Related papers: PDE propagation, sampling, and the Fourier ratio
In solving partial differential equations (PDEs), Fourier Neural Operators (FNOs) have exhibited notable effectiveness. However, FNO is observed to be ineffective with large Fourier kernels that parameterize more frequencies. Current…
This paper studies discretization of time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Most of the analysis in the literature has been performed on fully-discrete…
In this work, we investigate a numerical procedure for recovering a space-dependent diffusion coefficient in a (sub)diffusion model from the given terminal data, and provide a rigorous numerical analysis of the procedure. By exploiting…
In many applications in compressed sensing, the measurement matrix is a Fourier matrix, i.e., it measures the Fourier transform of the underlying signal at some specified `base' frequencies $\{u_i\}_{i=1}^M$, where $M$ is the number of…
Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar geometry. A…
Image signals typically are defined on a rectangular two-dimensional grid. However, there exist scenarios where this is not fulfilled and where the image information only is available for a non-regular subset of pixel position. For…
Photonic time-stretch overcomes the speed limitation of conventional digitizers, and enables the observation of non-repetitive and statistically rare phenomena that occur on short timescales. In most of the time-stretch applications, large…
Multi-channel short-time Fourier transform (STFT) domain-based processing of reverberant microphone signals commonly relies on power-spectral-density (PSD) estimates of early source images, where early refers to reflections contained within…
Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a…
We present an algorithm for the forward propagation of intervals through the discrete Fourier transform. The algorithm yields best-possible bounds when computing the amplitude of the Fourier transform for real and complex valued sequences.…
We propose and analyze an extended Fourier pseudospectral (eFP) method for the spatial discretization of the Gross-Pitaevskii equation (GPE) with low regularity potential by treating the potential in an extended window for its discrete…
Nonlinear and dispersive transmission impairments in coherent fiber-optic communication systems are often compensated by reverting the nonlinear Schr\"odinger equation, which describes the evolution of the signal in the link, numerically.…
We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our…
We analyze the propagation properties of the numerical versions of one and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both…
Low-count positron emission tomography (PET) reconstruction is a challenging inverse problem due to severe degradations arising from Poisson noise, photon scarcity, and attenuation correction errors. Existing deep learning methods typically…
Diffusion (score-based) generative models have been widely used for modeling various types of complex data, including images, audios, and point clouds. Recently, the deep connection between forward-backward stochastic differential equations…
Generative compressed sensing uses the range of a pretrained generator as a nonlinear model for recovering structured signals from limited measurements. We study a conditional version of this problem for image recovery from subsampled…
We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations for the solution of three dimensional frequency domain electromagnetic scattering problems with Perfectly Electric Conducting (PEC) boundary…
This paper deals with the problem of efficient sampling from a stochastic differential equation, given the drift function and the diffusion matrix. The proposed approach leverages a recent model for probabilities \cite{rudi2021psd} (the…
In the task of predicting spatio-temporal fields in environmental science using statistical methods, introducing statistical models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest.…