Fast polynomial transforms based on Toeplitz and Hankel matrices

Townsend, Alex; Webb, Marcus; Olver, Sheehan

doi:10.1090/mcom/3277

Cited by 57 publications

(55 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is known how to compute the Legendre expansion coefficients F leg from f in O(n 2 (log n) 2 log(1/ )) operations [31]. 5 Using the fact that [23, (18.7.9) & (18.9.7)] (j + 1 2 )P j (x) = (j + 1)(j + 2) (j + 3/2)C…”

Section: 4mentioning

confidence: 99%

Fast Poisson solvers for spectral methods

Fortunato

Townsend

2019

IMA Journal of Numerical Analysis

Self Cite

View full text Add to dashboard Cite

Poisson's equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here, we derive spectral methods for solving Poisson's equation on a square, cylinder, solid sphere, and cube that have an optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom required to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of finite difference matrices, our solver exploits a separated spectra property that holds for our spectral discretizations. Without parallelization, we can solve Poisson's equation on a square with 100 million degrees of freedom in under two minutes on a standard laptop. Throughout this paper, "optimal complexity" means a computational complexity that is optimal up to polylogarithmic factors. 1

show abstract

Section: 4mentioning

confidence: 99%

Fast Poisson solvers for spectral methods

Fortunato

Townsend

2019

IMA Journal of Numerical Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…If K 1 is the degree required to represent functions in x 1 and K 2 the degree for functions in x 2 , then O K 2 (K 1 + K 2 ) + K 3 operations are required to produce the expansion. We then apply the algorithm of [35] (as in one dimension) to convert each one-dimensional Chebyshev expansion into its corresponding Legendre expansion which costs an additional O K K 1 (log K 1 ) 2 + K 2 (log K 2 ) 2 operations. Compared with the O K 1 K 2 (K 1 + K 2 ) + K 3 2 operations of the SVD, this more sophisticated approach can be significantly less expensive when K is small compared to K 1 and K 2 .…”

Section: A Simple Spectral Methodsmentioning

confidence: 99%

An iterative domain decomposition, spectral finite element method on non-conforming meshes suitable for high frequency Helmholtz problems

Galagusz

McFee

2019

Journal of Computational Physics

View full text Add to dashboard Cite

The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree and non-conforming mesh refinement, including multiple hanging nodes per edge. Rather than globally assemble the finite element system, we describe an iterative domain decomposition method which can use element-wise fast solvers for elements of large degree. Since continuity between elements is enforced through moment equations, the resulting constraint equations are hierarchical. We show that, for high frequency problems, a subset of these constraints should be directly enforced, providing the coarse space in the dual-primal domain decomposition method. The subset of constraints is chosen based on a dispersion criterion involving mesh size and wavenumber. By increasing the size of the coarse space according to this criterion, the number of iterations in the domain decomposition method depends only weakly on the wavenumber. We demonstrate this convergence behaviour on standard domain decomposition test problems and conclude the paper with application of the method to electromagnetic problems in two dimensions. These examples include beam steering by lenses and photonic crystal waveguides, as well as radar cross section computation for dielectric, perfect electric conductor, and electromagnetic cloak scatterers.

show abstract

“…It is clear from the outset that it is some combination of regularity and decay at infinity for the function f which will determine the regularity of F -if for example f only decays algebraically at infinity, then F will be unbounded in (−1, 1)! With regards to computation, the first N Jacobi coefficients of a given function can be approximated in O N(log N) 2 operations using fast polynomial transform techniques described in [32]. An efficient and straightforward Julia implementation exists in the software package, APPROXFUN [25].…”

Section: Expansion Coefficientsmentioning

confidence: 99%

Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

Iserles

Webb

2019

Found Comput Math

Self Cite

View full text Add to dashboard Cite

Orthogonal systems in L 2 (R), once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation matrix is skewsymmetric, tridiagonal and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first N coefficients of the expansion can be computed to high accuracy in O (N log 2 N) operations. We consider two settings, one approximating a function f directly in (−∞, ∞) and the other approximatingIn each setting we prove that there is a single family, parametrised by α, β > −1, of orthogonal systems with a skew-symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where α, β = ±1/2 are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz-plus-Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials.

show abstract

Fast polynomial transforms based on Toeplitz and Hankel matrices

Cited by 57 publications

References 36 publications

Fast Poisson solvers for spectral methods

Fast Poisson solvers for spectral methods

An iterative domain decomposition, spectral finite element method on non-conforming meshes suitable for high frequency Helmholtz problems

Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

Contact Info

Product

Resources

About