## Publications

61 results found

Fasondini M, Olver S, Xu Y, 2023, Orthogonal polynomials on a class of planar algebraic curves, *Studies in Applied Mathematics*, Pages: 1-37, ISSN: 0022-2526

We construct bivariate orthogonal polynomials (OPs) on algebraiccurves of the form ym = φ(x) in R2 where m = 1, 2 and φ is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connectioncoefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrixis shown to be banded and the connection coefficients and Jacobi matrices for OPsof degree 0, . . . , N are computed via the Lanczos algorithm in O(Nd4) operations.

Fasondini M, Olver S, Xu Y, 2023, Orthogonal polynomials on planar cubic curves, *Foundations of Computational Mathematics*, Vol: 23, Pages: 1-31, ISSN: 1615-3375

Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to an appropriate weight function defined ona cubic curve, an explicit basis of orthogonal polynomials is constructed in termsof two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and squareroot singularities, and demonstrate their usage for solving differential equationswith singular solutions.

Gutleb TS, Carrillo JA, Olver S, 2022, Computation of power law equilibrium measures on balls of arbitrary dimension, *Constructive Approximation*, ISSN: 0176-4276

We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on d-dimensional ball domains, providing a substantial generalization of the work started in Gutleb et al. (Math Comput 9:2247–2281, 2022) for the one-dimensional case based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension d, in stark constrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension. We present several numerical experiments to showcase the accuracy and applicability of the method and discuss how our method compares with alternative numerical approaches and conjectured analytical solutions which exist for certain special cases. Finally, we discuss how our method can be used to explore the analytically poorly understood gap formation boundary to spherical shell support.

Gutleb T, Carrillo J, Olver S, 2022, Computing equilibrium measures with power law kernels, *Mathematics of Computation*, Vol: 91, Pages: 2247-2281, ISSN: 0025-5718

We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the formKpx ´ yq “ |x´y|αα ´|x´y|ββusing recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomialbases. The proposed method reduces what is na¨ıvely a difficult to approachoptimization problem over a measure space to a straightforward optimization problem over one or two variables fixing the support of the equilibriummeasure. The structure and rapid convergence properties of the obtained operators results in high computational efficiency in the individual optimizationsteps. We discuss stability and convergence of the method under a Tikhonovregularization and use an implementation to showcase comparisons with analytically known solutions as well as discrete particle simulations. Finally, wenumerically explore open questions with respect to existence and uniquenessof equilibrium measures as well as gap forming behaviour in parameter rangesof interest for power law kernels, where the support of the equilibrium measuresplits into two intervals.

Snowball B, Olver S, 2021, Sparse spectral methods for partial differential equations on spherical caps, *Transactions of Mathematics and Its Applications*, Vol: 5, Pages: 1-37, ISSN: 2398-4945

In recent years, sparse spectral methods for solving partial differential equationshave been derived using hierarchies of classical orthogonal polynomials on intervals,disks, disk-slices and triangles. In this work we extend the methodology to a hierarchyof non-classical multivariate orthogonal polynomials on spherical caps. The entries ofdiscretisations of partial differential operators can be effectively computed using for-mulae in terms of (non-classical) univariate orthogonal polynomials. We demonstratethe results on partial differential equations involving the spherical Laplacian and bihar-monic operators, showing spectral convergence with discretisations that can be madewell-conditioned using a simple preconditioner.

Olver S, Xu Y, 2021, Non-homogeneous wave equation on a cone, *Integral Transforms and Special Functions*, Vol: 32, Pages: 604-619, ISSN: 1065-2469

The wave equation(∂tt−c2∆x)u(x, t) =e−tf(x, t) in the cone{(x, t) :‖x‖ ≤t, x∈Rd, t∈R+}is shown to have a unique solution if u and its partial derivatives in x are in L2(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.

Webb M, Olver S, 2021, Spectra of Jacobi operators via connection coefficient matrices, *Communications in Mathematical Physics*, Vol: 382, Pages: 657-707, ISSN: 0010-3616

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.

Olver S, Xu Y, 2021, Orthogonal structure on a quadratic curve, *IMA Journal of Numerical Analysis*, Vol: 41, Pages: 206-246, ISSN: 0272-4979

Orthogonal polynomials on quadratic curves in the plane are studied.These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and twolines. For an integral with respect to an appropriate weight function defined onany quadratic curve, an explicit basis of orthogonal polynomials is constructedin terms of two families of orthogonal polynomials in one variable. Convergenceof the Fourier orthogonal expansions is also studied in each case. We discussapplications to the Fourier extension problem, interpolation of functions with sin-gularities or near singularities, and the solution of Schr ̈odinger’s equation withnon-differentiable or nearly-non-differentiable potentials.

