), overcoming the restriction to
density-density interaction without increasing the complexity of the
computational algorithm. Our approach drastically reduces the problem of the
high-dimensional Gutzwiller minimization by mapping it to a minimization only
in the variational density matrix, in the spirit of the Levy and Lieb
formulation of DFT. For fixed density the Gutzwiller renormalization matrix is
determined as a fixpoint of a proper functional, whose evaluation only requires
ground-state calculations of matrices defined in the Gutzwiller variational
space. Furthermore, the proposed method is able to account for the symmetries
of the variational function in a controlled way, reducing the number of
variational parameters. After a detailed description of the method we present
calculations for multi-band Hubbard models with full (rotationally invariant)
Hund's rule on-site interaction. Our analysis shows that the numerical
algorithm is very efficient, stable and easy to implement. For these reasons
this method is particularly suitable for first principle studies -- e.g., in
combination with DFT -- of many complex real materials, where the full
intra-atomic interaction is important to obtain correct results.Comment: 19 pages, 7 figure
On Polynomial Multiplication in Chebyshev Basis
In a recent paper Lima, Panario and Wang have provided a new method to
multiply polynomials in Chebyshev basis which aims at reducing the total number
of multiplication when polynomials have small degree. Their idea is to use
Karatsuba's multiplication scheme to improve upon the naive method but without
being able to get rid of its quadratic complexity. In this paper, we extend
their result by providing a reduction scheme which allows to multiply
polynomial in Chebyshev basis by using algorithms from the monomial basis case
and therefore get the same asymptotic complexity estimate. Our reduction allows
to use any of these algorithms without converting polynomials input to monomial
basis which therefore provide a more direct reduction scheme then the one using
conversions. We also demonstrate that our reduction is efficient in practice,
and even outperform the performance of the best known algorithm for Chebyshev
basis when polynomials have large degree. Finally, we demonstrate a linear time
equivalence between the polynomial multiplication problem under monomial basis
and under Chebyshev basis
New Algorithms for Computing a Single Component of the Discrete Fourier Transform
This paper introduces the theory and hardware implementation of two new
algorithms for computing a single component of the discrete Fourier transform.
In terms of multiplicative complexity, both algorithms are more efficient, in
general, than the well known Goertzel Algorithm.Comment: 4 pages, 3 figures, 1 table. In: 10th International Symposium on
Communication Theory and Applications, Ambleside, U
FFT Interpolation from Nonuniform Samples Lying in a Regular Grid
This paper presents a method to interpolate a periodic band-limited signal
from its samples lying at nonuniform positions in a regular grid, which is
based on the FFT and has the same complexity order as this last algorithm. This
kind of interpolation is usually termed "the missing samples problem" in the
literature, and there exists a wide variety of iterative and direct methods for
its solution. The one presented in this paper is a direct method that exploits
the properties of the so-called erasure polynomial, and it provides a
significant improvement on the most efficient method in the literature, which
seems to be the burst error recovery (BER) technique of Marvasti's et al. The
numerical stability and complexity of the method are evaluated numerically and
compared with the pseudo-inverse and BER solutions.Comment: Submitted to the IEEE Transactions on Signal Processin
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