Related papers: Fourier spectral methods for nonlocal models
Fourier-encoded implicit neural representations (INRs) have shown strong capability in modeling continuous signals from discrete samples. However, conventional Fourier feature mappings use a fixed set of frequencies over the entire spatial…
Integral-equation-based fast direct solvers for electromagnetic scattering can substantially reduce computational costs, especially in the presence of multiple excitations. We recently proposed a new high-frequency fast direct solver…
In this work we propose a unified Fourier Spectral Transformer network that integrates the strengths of classical spectral methods and attention based neural architectures. By transforming the original PDEs into spectral ordinary…
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…
Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate…
Diffusion models provide a principled framework for generative modeling via stochastic differential equations and time-reversed dynamics. Extending spectral diffusion approaches to spherical data, however, raises nontrivial geometric and…
The topic of these notes could be easily expanded into a full one-semester course. Nevertheless, we shall try to give some flavour along with theoretical bases of spectral and pseudo-spectral methods. The main focus is made on Fourier-type…
We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new H\"ormander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties…
We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since…
Traditional resolvent analysis is a powerful framework for identifying the most amplified input-output structures in fluid flows from a stationary base state. Extending this resolvent analysis to periodic base flows poses computational…
The fast algorithms in Fourier optics have invigorated multifunctional device design and advanced imaging technologies. However, the necessity for fast computations has led to limitations in the widely used conventional Fourier methods,…
This paper presents a novel, efficient, high-order accurate, and stable spectral element-based model for computing the complete three-dimensional linear radiation and diffraction problem for floating offshore structures. We present a…
We give a new fast method for evaluating sprectral approximations of nonlinear polynomial functionals. We prove that the new algorithm is convergent if the functions considered are smooth enough, under a general assumption on the spectral…
Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an…
When using Laguerre and Hermite spectral methods to numerically solve PDEs in unbounded domains, the number of collocation points assigned inside the region of interest is often insufficient, particularly when the region is expanded or…
Nonlinear spectral problems arise across a range of fields, including mechanical vibrations, fluid-solid interactions, and photonic crystals. Discretizing infinite-dimensional nonlinear spectral problems often introduces significant…
In this note is presented a method, given nodal values on multidimensional nonconforming spectral elements, for calculating global Fourier-series coefficients. This method is ``exact'' in that given the approximation inherent in the…
We apply pseudo-spectral methods to integrate functional flow equations with high accuracy, extending earlier work on functional fixed point equations \cite{Borchardt:2015rxa}. The advantages of our method are illustrated with the help of…
Differentiable simulations of optical systems can be combined with deep learning-based reconstruction networks to enable high performance computational imaging via end-to-end (E2E) optimization of both the optical encoder and the deep…
In the paper, a novel algorithm employing pseudo-spectral approach is developed for the PKN model of hydrofracturing. The respective solvers based on this approach compute both the solution and its temporal derivative. In comparison with…