Related papers: Introduction to spectral methods
We derive, similar to Lau and Riha, a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the…
These notes are meant as an introduction to the theory of nonlinear spectral theory. We will discuss the variational form of nonlninear eigenvalue problems and the corresponding non-linear Euler--Lagrange equations, as well as connections…
Many common methods for data analysis rely on linear algebra. We provide new results connecting data analysis error to numerical accuracy, which leads to the first meaningful stopping criterion for two way spectral partitioning. More…
Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by…
In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, ``[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless…
The calculation of a three-dimensional underwater acoustic field has always been a key problem in computational ocean acoustics. Traditionally, this solution is usually obtained by directly solving the acoustic Helmholtz equation using a…
An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
We have developed a method for constructing spectral approximations for convolution operators of Fredholm type. The algorithm we propose is numerically stable and takes advantage of the recurrence relations satisfied by the entries of such…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…
How do the geometric properties of a domain impact the spectrum of an operator defined on it? How do we compute accurate and reliable approximations of these spectra? The former question is studied in spectral geometry, and the latter is a…
This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss-Hermite…
In this paper, we propose a sparse spectral-Galerkin approximation scheme for solving the second-order partial differential equations on an arbitrary tetrahedron. Generalized Koornwinder polynomials are introduced on the reference…
In this paper we discuss the stability of an alternative pollution-free procedure for computing spectra. The main difference with the Galerkin method lies in the fact that it gives rise to a weak approximate problem which is quadratic in…
Physicists routinely need probabilistic models for a number of tasks such as parameter inference or the generation of new realizations of a field. Establishing such models for highly non-Gaussian fields is a challenge, especially when the…
We analyze randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied include the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for these tasks. Our…
Fractional $q$-extensions of some classical $q$-orthogonal polynomials are introduced and some of the main properties of the new defined functions are given. Next, a fractional $q$-difference equation of Gauss type is introduced and solved…
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…
We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems for functional differential equations. In particular, we prove that the numerical collocation solution…
This is to review some recent progress in PDE. The emphasis is on (energy) supercritical nonlinear Schr\"odinger equations. The methods are applicable to other nonlinear equations.
We study the elastic time-harmonic wave scattering problems on unbounded domains with boundaries composed of finite collections of disjoints finite open arcs (or cracks) in two dimensions. Specifically, we present a fast spectral Galerkin…