12,102 research outputs found
Parts formulas involving the Fourier-Feynman transform associated with Gaussian process on Wiener space
In this paper, using a very general Cameron--Storvick theorem on the Wiener
space , we establish various integration by parts formulas involving
generalized analytic Feynman integrals, generalized analytic Fourier--Feynman
transforms, and the first variation (associated with Gaussian processes) of
functionals on having the form
for
scale almost every , where denotes
the Paley--Wiener--Zygmund stochastic integral , and
is an orthogonal set of nonzero functions in
. The Gaussian processes used in this paper are not stationary
Approximation of integral operators using convolution-product expansions
We consider a class of linear integral operators with impulse responses
varying regularly in time or space. These operators appear in a large number of
applications ranging from signal/image processing to biology. Evaluating their
action on functions is a computation-ally intensive problem necessary for many
practical problems. We analyze a technique called convolution-product
expansion: the operator is locally approximated by a convolution, allowing to
design fast numerical algorithms based on the fast Fourier transform. We design
various types of expansions, provide their explicit rates of approximation and
their complexity depending on the time varying impulse response smoothness.
This analysis suggests novel wavelet based implementations of the method with
numerous assets such as optimal approximation rates, low complexity and storage
requirements as well as adaptivity to the kernels regularity. The proposed
methods are an alternative to more standard procedures such as panel
clustering, cross approximations, wavelet expansions or hierarchical matrices
Accelerating the 2-point and 3-point galaxy correlation functions using Fourier transforms
Though Fourier Transforms (FTs) are a common technique for finding
correlation functions, they are not typically used in computations of the
anisotropy of the two-point correlation function (2PCF) about the line of sight
in wide-angle surveys because the line-of-sight direction is not constant on
the Cartesian grid. Here we show how FTs can be used to compute the multipole
moments of the anisotropic 2PCF. We also show how FTs can be used to accelerate
the 3PCF algorithm of Slepian & Eisenstein (2015). In both cases, these FT
methods allow one to avoid the computational cost of pair counting, which
scales as the square of the number density of objects in the survey. With the
upcoming large datasets of DESI, Euclid, and LSST, FT techniques will therefore
offer an important complement to simple pair or triplet counts.Comment: 5 pages, no figures, accepted MNRAS Letters; v2 matches accepted
versio
Noise and Signal for Spectra of Intermittent Noiselike Emission
We show that intermittency of noiselike emission, after propagation through a
scattering medium, affects the distribution of noise in the observed
correlation function. Intermittency also affects correlation of noise among
channels of the spectrum, but leaves the average spectrum, average correlation
function, and distribution of noise among channels of the spectrum unchanged.
Pulsars are examples of such sources: intermittent and affected by interstellar
propagation. We assume that the source emits Gaussian white noise, modulated by
a time-envelope. Propagation convolves the resulting time series with an
impulse-response function that represents effects of dispersion, scattering,
and absorption. We assume that this propagation kernel is shorter than the time
for an observer to accumulate a single spectrum. We show that rapidly-varying
intermittent emission tends to concentrate noise near the central lag of the
correlation function. We derive mathematical expressions for this effect, in
terms of the time envelope and the propagation kernel. We present examples,
discuss effects of background noise, and compare our results with observations.Comment: 30 pages, 4 figure
A space of generalized Brownian motion path-valued continuous functions with application
In this paper, we introduce the paths space
which is consists of generalized Brownian motion path-valued continuous
functions on . We next present several relevant examples of the paths
space integral. We then discuss the concept of the analytic Feynman integration
theory for functionals on the paths space
Fast and Stable Pascal Matrix Algorithms
In this paper, we derive a family of fast and stable algorithms for
multiplying and inverting Pascal matrices that run in time and are closely related to De Casteljau's algorithm for B\'ezier curve
evaluation. These algorithms use a recursive factorization of the triangular
Pascal matrices and improve upon the cripplingly unstable fast
Fourier transform-based algorithms which involve a Toeplitz matrix
factorization. We conduct numerical experiments which establish the speed and
stability of our algorithm, as well as the poor performance of the Toeplitz
factorization algorithm. As an example, we show how our formulation relates to
B\'ezier curve evaluation
Rigid-Motion Scattering for Texture Classification
A rigid-motion scattering computes adaptive invariants along translations and
rotations, with a deep convolutional network. Convolutions are calculated on
the rigid-motion group, with wavelets defined on the translation and rotation
variables. It preserves joint rotation and translation information, while
providing global invariants at any desired scale. Texture classification is
studied, through the characterization of stationary processes from a single
realization. State-of-the-art results are obtained on multiple texture data
bases, with important rotation and scaling variabilities.Comment: 19 pages, submitted to International Journal of Computer Visio
A bootstrapping approach to jump inequalities and their applications
The aim of this paper is to present an abstract and general approach to jump
inequalities in harmonic analysis. Our principal conclusion is the refinement
of -variational estimates, previously known for , to end-point results
for the jump quasi-seminorm corresponding to . This is applied to the
dimension-free results recently obtained by the first two authors in
collaboration with Bourgain and Wr\'obel (arXiv:1708.04639 and
arXiv:1804.07679), and also to operators of Radon type treated by Jones,
Seeger, and Wright.Comment: 25 pages, small correction
Spatio-spectral concentration of convolutions
Differential equations may possess coefficients that vary on a spectrum of
scales. Because coefficients are typically multiplicative in real space, they
turn into convolution operators in spectral space, mixing all wavenumbers.
However, in many applications, only the largest scales of the solution are of
interest and so the question turns to whether it is possible to build effective
coarse-scale models of the coefficients in such a manner that the large scales
of the solution are left intact. Here we apply the method of numerical
homogenization to deterministic linear equations to generate sub-grid-scale
models of coefficients at desired frequency cutoffs. We use the Fourier basis
to project, filter and compute correctors for the coefficients. The method is
tested in 1D and 2D scenarios and found to reproduce the coarse scales of the
solution to varying degrees of accuracy depending on the cutoff. We relate this
method to mode-elimination Renormalization Group (RG) and discuss the
connection between accuracy and the cutoff wavenumber. The tradeoff is governed
by a form of the uncertainty principle for convolutions, which states that as
the convolution operator is squeezed in the spectral domain, it broadens in
real space. As a consequence, basis sparsity is a high virtue and the choice of
the basis can be critical.Comment: 24 pages, 6 figures, in press, Journal of Computational Physic
Haptic Assembly Using Skeletal Densities and Fourier Transforms
Haptic-assisted virtual assembly and prototyping has seen significant
attention over the past two decades. However, in spite of the appealing
prospects, its adoption has been slower than expected. We identify the main
roadblocks as the inherent geometric complexities faced when assembling objects
of arbitrary shape, and the computation time limitation imposed by the
notorious 1 kHz haptic refresh rate. We addressed the first problem in a recent
work by introducing a generic energy model for geometric guidance and
constraints between features of arbitrary shape. In the present work, we
address the second challenge by leveraging Fourier transforms to compute the
constraint forces and torques. Our new concept of 'geometric energy' field is
computed automatically from a cross-correlation of 'skeletal densities' in the
frequency domain, and serves as a generalization of the manually specified
virtual fixtures or heuristically identified mating constraints proposed in the
literature. The formulation of the energy field as a convolution enables
efficient computation using fast Fourier transforms (FFT) on the graphics
processing unit (GPU). We show that our method is effective for low-clearance
assembly of objects of arbitrary geometric and syntactic complexity.Comment: A shorter version was presented in ASME Computers and Information in
Engineering Conference (CIE'2015) (Best Paper Award
- …