arXiv subjects

Sheehan Olver

Publications and source records attributed to Sheehan Olver.

At least 37 records · Page 2Linked to original sources

arXiv2020

Orthogonal polynomials on planar cubic curves

Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions.

math.NA↗

arXiv2020

Spectra of Jacobi operators via connection coefficient matrices

We address the computational spectral theory of Jacobi operators that are compact perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In particular, for finite-rank perturbation we show that the computation of the spectrum can be reduced to a polynomial root finding problem, from a polynomial that is derived explicitly from the entries of a connection coefficient matrix. A formula for the spectral measure of the operator is also derived explicitly from these entries. The analysis is extended to trace-class perturbations. We address issues of computability in the framework of the Solvability Complexity Index, proving that the spectrum of compact perturbations of the free Jacobi operator is computable in finite time with guaranteed error control in the Hausdorff metric on sets.

math.SP↗

arXiv2020

Non-homogeneous wave equation on a cone

The wave equation $\left(\partial_{tt} - c^2 Δ_x\right) u(x,t) = e^{-t} f(x,t)$ is shown to have a unique solution if $u$ and its partial derivatives in $x$ are in $L^2(e^{-t})$ on the cone, and the solution can be explicit given in the Fourier series of orthogonal polynomials on the cone. This provides a particular solution for the boundary value problems of the non-homogeneous wave equation on the cone, which can be combined with a solution to the homogeneous wave equation in the cone to obtain the full solution.

math.CA↗

arXiv2020

Sparse spectral and p-finite element methods for partial differential equations on disk slices and trapeziums

Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work we extend this methodology to a hierarchy of non-classical orthogonal polynomials on disk slices (e.g. a half-disk) and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being defined by an algebraic curve, and that the entries of partial differential operators can be determined using formulae in terms of (non-classical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and Biharmonic equations.

math.NA↗

arXiv2020

Orthogonal structure on a quadratic curve

Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. As an application, we see that the resulting bases can be used to interpolate functions on the real line with singularities of the form $|x|$, $\sqrt{x^2+ ε^2}$, or $1/x$, with exponential convergence.

math.NA↗

arXiv2019

Orthogonal polynomials in and on a quadratic surface of revolution

We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

math.CA↗

arXiv2019

A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle

We introduce and analyse a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to achieve high efficiency and exponential convergence. The discussion is followed by a demonstration of the method on example Volterra integral equations of the first and second kind with known analytic solutions as well as an application-oriented numerical experiment. We prove convergence for both first and second kind problems, where the former builds on connections with Toeplitz operators.

math.NA↗

arXiv2019

A sparse spectral method on triangles

In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse differentiation operators. This allows us to derive a practical spectral method for solving linear partial differential equations on triangles with sparse discretizations. We can thereby rapidly solve partial differential equations using polynomials with degrees in the thousands, resulting in sparse discretizations with as many as several million degrees of freedom.

math.NA↗

arXiv2018

Orthogonal structure on a wedge and on the boundary of a square

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.

math.CA↗

arXiv2018

Recurrence relations for orthogonal polynomials on a triangle

This paper derives sparse recurrence relations between orthogonal polynomials on a triangle and their partial derivatives, which are analogous to recurrence relations for Jacobi polynomials. We derive these recurrences in a systematic fashion by introducing ladder operators that map an orthogonal polynomial to another by incrementing or decrementing its associated parameters by one.

math.CA↗

arXiv2017

A fast and spectrally convergent algorithm for rational-order fractional integral and differential equations

A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for ordinary differential equations [Olver & Townsend 2013], and involves constructing two different bases, one for the domain of the operator and one for the range of the operator. The bases are constructed from direct sums of suitably weighted ultraspherical or Jacobi polynomial expansions, for which explicit representations of fractional integrals and derivatives are known, and are carefully chosen so that the resulting operators are banded or almost-banded. Geometric convergence is demonstrated for numerous model problems when the variable coefficients and right-hand side are sufficiently smooth.

math.NA↗

arXiv2017

Evidence of the Poisson/Gaudin-Mehta phase transition for banded matrices on global scales

We prove that the Poisson/Gaudin--Mehta phase transition conjectured to occur when the bandwidth of an $N \times N$ symmetric banded matrix grows like $\sqrt N$ is observable as a critical point in the fourth moment of the level density for a wide class of symmetric banded matrices. A second critical point when the bandwidth grows like ${2 \over 5} N$ leads to a new conjectured phase transition in the eigenvalue localization, whose existence we demonstrate in numerical experiments.

math.PR↗

arXiv2016

A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface.

math.NA↗

arXiv2016

Tensor calculus in polar coordinates using Jacobi polynomials

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r=0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.

math.NA↗

arXiv2016

Fast polynomial transforms based on Toeplitz and Hankel matrices

Many standard conversion matrices between coefficients in classical orthogonal polynomial expansions can be decomposed using diagonally-scaled Hadamard products involving Toeplitz and Hankel matrices. This allows us to derive $\smash{\mathcal{O}(N(\log N)^2)}$ algorithms, based on the fast Fourier transform, for converting coefficients of a degree $N$ polynomial in one polynomial basis to coefficients in another. Numerical results show that this approach is competitive with state-of-the-art techniques, requires no precomputational cost, can be implemented in a handful of lines of code, and is easily adapted to extended precision arithmetic.

math.NA↗

arXiv2015

Numerical Methods for the Computation of the Confluent and Gauss Hypergeometric Functions

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss-Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide 'roadmaps' with our recommendation for which methods should be used in each situation.

math.NA↗

arXiv2015

Numerical Methods for the Discrete Map $Z^a$

As a basic example in nonlinear theories of discrete complex analysis, we explore various numerical methods for the accurate evaluation of the discrete map $Z^a$ introduced by Agafonov and Bobenko. The methods are based either on a discrete Painlevé equation or on the Riemann-Hilbert method. In the latter case, the underlying structure of a triangular Riemann-Hilbert problem with a non-triangular solution requires special care in the numerical approach. Complexity and numerical stability are discussed, the results are illustrated by numerical examples

math.NA↗

arXiv2015

The automatic solution of partial differential equations using a global spectral method

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank $2$, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree $(n_x,n_y)$ is computed in $\mathcal{O}((n_x n_y)^{3/2})$ operations. Partial differential operators of splitting rank $\geq 3$ are solved via a linear system involving a block-banded matrix in $\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3}))$ operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under $60$ seconds. An experimental implementation in the Julia language can currently perform the same solve in $10$ seconds.

math.NA↗