School of Mathematics and Statistics

The University of Sydney

NSW 2006

Australia

Telephone | +61 2 9351 5782 |

Sheehan.Olver@sydney.edu.au |

## Book: Riemann–Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions | |

Riemann–Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann–Hilbert problem. This book addresses the applied and computational theory of Riemann–Hilbert problems, includes an introduction to computational complex analysis, an introduction to the applied theory of Riemann–Hilbert problems from an analytical and numerical perspective, and a discussion of applications to integrable systems, differential equations, and special function theory. | |

T. Trogdon & S. Olver (2015), Riemann–Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions, SIAM. |

## Random matrix theory and orthogonal polynomials | |

The core of random matrix theory is spectral analysis of large random matrices. Such matrices are crucial to the study of large systems of particles that repulse each other. By developing numerical methods for complex analytical structures that underly random matrices, finite dimensional statistics and statistics of algebraic manipulations of random matrices are calculable. | |

- S. Olver & A. Townsend (2013), Fast inverse transform sampling in one and two dimensions, arXiv:1307.1223.
- S. Olver & N. Raj Rao (2012), Numerical computation of convolutions in free probability theory, arXiv:1203.1958v2.
- S. Olver, N. Raj Rao & T. Trogdon (2015), Sampling unitary ensembles,
*Rand. Mat.: Th. Appl.*,**4**: 1550002 - A. Townsend, T. Trogdon & S. Olver (2016), Fast computation of Gauss quadrature nodes and weights on the whole real line,
*IMA J. Numer. Anal.*,**36**: 337–358 - T. Trogdon & S. Olver (2016), A Riemann–Hilbert approach to Jacobi operators and Gaussian quadrature,
*IMA J. Numer. Anal.*,**36**: 174–196 - P. Deift, G. Menon, S. Olver & T. Trogdon (2014), Universality in numerical computations with random data,
*Proc. Nat. Acad. Sci.*,**111**: 14973–14978 - S. Olver & T. Trogdon (2014), Numerical solution of Riemann–Hilbert problems: random matrix theory and orthogonal polynomials,
*Const. Approx.*,**39**: 101–149. - S. Olver (2011), Computation of equilibrium measures,
*J. Approx. Theory*,**163**: 1185–1207. - T. Claeys & S. Olver (2012), Numerical study of higher order analogues of the Tracy–Widom distribution, in
*Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, Contemporary Mathematics*,**578**: 83–99.
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## Spectral methods | |

Spectral methods are numerical methods for solving differential and singular integral equations globally. They have the remarkable property that they converge to the true solution exponentially fast. By using specially constructed bases, spectral methods can be designed that involve only sparse, well-conditioned linear systems, allowing for efficient computations that require as many as a million unknowns. | |

- N. Hale & S. Olver (2016), A fast and spectrally convergent algorithm for fractional integral and differential equations with half-integer order terms, arXiv:1611.08028.
- A. Townsend, M. Webb & S. Olver (2016), Fast polynomial transforms based on Toeplitz and Hankel matrices, arXiv:1604.07486.
- R.M. Slevinsky & S. Olver (2017), A fast and well-conditioned spectral method for singular integral equations,
*J. Comp. Phys.*,**332**: 290–315. - G.M. Vasil,
K.J. Burns,
D. Lecoanet,
S. Olver,
B.P. Brown &
J.S. Oishi (2016), Tensor calculus in polar coordinates using Jacobi polynomials,
*J. Comp. Phys.*,**325**: 53–73 - A. Townsend & S. Olver (2015),
The automatic solution of partial differential equations using a global spectral method, in
*J. Comp. Phys.*,**299**: 106–123 - R.M. Slevinsky & S. Olver (2015), On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods,
*SIAM J. Sci. Comp.*,**37**: A676–A700 - S. Olver & A. Townsend (2013), A fast and well-conditioned spectral method,
*SIAM Review*,**55**: 462–489. - S. Olver (2009), On the convergence rate of a modified Fourier series,
*Maths Comp.*,**78**: 1629–1645. - S. Olver & A. Townsend (2014), A practical framework for infinite-dimensional linear algebra,
*Proceedings of the 1st First Workshop for High Performance Technical Computing in Dynamic Languages*, 57–62.
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## Integrable systems and Riemann–Hilbert problems | |

Important physical equations — including shallow water waves, nonlinear optics and others — have the property that they are integrable. One aspect of integrability is that the equations can be reduced to Riemann–Hilbert problems: boundary value problems in the complex plane. By solving Riemann–Hilbert problems numerically, solutions to integrable systems can be calculated accurately for arbitrarily large time. | |

