Fast Algorithms for the computation of Fourier Extensions of arbitrary length

Fourier series of smooth, non-periodic functions on $[-1,1]$ are known to exhibit the Gibbs phenomenon, and exhibit overall slow convergence. One way of overcoming these problems is by using a Fourier series on a larger domain, say $[-T,T]$ with $T>1$, a technique called Fourier extension or Fourier continuation. When constructed as the discrete least squares minimizer in equidistant points, the Fourier extension has been shown shown to converge geometrically in the truncation parameter $N$. A fast ${\mathcal O}(N \log^2 N)$ algorithm has been described to compute Fourier extensions for the case where $T=2$, compared to ${\mathcal O}(N^3)$ for solving the dense discrete least squares problem. We present two ${\mathcal O}(N\log^2 N )$ algorithms for the computation of these approximations for the case of general $T$, made possible by exploiting the connection between Fourier extensions and Prolate Spheroidal Wave theory. The first algorithm is based on the explicit computation of so-called periodic discrete prolate spheroidal sequences, while the second algorithm is purely algebraic and only implicitly based on the theory

Fast, Dense Feature SDM on an iPhone

In this paper, we present our method for enabling dense SDM to run at over 90 FPS on a mobile device. Our contributions are two-fold. Drawing inspiration from the FFT, we propose a Sparse Compositional Regression (SCR) framework, which enables a significant speed up over classical dense regressors. Second, we propose a binary approximation to SIFT features. Binary Approximated SIFT (BASIFT) features, which are a computationally efficient approximation to SIFT, a commonly used feature with SDM. We demonstrate the performance of our algorithm on an iPhone 7, and show that we achieve similar accuracy to SDM

Detection of variable frequency signals using a fast chirp transform

The detection of signals with varying frequency is important in many areas of physics and astrophysics. The current work was motivated by a desire to detect gravitational waves from the binary inspiral of neutron stars and black holes, a topic of significant interest for the new generation of interferometric gravitational wave detectors such as LIGO. However, this work has significant generality beyond gravitational wave signal detection. We define a Fast Chirp Transform (FCT) analogous to the Fast Fourier Transform (FFT). Use of the FCT provides a simple and powerful formalism for detection of signals with variable frequency just as Fourier transform techniques provide a formalism for the detection of signals of constant frequency. In particular, use of the FCT can alleviate the requirement of generating complicated families of filter functions typically required in the conventional matched filtering process. We briefly discuss the application of the FCT to several signal detection problems of current interest

Quantum computation and analysis of Wigner and Husimi functions: toward a quantum image treatment

We study the efficiency of quantum algorithms which aim at obtaining phase space distribution functions of quantum systems. Wigner and Husimi functions are considered. Different quantum algorithms are envisioned to build these functions, and compared with the classical computation. Different procedures to extract more efficiently information from the final wave function of these algorithms are studied, including coarse-grained measurements, amplitude amplification and measure of wavelet-transformed wave function. The algorithms are analyzed and numerically tested on a complex quantum system showing different behavior depending on parameters, namely the kicked rotator. The results for the Wigner function show in particular that the use of the quantum wavelet transform gives a polynomial gain over classical computation. For the Husimi distribution, the gain is much larger than for the Wigner function, and is bigger with the help of amplitude amplification and wavelet transforms. We also apply the same set of techniques to the analysis of real images. The results show that the use of the quantum wavelet transform allows to lower dramatically the number of measurements needed, but at the cost of a large loss of information.Comment: Revtex, 13 pages, 16 figure

Toward a Robust Sparse Data Representation for Wireless Sensor Networks

Compressive sensing has been successfully used for optimized operations in wireless sensor networks. However, raw data collected by sensors may be neither originally sparse nor easily transformed into a sparse data representation. This paper addresses the problem of transforming source data collected by sensor nodes into a sparse representation with a few nonzero elements. Our contributions that address three major issues include: 1) an effective method that extracts population sparsity of the data, 2) a sparsity ratio guarantee scheme, and 3) a customized learning algorithm of the sparsifying dictionary. We introduce an unsupervised neural network to extract an intrinsic sparse coding of the data. The sparse codes are generated at the activation of the hidden layer using a sparsity nomination constraint and a shrinking mechanism. Our analysis using real data samples shows that the proposed method outperforms conventional sparsity-inducing methods.Comment: 8 page

Global integration of the Schr\"odinger equation: a short iterative scheme within the wave operator formalism using discrete Fourier transforms

A global solution of the Schr\"odinger equation for explicitly time-dependent Hamiltonians is derived by integrating the non-linear differential equation associated with the time-dependent wave operator. A fast iterative solution method is proposed in which, however, numerous integrals over time have to be evaluated. This internal work is done using a numerical integrator based on Fast Fourier Transforms (FFT). The case of a transition between two potential wells of a model molecule driven by intense laser pulses is used as an illustrative example. This application reveals some interesting features of the integration technique. Each iteration provides a global approximate solution on grid points regularly distributed over the full time propagation interval. Inside the convergence radius, the complete integration is competitive with standard algorithms, especially when high accuracy is required.Comment: 25 pages, 14 figure

Fast and Exact Spin-s Spherical Harmonic Transforms

We demonstrate a fast spin-s spherical harmonic transform algorithm, which is flexible and exact for band-limited functions. In contrast to previous work, where spin transforms are computed independently, our algorithm permits the computation of several distinct spin transforms simultaneously. Specifically, only one set of special functions is computed for transforms of quantities with any spin, namely the Wigner d-matrices evaluated at {\pi}/2, which may be computed with efficient recursions. For any spin the computation scales as O(L^3) where L is the band-limit of the function. Our publicly available numerical implementation permits very high accuracy at modest computational cost. We discuss applications to the Cosmic Microwave Background (CMB) and gravitational lensing.Comment: 22 pages, preprint format, 5 figure

First-principle molecular dynamics with ultrasoft pseudopotentials: parallel implementation and application to extended bio-inorganic system

We present a plane-wave ultrasoft pseudopotential implementation of first-principle molecular dynamics, which is well suited to model large molecular systems containing transition metal centers. We describe an efficient strategy for parallelization that includes special features to deal with the augmented charge in the contest of Vanderbilt's ultrasoft pseudopotentials. We also discuss a simple approach to model molecular systems with a net charge and/or large dipole/quadrupole moments. We present test applications to manganese and iron porphyrins representative of a large class of biologically relevant metallorganic systems. Our results show that accurate Density-Functional Theory calculations on systems with several hundred atoms are feasible with access to moderate computational resources.Comment: 29 pages, 4 Postscript figures, revtex

A Fast Poisson Solver of Second-Order Accuracy for Isolated Systems in Three-Dimensional Cartesian and Cylindrical Coordinates

We present an accurate and efficient method to calculate the gravitational potential of an isolated system in three-dimensional Cartesian and cylindrical coordinates subject to vacuum (open) boundary conditions. Our method consists of two parts: an interior solver and a boundary solver. The interior solver adopts an eigenfunction expansion method together with a tridiagonal matrix solver to solve the Poisson equation subject to the zero boundary condition. The boundary solver employs James's method to calculate the boundary potential due to the screening charges required to keep the zero boundary condition for the interior solver. A full computation of gravitational potential requires running the interior solver twice and the boundary solver once. We develop a method to compute the discrete Green's function in cylindrical coordinates, which is an integral part of the James algorithm to maintain second-order accuracy. We implement our method in the {\tt Athena++} magnetohydrodynamics code, and perform various tests to check that our solver is second-order accurate and exhibits good parallel performance.Comment: 24 pages, 13 figures; accepted for publication in ApJ