arXiv subjects

Sheehan Olver

Publications and source records attributed to Sheehan Olver.

49 records · Page 3Linked to original sources

arXiv2014

Fast computation of Gauss quadrature nodes and weights on the whole real line

A fast and accurate algorithm for the computation of Gauss-Hermite and generalized Gauss-Hermite quadrature nodes and weights is presented. The algorithm is based on Newton's method with carefully selected initial guesses for the nodes and a fast evaluation scheme for the associated orthogonal polynomial. In the Gauss-Hermite case the initial guesses and evaluation scheme rely on explicit asymptotic formulas. For generalized Gauss-Hermite, the initial guesses are furnished by sampling a certain equilibrium measure and the associated polynomial evaluated via a Riemann-Hilbert reformulation. In both cases the $n$-point quadrature rule is computed in $\mathcal{O}(n)$ operations to an accuracy that is close to machine precision. For sufficiently large $n$, some of the quadrature weights have a value less than the smallest positive normalized floating-point number in double precision and we exploit this fact to achieve a complexity as low as $\mathcal{O}(\sqrt{n})$.

math.NA↗

arXiv2014

A practical framework for infinite-dimensional linear algebra

We describe a framework for solving a broad class of infinite-dimensional linear equations, consisting of almost banded operators, which can be used to resepresent linear ordinary differential equations with general boundary conditions. The framework contains a data structure on which row operations can be performed, allowing for the solution of linear equations by the adaptive QR approach. The algorithm achieves $O(n^{\rm opt})$ complexity, where $n^{\rm opt}$ is the number of degrees of freedom required to achieve a desired accuracy, which is determined adaptively. In addition, special tensor product equations, such as partial differential equations on rectangles, can be solved by truncating the operator in the $y$-direction with $n_y$ degrees of freedom and using a generalized Schur decomposition to upper triangularize, before applying the adaptive QR approach to the $x$-direction, requiring $O(n_y^2 n_x^{\rm opt})$ operations. The framework is implemented in the ApproxFun package written in the Julia programming language, which achieves highly competitive computational costs by exploiting unique features of Julia.

math.NA↗

arXiv2014

Universality in Numerical Computations with Random Data. Case Studies

The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the halting time, i.e., the histogram for the halting times, centered by the sample average and scaled by the sample variance, collapses to a universal curve, independent of the input data distribution, as the dimension increases. Thus, up to two components, the sample average and the sample variance, the statistics for the halting time are universally prescribed. The case studies include six standard numerical algorithms, as well as a model of neural computation and decision making. A link to relevant software is provided in for the reader who would like to do computations of his'r own.

math.NA↗

arXiv2014

On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods

We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the $\sinh$ map, and allows the treatment of a finite number of singularities in the complex plane. In the case where locations are unknown, the so-called Sinc-Padé approximants are used to provide approximate results. This adaptive method is shown to have almost the same convergence properties. We use the conformal maps to generate high accuracy solutions to several challenging integrals, nonlinear waves, and multidimensional integrals.

math.NA↗

arXiv2014

Sampling unitary invariant ensembles

We develop an algorithm for sampling from the unitary invariant random matrix ensembles. The algorithm is based on the representation of their eigenvalues as a determinantal point process whose kernel is given in terms of orthogonal polynomials. Using this algorithm, statistics beyond those known through analysis are calculable through Monte Carlo simulation. Unexpected phenomena are observed in the simulations.

math-ph↗

arXiv2013

A Riemann--Hilbert approach to Jacobi operators and Gaussian quadrature

The computation of the entries of Jacobi operators associated with orthogonal polynomials has important applications in numerical analysis. From truncating the operator to form a Jacobi matrix, one can apply the Golub--Welsh algorithm to compute the Gaussian quadrature weights and nodes. Furthermore, the entries of the Jacobi operator are the coefficients in the three-term recurrence relationship for the polynomials. This provides an efficient method for evaluating the orthogonal polynomials. Here, we present an $\mathcal O(N)$ method to compute the first $N$ rows of Jacobi operators from the associated weight. The method exploits the Riemann--Hilbert representation of the polynomials by solving a deformed Riemann--Hilbert problem numerically. We further adapt this computational approach to certain entire weights that are beyond the reach of current asymptotic Riemann--Hilbert techniques.

math.NA↗

arXiv2013

Change of variable formulæ for regularizing slowly decaying and oscillatory Cauchy and Hilbert transforms