- S. Olver (2014), Change of variable formulæ for regularizing slowly decaying and oscillatory Cauchy and Hilbert transforms,
*Anal. Appl.*,**12**: 369–384. - S. Olver & T. Trogdon (2014), Nonlinear steepest descent and the numerical solution of Riemann–Hilbert problems,
*Comm. Pure Appl. Maths*,**67**: 1353–1389. - T. Trogdon & S. Olver (2013), Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations,
*Proc. Royal Soc. A*,**469**: 20120330. - T. Trogdon, S. Olver & B. Deconinck (2012), Numerical inverse scattering for the Korteweg–de Vries and modified Korteweg–de Vries equations,
*Physica D*,**241**: 1003–1025. - S. Olver (2012), A general framework for solving Riemann–Hilbert problems numerically,
*Numer. Math.*,**122**: 305–340. - S. Olver (2011), Numerical solution of Riemann–Hilbert problems: Painlevé II,
*Found. Comput. Maths*,**11**: 153–179. - S. Olver (2011), Computing the Hilbert transform and its inverse,
*Maths Comp.*,**80**: 1745–1767. - F. Bornemann, A. Its, S. Olver & G. Wechslberger (2015), Numerical Methods for the Discrete Map Z
^{a}, to appear in: A. Bobenko,*Advances in Discrete Differential Geometry*, Springerâ€“Verlag.
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## Oscillatory integrals and differental equations, special functions | |

High oscillation plagues traditional numerical methods, because the oscillations must be resolved. These difficulties are avoidable by incorporating asymptotics into numerical schemes, so that the oscillations are completely removed. | |

- J.W. Pearson, S. Olver & M.A. Porter (2016), Numerical methods for the computation of the confluent and Gauss hypergeometric functions, to appear in
*Numer. Alg.* - D. Huybrechs & S. Olver (2012), Superinterpolation in highly oscillatory quadrature,
*Found. Comput. Maths*,**12**: 203–228. - S. Olver (2010), Fast, numerically stable computation of oscillatory integrals with stationary points,
*BIT Numer. Math.*,**50**: 149–171. - S. Olver (2010), Shifted GMRES for oscillatory integrals,
*Numer. Math.*,**114**: 607–628. - S. Olver (2009), GMRES for the differentiation operator,
*SIAM J. Numer. Anal.*,**47**: 3359–3373. - S. Olver (2007), Moment-free numerical approximation of highly oscillatory integrals with stationary points,
*Euro. J. Appl. Maths*,**18**: 435–447. - S. Olver (2007), Numerical approximation of vector-valued highly oscillatory integrals,
*BIT Numer. Math.*,**47**: 637–655. - S. Olver (2006), On the quadrature of multivariate highly oscillatory integrals over non-polytope domains,
*Numer. Math.*,**103**: 643–665. - S. Olver (2006), Moment-free numerical integration of highly oscillatory functions,
*IMA J. Numer. Anal.*,**26**: 213–227. - S. Olver (2015), Levin quadrature,
*Encyclopedia of Applied and Computational Mathematics*, B. Engquist (ed.), Springer, 785–786. - S. Olver (2011), GMRES for oscillatory matrix-valued differential equations,
*Spectral and High Order Methods for Partial Differential Equations*, J.S. Hesthaven and E. M. Rønquist (eds.), Springer, 267–274. - D. Huybrechs & S. Olver (2009), Highly oscillatory quadrature,
*Highly Oscillatory Problems*, London Mathematical Society Lecture Note Series 366, Cambridge University Press, 25–50. - D. Huybrechs & S. Olver (2009), Rapid function approximation by modified Fourier series,
*Highly Oscillatory Problems*, London Mathematical Society Lecture Note Series 366, Cambridge University Press, 51–71. - S. Olver (2007), Numerical quadrature of highly oscillatory integrals using derivatives,
*Algorithms for Approximation*, A. Iske and J. Levesley (eds.), Springer–Verlag, pp. 381–388. - A. Iserles, S. P. Nørsett & S. Olver (2006), Highly oscillatory quadrature: The story so far,
*Proceedings of ENuMath, Santiago de Compostela*, A. Bermudez de Castro*et al*, (eds.), Springer–Verlag, 97–118. - S. Olver (2008)
*Numerical Approximation of Highly Oscillatory Integrals*, PhD Thesis, University of Cambridge. - S. Olver (2006) Numerical approximation of highly oscillatory integrals, Smith-Knight/Rayleigh-Knight Essay, Class 1.
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- SingularIntegralEquations.jl, a
*Julia*package for solving singular integral equations - ApproxFun.jl, a
*Julia*package for function approximation, random number generation and solving differential equations - RHPackage, a
*Mathematica*package for computing solutions to Riemann–Hilbert problems

*ApproxFun*and solution of differential equations in*Julia*- Numerical free probability
- Numerical calculation of random matrix distributions and orthogonal polynomials
- Numerical inverse scattering for Korteweg–de Vries and nonlinear Schrödinger equations
- Numerical solution of Riemann–Hilbert problems: Painlevé II
- Numerically approximation of highly oscillatory integrals, Leslie Fox Second Prize Talk

- MATH3076/3976 Mathematical Computing
- Computational Projects in Applied Mathematics
- Numerical Complex Analysis