35,899 research outputs found
A new Truncated Fourier Transform algorithm
Truncated Fourier Transforms (TFTs), first introduced by Van der Hoeven,
refer to a family of algorithms that attempt to smooth "jumps" in complexity
exhibited by FFT algorithms. We present an in-place TFT whose time complexity,
measured in terms of ring operations, is comparable to existing not-in-place
TFT methods. We also describe a transformation that maps between two families
of TFT algorithms that use different sets of evaluation points.Comment: 8 pages, submitted to the 38th International Symposium on Symbolic
and Algebraic Computation (ISSAC 2013
A Fast Method for Numerical Realization of Fourier Tools
This chapter presents new application of author’s recent algorithms for fast summations of truncated Fourier series. A complete description of this method is given, and an algorithm for numerical implementation with a given accuracy for the Fourier transform is proposed
Fast Ewald summation for free-space Stokes potentials
We present a spectrally accurate method for the rapid evaluation of
free-space Stokes potentials, i.e. sums involving a large number of free space
Green's functions. We consider sums involving stokeslets, stresslets and
rotlets that appear in boundary integral methods and potential methods for
solving Stokes equations. The method combines the framework of the Spectral
Ewald method for periodic problems, with a very recent approach to solving the
free-space harmonic and biharmonic equations using fast Fourier transforms
(FFTs) on a uniform grid. Convolution with a truncated Gaussian function is
used to place point sources on a grid. With precomputation of a scalar grid
quantity that does not depend on these sources, the amount of oversampling of
the grids with Gaussians can be kept at a factor of two, the minimum for
aperiodic convolutions by FFTs. The resulting algorithm has a computational
complexity of O(N log N) for problems with N sources and targets. Comparison is
made with a fast multipole method (FMM) to show that the performance of the new
method is competitive.Comment: 35 pages, 15 figure
A fast analysis-based discrete Hankel transform using asymptotic expansions
A fast and numerically stable algorithm is described for computing the
discrete Hankel transform of order as well as evaluating Schl\"{o}milch and
Fourier--Bessel expansions in
operations. The algorithm is based on an asymptotic expansion for Bessel
functions of large arguments, the fast Fourier transform, and the Neumann
addition formula. All the algorithmic parameters are selected from error bounds
to achieve a near-optimal computational cost for any accuracy goal. Numerical
results demonstrate the efficiency of the resulting algorithm.Comment: 22 page
Fast Ewald summation for electrostatic potentials with arbitrary periodicity
A unified treatment for fast and spectrally accurate evaluation of
electrostatic potentials subject to periodic boundary conditions in any or none
of the three space dimensions is presented. Ewald decomposition is used to
split the problem into a real space and a Fourier space part, and the FFT based
Spectral Ewald (SE) method is used to accelerate the computation of the latter.
A key component in the unified treatment is an FFT based solution technique for
the free-space Poisson problem in three, two or one dimensions, depending on
the number of non-periodic directions. The cost of calculations is furthermore
reduced by employing an adaptive FFT for the doubly and singly periodic cases,
allowing for different local upsampling rates. The SE method will always be
most efficient for the triply periodic case as the cost for computing FFTs will
be the smallest, whereas the computational cost for the rest of the algorithm
is essentially independent of the periodicity. We show that the cost of
removing periodic boundary conditions from one or two directions out of three
will only marginally increase the total run time. Our comparisons also show
that the computational cost of the SE method for the free-space case is
typically about four times more expensive as compared to the triply periodic
case. The Gaussian window function previously used in the SE method, is here
compared to an approximation of the Kaiser-Bessel window function, recently
introduced. With a carefully tuned shape parameter that is selected based on an
error estimate for this new window function, runtimes for the SE method can be
further reduced. Keywords: Fast Ewald summation, Fast Fourier transform,
Arbitrary periodicity, Coulomb potentials, Adaptive FFT, Fourier integral,
Spectral accuracy.Comment: 21 pages, 11 figure
Frequency Analysis of Gradient Estimators in Volume Rendering
Gradient information is used in volume rendering to classify and color samples along a ray. In this paper, we present an analysis of the theoretically ideal gradient estimator and compare it to some commonly used gradient estimators. A new method is presented to calculate the gradient at arbitrary sample positions, using the derivative of the interpolation filter as the basis for the new gradient filter. As an example, we will discuss the use of the derivative of the cubic spline. Comparisons with several other methods are demonstrated. Computational efficiency can be realized since parts of the interpolation computation can be leveraged in the gradient estimatio
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