相关论文: Introduction to spectral methods
The normal mode model is one of the most popular approaches for solving underwater sound propagation problems. Among other methods, the finite difference method is widely used in classic normal mode programs. In many recent studies, the…
Spectral polynomial approximation of smooth functions allows real-time manipulation of and computation with them, as in the Chebfun system. Extension of the technique to two-dimensional and three-dimensional functions on hyperrectangles has…
We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional…
We consider capillary surfaces that are constructed by bounded generating curves. This class of surfaces includes radially symmetric and lower dimensional fluid-fluid interfaces. We use the arc-length representation of the differential…
The aim of the present work is to introduce a method based on Chebyshev polynomials for the numerical solution of a system of Cauchy type singular integral equations of the first kind on a finite segment. Moreover, an estimation error is…
In this paper, we design and analyze a novel spectral method for the subdiffusion equation. As it has been known, the solutions of this equation are usually singular near the initial time. Consequently, direct application of the traditional…
The Euclidean Algorithm is the often forgotten key to rational approximation techniques, including Taylor, Lagrange, Hermite, osculating, cubic spline, Chebyshev, Pade and other interpolation schemes. A unified view of these various…
The advection-diffusion and wave equations are the fundamental equations governing any physical law and therefore arise in many areas of physics and astrophysics. For complex problems and geometries, only numerical simulations can give…
We discuss the general method of Grushin problems, closely related to Shur complements, Feshbach projections and effective Hamiltonians, and describe various appearances in spectral theory, pdes, mathematical physics and numerical problems.
We propose and investigate two new methods to approximate $f({\bf A}){\bf b}$ for large, sparse, Hermitian matrices ${\bf A}$. The main idea behind both methods is to first estimate the spectral density of ${\bf A}$, and then find…
In this paper, we present an efficient fully spectral approximation scheme for exploring the one-dimensional steady-state neutron transport equation. Our methodology integrates the spectral-(Petrov-)Galerkin scheme in the spatial dimension…
Spectral approximation by polynomials on the unit ball is studied in the frame of the Sobolev spaces $W^{s}_p(\ball)$, $1<p<\infty$. The main results give sharp estimates on the order of approximation by polynomials in the Sobolev spaces…
Neural integral equations are deep learning models based on the theory of integral equations, where the model consists of an integral operator and the corresponding equation (of the second kind) which is learned through an optimization…
Classical approximation bases such as Chebyshev polynomials provide principled and interpretable representations, but their multivariate tensor-product constructions scale exponentially with dimension and impose axis-aligned structure that…
The purpose of this work is to study spectral methods to approximate the eigenvalues of nonlocal integral operators. Indeed, even if the spatial domain is an interval, it is very challenging to obtain closed analytical expressions for the…
Three-dimensional numerical models for underwater sound propagation are popular in computational ocean acoustics. For horizontally slowly varying waveguide environments, an adiabatic mode-parabolic equation hybrid theory can be used for…
Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a…
Spectroscopy is one of the most important tools that an astronomer has for studying the universe. This chapter begins by discussing the basics, including the different types of optical spectrographs, with extension to the ultraviolet and…
The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a harmonic point source. Its solution consists of a set of discrete modes radiating into the upper…
In a mixed generalized linear model, the goal is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two…