Fast algorithms using orthogonal polynomials

Abstract

We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal complexity, including in multivariate settings such as on triangles and for spherical harmonics. Ordinary and partial differential equations can be solved via sparse linear algebra when set up using orthogonal polynomials as a basis, provided that care is taken with the weights of orthogonality. A similar idea, together with low-rank approximation, gives an efficient method for solving singular integral equations. These techniques can be combined to produce high-performance codes for a wide range of problems that appear in applications.

Applying an iterative method numerically to solve n × n matrix Wiener–Hopf equations with exponential factors

“…The iterative method has been implemented in JULIA using APPROXFUN.JL [45] and SINGULARINTEGRALEQUATIONS.JL [33,46]. By precomputing the moments of the Cauchy transforms evaluated at points of interest the associated computational expense may be decoupled from the iterative procedure.…”

“…where n c is the number of basis functions used; the coefficients a j may be found via fast transforms available in APPROXFUN.JL [45]. The Cauchy transform (3.10) may then be computed via a related basis, the 'vanishing basis' of Chebyshev polynomials of the first kind, given by…”

Fast global spectral methods for three-dimensional partial differential equations

“…Whilst the efficient approximation of trivariate functions on cubes by combining tensorized Chebyshev interpolation and lowrank approximations has been studied in Chebfun3 [30] and Chebfun3F [15], there is no generalization of Chebop2 for three-dimensional PDEs on cubes. So far, the ideas of Chebop2 have been extended to triangular domains [47] and to disks [64,45]. For the special case of the Poisson equation with homogeneous Dirichlet boundary conditions, extensions to three-dimensional spheres [58], cylinders and unit cubes [22] have been developed.…”

“…The global spectral method can be used to compute solutions numerically to very high accuracy. It is not to be confused with so called spectral element methods [21,31,34] and p-and hp-finite element methods [2,45,66], where u is represented as sum of individual elements, each of which are approximated using truncated series expansions, instead of a global approximation of u. There also exist solvers relying on using domain decompositions in combination with truncated series expansions [28,48].…”

“…We denote the eigenvalue obtained for n by λ[n]. For both algorithms we plot |λ[n] − λ[45]| since the exact λ is not known.…”