Formulæ are derived for expressing Cauchy and Hilbert transforms of a function $f$ in terms of Cauchy and Hilbert transforms of $f(x^r)$. When $r$ is an integer, this corresponds to evaluating the Cauchy transform of $f(x^r)$ at all choices of $z^{1/r}$. Related formuæ for rational $r$ result in a reduction to a generalized Cauchy transform living on a Riemann surface, which in turn is reducible to the standard Cauchy transform. These formulæ are used to regularize the behaviour of functions that are slowly decaying or oscillatory, in order to facilitate numerical computation and extend asymptotic results.

math.CV↗

arXiv2013

Numerical computation of convolutions in free probability theory

We develop a numerical approach for computing the additive, multiplicative and compressive convolution operations from free probability theory. We utilize the regularity properties of free convolution to identify (pairs of) `admissible' measures whose convolution results in a so-called `invertible measure' which is either a smoothly-decaying measure supported on the entire real line (such as the Gaussian) or square-root decaying measure supported on a compact interval (such as the semi-circle). This class of measures is important because these measures along with their Cauchy transforms can be accurately represented via a Fourier or Chebyshev series expansion, respectively. Thus, knowledge of the functional inverse of their Cauchy transform suffices for numerically recovering the invertible measure via a non-standard yet well-behaved Vandermonde system of equations. We describe explicit algorithms for computing the inverse Cauchy transform alluded to and recovering the associated measure with spectral accuracy. Convergence is guaranteed under broad assumptions on the input measures.

math.PR↗

arXiv2013

Fast inverse transform sampling in one and two dimensions

We develop a computationally efficient and robust algorithm for generating pseudo-random samples from a broad class of smooth probability distributions in one and two dimensions. The algorithm is based on inverse transform sampling with a polynomial approximation scheme using Chebyshev polynomials, Chebyshev grids, and low rank function approximation. Numerical experiments demonstrate that our algorithm outperforms existing approaches.

math.NA↗

arXiv2012

Numerical solution of Riemann--Hilbert problems: random matrix theory and orthogonal polynomials

In recent developments, a general approach for solving Riemann--Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with an approach to compute Fredholm determinants, we are able to calculate level densities and gap statistics for general finite-dimensional unitary ensembles. We also include a description of how to compute the Hastings--McLeod solution of the homogeneous Painlevé II equation.

math-ph↗

arXiv2012

A fast and well-conditioned spectral method

A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes O(m^2n) operations, where m is the number of Chebyshev points needed to resolve the coefficients of the differential operator and n is the number of Chebyshev coefficients needed to resolve the solution to the differential equation. We prove stability of the method by relating it to a diagonally preconditioned system which has a bounded condition number, in a suitable norm. For Dirichlet boundary conditions, this implies stability in the standard 2-norm. An adaptive QR factorization is developed to efficiently solve the resulting linear system and automatically choose the optimal number of Chebyshev coefficients needed to represent the solution. The resulting algorithm can efficiently and reliably solve for solutions that require as many as a million unknowns.

math.NA↗

arXiv2012

Nonlinear steepest descent and the numerical solution of Riemann-Hilbert problems

The effective and efficient numerical solution of Riemann-Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann-Hilbert problems, the resulting numerical methods have been shown numerically to retain accuracy as values of certain parameters become arbitrarily large. Remarkably, this numerical approach does not require knowledge of local parametrices; rather, the deformed contour is scaled near stationary points at a specific rate. The primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved. To do so, we first construct a general theoretical framework for the numerical solution of Riemann-Hilbert problems. Second, we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes. In particular, we prove sufficient conditions for numerical methods to retain accuracy. Finally, we compute solutions to the homogeneous Painlevé II equation and the modified Korteweg-de Vries equations to explicitly demonstrate the practical validity of the theory.

math.NA↗

arXiv2011

Numerical study of higher order analogues of the Tracy-Widom distribution

We study a family of distributions that arise in critical unitary random matrix ensembles. They are expressed as Fredholm determinants and describe the limiting distribution of the largest eigenvalue when the dimension of the random matrices tends to infinity. The family contains the Tracy-Widom distribution and higher order analogues of it. We compute the distributions numerically by solving a Riemann-Hilbert problem numerically, plot the distributions, and discuss several properties that they appear to exhibit.

math-ph↗