Related papers: An $O(N^3)$ algorithm for the calculation of far f…
We consider the problem of finding the Discrete Fourier Transform (DFT) of $N-$ length signals with known frequency support of size $k$. When $N$ is a power of 2 and the frequency support is a spectral set, we provide an $O(k \log k)$…
Computation of the spherical harmonic rotation coefficients or elements of Wigner's d-matrix is important in a number of quantum mechanics and mathematical physics applications. Particularly, this is important for the Fast Multipole Methods…
This paper is concerned with a nonlinear imaging problem, which aims to reconstruct a locally perturbed, perfectly reflecting, infinite plane from intensity-only (or phaseless) far-field or near-field data. A recursive Newton iteration…
Scalar diffraction calculations such as the angular spectrum method (ASM) and Fresnel diffraction, are widely used in the research fields of optics, X-rays, electron beams, and ultrasonics. It is possible to accelerate the calculation using…
The article presents a computationally effective algorithm for calculating the multiresolution discrete Fourier transform (MrDFT). The algorithm is based on the idea of reducing the computational complexity which was introduced by Wen and…
We present a spectrally accurate fast algorithm for evaluating the solution to the scalar wave equation in free space driven by a large collection of point sources in a bounded domain. With $M$ sources temporally discretized by $N_t$ time…
Given $N_{\textrm{tot}}$ applications of a unitary operation with an unknown phase $\theta$, a large-scale fault-tolerant quantum system can {reduce} an estimate's {error} scaling from $\mathcal{O} \left[ 1 / \sqrt{N_{\textrm{tot}}}…
In this paper we present a new highly efficient calculation method for the far field amplitude pattern that arises from scattering problems governed by the d-dimensional Helmholtz equation and, by extension, Schr\"odinger's equation. The…
Nonlinear Fourier division Multiplexing (NFDM) can be realized from modulating the discrete nonlinear spectrum of an $N$-solitary waveform. To generate an $N$-solitary waveform from desired discrete spectrum (eigenvalue and discrete…
The graph fractional Fourier transform (GFRFT) for unitary graph Fourier transform (GFT) matrices can be interpreted through the scalar function $e^{j\alpha\theta}$ on the unit circle. Under the principal branch, its Fourier-series…
Advanced techniques using Neural Radiance Fields (NeRF), Signed Distance Fields (SDF), and Occupancy Fields have recently emerged as solutions for 3D indoor scene reconstruction. We introduce a novel two-phase learning approach, H2O-SDF,…
We present Factor Fields, a novel framework for modeling and representing signals. Factor Fields decomposes a signal into a product of factors, each represented by a classical or neural field representation which operates on transformed…
We outline a novel numerical method, called Ultrafast Ultrafast (UF$^2$), for calculating the $n^\text{th}$-order wavepackets required for calculating n-wave mixing signals. The method is simple to implement, and we demonstrate that it is…
Visibility information is critical in game AI applications, but the computational cost of raycasting-based methods poses a challenge for real-time systems. To address this challenge, we propose a novel method that represents a partitioned…
The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT…
The integro-differential formulation of the RTE and its solution by iterations on the source has been extended here to handle anisotropic scattering. The iterative part of the method is O(N ln N ), thanks to an efficient use of H-matrices.…
The one-dimensional (1D) fractional Fourier transform (FRFT) generalizes the Fourier transform, offering significant advantages in the time-frequency analysis of non-stationary signals. While various 2D extensions exist, such as the 2D…
The $O(N)$ model with scalar quartic interactions at its ultraviolet fixed point, and the $O(N)$ model with scalar cubic interactions at its infra-red fixed point are conjectured to be equivalent. This has been checked by comparing various…
The derivation of cosmological parameters from astrophysical data sets routinely involves operations counts which scale as O(N^3) where N is the number of data points. Currently planned missions, including MAP and Planck, will generate sky…
We describe recent progress in developing practical ab initio methods for which the computer effort is proportional to the number of atoms: linear scaling or O(N) methods. It is shown that the locality property of the density matrix gives a…