Arakaki T, Bolewski J, Deits R, et al., 2020, JuliaPy/pyjulia: PyJulia v0.5.6

python interface to julia

Gutleb TS, Carrillo JA, Olver S, 2020, Computing equilibrium measures with power law Kernels, Publisher: arXiv

We introduce a method to numerically compute equilibrium measures forproblems with attractive-repulsive power law kernels of the form $K(x-y) =\frac{|x-y|^\alpha}{\alpha}-\frac{|x-y|^\beta}{\beta}$ using recursivelygenerated banded and approximately banded operators acting on expansions inultraspherical polynomial bases. The proposed method reduces what is naively adifficult to approach optimization problem over a measure space to astraightforward optimization problem over one or two variables fixing thesupport of the equilibrium measure. The structure and rapid convergenceproperties of the obtained operators results in high computational efficiencyin the individual optimization steps. We discuss stability and convergence ofthe method under a Tikhonov regularization and use an implementation toshowcase comparisons with analytically known solutions as well as discreteparticle simulations. Finally, we numerically explore open questions withrespect to existence and uniqueness of equilibrium measures as well as gapforming behaviour in parameter ranges of interest for power law kernels, wherethe support of the equilibrium measure splits into two intervals.

Olver S, Townsend A, Vasil G, 2020, Recurrence relations for a family of orthogonal polynomials on a triangle, ICOSAHOM 2018, Publisher: Springer Verlag, ISSN: 1439-7358

This paper derives sparse recurrence relations between orthogonal polynomials on a triangle and their partialderivatives, which are analogous to recurrence relations for Jacobi polynomials. We derive these recurrences in asystematic fashion by introducing ladder operators that map an orthogonal polynomial to another by incrementingor decrementing its associated parameters by one. We apply the results to efficiently calculating the Laplacian ofpolynomial approximations of functions on the triangle, using polynomial degrees in the thousands, i.e., millions ofdegrees of freedom.

Rackauckas C, Singhvi A, Ma Y, et al., 2020, SciML/DifferentialEquations.jl: v6.15.0

SciML/DifferentialEquations.jl: v6.15.0

Snowball B, Olver S, 2020, Sparse spectral and p-finite element methods for partial differential equations on disk slices and trapeziums, *Studies in Applied Mathematics*, Vol: 145, Pages: 3-35, ISSN: 0022-2526

Sparse spectral methods for solving partial differential equations have been derivedin 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-classicalorthogonal polynomials on disk slices and trapeziums. This builds on the observationthat 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 formulaein terms of (non-classical) univariate orthogonal polynomials. We apply the frameworkto solving the Poisson, variable coefficient Helmholtz, and Biharmonic equations. Inthis paper we focus on constant Dirichlet boundary conditions, as well as zero Dirichletand Neumann boundary conditions, with other types of boundary conditions requiringfuture work.

Gutleb T, Olver S, 2020, A sparse spectral method for Volterra integral equations using orthogonal polynomials on the triangle, *SIAM Journal on Numerical Analysis*, Vol: 58, Pages: 1993-2018, ISSN: 0036-1429

We introduce and analyze 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 or without known analytic solutions as well as an application-oriented numerical experiment. We prove convergence for both first and second kindproblems, where the former builds on connections with Toeplitz operators.

Sanders DP, Benet L, Agarwal K, et al., 2020, JuliaIntervals/IntervalArithmetic.jl: v0.17.5

IntervalArithmetic v0.17.5Diff since v0.17.4Closed issues:Underflow handling (#302)Display Interval{Float32} differently (#387)Merged pull requests:Completes ieee1788-constructors.jl (#386) (@krish8484)Completes mpfi.jl and fi.lib.jl (#390) (@krish8484)Display Interval{Float32} with f0 (#391) (@krish8484)

Olver S, Xu Y, 2020, Orthogonal polynomials in and on a quadratic surface of revolution, *Mathematics of Computation*, Vol: 89, Pages: 2848-2865, ISSN: 0025-5718

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

Olver S, Slevinsky RM, Townsend A, 2020, Fast algorithms using orthogonal polynomials, *Acta Numerica*, Vol: 29, Pages: 573-699, ISSN: 0962-4929

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.

Ho K, Olver S, Kelman T, et al., 2019, JuliaMatrices/LowRankApprox.jl: v0.4.0

JuliaMatrices/LowRankApprox.jl: v0.4.0

Olver S, Townsend A, Vasil G, 2019, A sparse spectral method on triangles, *SIAM Journal on Scientific Computing*, Vol: 41, Pages: A3728-A3756, ISSN: 1064-8275

In this paper, we demonstrate that many of the computational tools for univariateorthogonal 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 derivea practical spectral method for solving linear partial differential equations on triangles with sparsediscretizations. We can thereby rapidly solve partial differential equations using polynomials withdegrees in the thousands, resulting in sparse discretizations with as many as several million degreesof freedom.

Chen J, Olver S, Nash J, et al., 2019, JuliaMath/RandomMatrices.jl: v0.5.0

JuliaMath/RandomMatrices.jl: v0.5.0

Olver SS, Xu Y, 2019, Orthogonal structure on a wedge and on the boundary of a square, *Foundations of Computational Mathematics*, Vol: 19, Pages: 561-589, ISSN: 1615-3375

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.

Olver S, Ho K, Kelman T, et al., 2019, JuliaMatrices/LowRankApprox.jl: v0.2.3

This Julia package provides fast low-rank approximation algorithms for BLAS/LAPACK-compatible matrices based on some of the latest technology in adaptive randomized matrix sketching. Currently implemented algorithms include:sketch methods:random Gaussianrandom subsetsubsampled random Fourier transformsparse random Gaussianpartial range finderpartial factorizations:QR decompositioninterpolative decompositionsingular value decompositionHermitian eigendecompositionCUR decompositionspectral norm estimationBy "partial", we mean essentially that these algorithms are early-terminating, i.e., they are not simply post-truncated versions of their standard counterparts. There is also support for "matrix-free" linear operators described only through their action on vectors. All methods accept a number of options specifying, e.g., the rank, estimated absolute precision, and estimated relative precision of approximation.Our implementation borrows heavily from the perspective espoused by N. Halko, P.G. Martinsson, J.A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53 (2): 217-288, 2011., except that we choose the interpolative decomposition (ID) as our basic form of approximation instead of matrix range projection. The reason is that the latter requires expensive matrix-matrix multiplication to contract to the relevant subspace, while the former can sometimes be computed much faster, depending on the accelerated sketching strategy employed.This package has been developed with performance in mind, and early tests have shown large speedups over similar codes written in MATLAB and Python (and even some in Fortran and C). For example, computing an ID of a Hilbert matrix of order 1024 to relative precision ~1e-15 takes:~0.02 s using LowRankApprox in Julia~0.07 s using SciPy in Python (calling a Fortran backend; see PyMatrixID)~0.3 s in MATLABThis difference can be attributed in part t

Olver S, Ho K, Kelman T, et al., 2019, JuliaMatrices/LowRankApprox.jl: Fix rectangular psvd

Fast low-rank matrix approximation in Julia

Olver S, Ho K, Kelman T, et al., 2018, JuliaMatrices/LowRankApprox.jl: Fix compile time

Fast low-rank matrix approximation in Julia

Hale N, Olver SS, 2018, A fast and spectrally convergent algorithm for rational-order fractional integral and differential equations, *SIAM Journal on Scientific Computing*, Vol: 40, Pages: A2456-A2491, ISSN: 1064-8275

A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-orderfractional integral and differential equations is described. The approach is related to the ultraspherical method for ordinarydifferential equations [27], and involves constructing two different bases, one for the domain of the operator and one for therange of the operator. The bases are constructed from direct sums of suitably weighted ultraspherical or Jacobi polynomialexpansions, for which explicit representations of fractional integrals and derivatives are known, and are carefully chosen so thatthe resulting operators are banded or almost-banded. Geometric convergence is demonstrated for numerous model problemswhen the variable coefficients and right-hand side are sufficiently smooth.

Townsend A, Webb M, Olver S, 2018, Fast polynomial transforms based on Toeplitz and Hankel matrices, *Mathematics of Computation*, Vol: 87, Pages: 1913-1934, ISSN: 0025-5718

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 algorithms with an observed complexity of $ \smash {\mathcal {O}(N\log ^2 \! N)}$, 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.

Olver SS, Swan A, 2018, Evidence of the Poisson/Gaudin–Mehta phase transition for banded matrices on global scales, *Random Matrices: Theory and Applications*, Vol: 7, Pages: 1850002-1-1850002-21, ISSN: 2010-3263

We prove that the Poisson/Gaudin–Mehta phase transition conjectured to occur when the bandwidth of an N×N symmetric band matrix grows like b=N−−√ is naturally observable in the rate of convergence of the level density to the Wigner semi-circle law. Specifically, we show for periodic and non-periodic band matrices the rate of convergence of the fourth moment of the level density is independent of the boundary conditions in the localized regime b≪N−−√ with a rate of O(1b) for both cases, whereas in the delocalized regime b≫N−−√ where boundary effects become important, the rate of convergence for the two ensembles differs significantly, slowing to O(bN) for non-periodic band matrices. Additionally, we examine the case of thick non-periodic band matrices b=cN, showing that the fourth moment is maximally deviated from the Wigner semi-circle law when b=25N, and provide numerical evidence that the eigenvector statistics also exhibit critical behavior at this point.

Slevinsky RM, Olver S, 2017, A fast and well-conditioned spectral method for singular integral equations, *Journal of Computational Physics*, Vol: 332, Pages: 290-315, ISSN: 0021-9991

Pearson JW, Olver S, Porter MA, 2017, Numerical methods for the computation of the confluent and Gauss hypergeometric functions, *NUMERICAL ALGORITHMS*, Vol: 74, Pages: 821-866, ISSN: 1017-1398

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Vasil GM, Burns KJ, Lecoanet D,
et al., 2016, Tensor calculus in polar coordinates using Jacobi polynomials, *Journal of Computational Physics*, Vol: 325, Pages: 53-73, ISSN: 0021-9991

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