Dr Sheehan Olver

F07 - Carslaw Building
The University of Sydney

Telephone 9351 5782
Fax 9351 4534

Website Personal web page

Research interests

· Integrable systems

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.

· Random matrix theory

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.

· Spectral methods

Spectral methods are numerical methods for solving differential 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.

· Oscillatory integrals and differental equations

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.

Teaching and supervision

Timetable

Awards and honours

2012 Adams Prize

2012 Carl–Erik Fröberg Prize

2013 Cherry Ripe Prize

Selected grants

2013

  • Numerical Methods for Inverse-Scattering and Stability of Nonlinear Waves; Miller P, Olver S; DVC Research/International Research Collaboration Award (IRCA).
  • A new class of fast and reliable spectral methods for partial differential equations; Olver S, Olver S; Australian Research Council (ARC)/Discovery Early Career Researcher Award (DECRA).

Selected publications

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Book Chapters

  • Claeys, T., Olver, S. (2012). Numerical study of higher order analogues of the Tracy-Widom distribution. In J. Arvesu, G. Lopez Lagomasino (Eds.), Contemporary Mathematics: Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, (pp. 83-98). Providence, Rhode Island, USA: American Mathematical Society.

Journals

  • Olver, S., Trogdon, T. (2014). Numerical Solution of Riemann–Hilbert Problems: Random Matrix Theory and Orthogonal Polynomials. Constructive Approximation, 39(1), 101-149. [More Information]
  • Olver, S., Townsend, A. (2013). A Fast and Well-Conditioned Spectral Method. SIAM Review, 55(3), 462-489. [More Information]
  • Trogdon, T., Olver, S. (2013). Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 469(2149), 1-22. [More Information]
  • Olver, S. (2012). A general framework for solving Riemann-Hilbert problems numerically. Numerische Mathematik, 122(2), 305-340. [More Information]
  • Trogdon, T., Olver, S., Deconinck, B. (2012). Numerical inverse scattering for the Korteweg-de Vries and modified Korteweg-de Vries equations. Physica D: Nonlinear Phenomena, 241(11), 1003-1025. [More Information]
  • Huybrechs, D., Olver, S. (2012). Superinterpolation in highly oscillatory quadrature. Foundations of Computational Mathematics, 12, 203-228. [More Information]
  • Olver, S. (2011). Computation of equilibrium measures. Journal Of Approximation Theory, 163(9), 1185-1207. [More Information]
  • Olver, S. (2011). Computing the Hilbert transform and its inverse. Mathematics of Computation, 80(275), 1745-1767.
  • Olver, S. (2011). Numerical Solution of Riemann–Hilbert Problems: Painlevé II. Foundations of Computational Mathematics, 11, 153-179. [More Information]
  • Olver, S. (2010). Fast, numerically stable computation of oscillatory integrals with stationary points. Bit (Lisse): numerical mathematics, 50(1), 149-171. [More Information]
  • Olver, S. (2010). Shifted GMRES for oscillatory integrals. Numerische Mathematik, 114, 607-628. [More Information]
  • Olver, S. (2009). GMRES for the differentiation operator. SIAM Journal on Numerical Analysis, 47(5), 3359-3373. [More Information]
  • Olver, S. (2009). On the convergence rate of a modified Fourier series. Mathematics of Computation, 78(267), 1629-1645.
  • Olver, S. (2007). Moment-free numerical approximation of highly oscillatory integrals with stationary points. European Journal of Applied Mathematics, 18(4), 435-447. [More Information]
  • Olver, S. (2007). Numerical approximation of vector-valued highly oscillatory integrals. Bit (Lisse): numerical mathematics, 47(3), 637-655. [More Information]
  • Olver, S. (2006). Moment-free numerical integration of highly oscillatory functions. IMA Journal of Numerical Analysis, 26, 213-227. [More Information]
  • Olver, S. (2006). On the quadrature of multivariate highly oscillatory integrals over non-polytope domains. Numerische Mathematik, 103, 643-665. [More Information]

2014

  • Olver, S., Trogdon, T. (2014). Numerical Solution of Riemann–Hilbert Problems: Random Matrix Theory and Orthogonal Polynomials. Constructive Approximation, 39(1), 101-149. [More Information]

2013

  • Olver, S., Townsend, A. (2013). A Fast and Well-Conditioned Spectral Method. SIAM Review, 55(3), 462-489. [More Information]
  • Trogdon, T., Olver, S. (2013). Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 469(2149), 1-22. [More Information]

2012

  • Olver, S. (2012). A general framework for solving Riemann-Hilbert problems numerically. Numerische Mathematik, 122(2), 305-340. [More Information]
  • Trogdon, T., Olver, S., Deconinck, B. (2012). Numerical inverse scattering for the Korteweg-de Vries and modified Korteweg-de Vries equations. Physica D: Nonlinear Phenomena, 241(11), 1003-1025. [More Information]
  • Claeys, T., Olver, S. (2012). Numerical study of higher order analogues of the Tracy-Widom distribution. In J. Arvesu, G. Lopez Lagomasino (Eds.), Contemporary Mathematics: Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, (pp. 83-98). Providence, Rhode Island, USA: American Mathematical Society.
  • Huybrechs, D., Olver, S. (2012). Superinterpolation in highly oscillatory quadrature. Foundations of Computational Mathematics, 12, 203-228. [More Information]

2011

  • Olver, S. (2011). Computation of equilibrium measures. Journal Of Approximation Theory, 163(9), 1185-1207. [More Information]
  • Olver, S. (2011). Computing the Hilbert transform and its inverse. Mathematics of Computation, 80(275), 1745-1767.
  • Olver, S. (2011). Numerical Solution of Riemann–Hilbert Problems: Painlevé II. Foundations of Computational Mathematics, 11, 153-179. [More Information]

2010

  • Olver, S. (2010). Fast, numerically stable computation of oscillatory integrals with stationary points. Bit (Lisse): numerical mathematics, 50(1), 149-171. [More Information]
  • Olver, S. (2010). Shifted GMRES for oscillatory integrals. Numerische Mathematik, 114, 607-628. [More Information]

2009

  • Olver, S. (2009). GMRES for the differentiation operator. SIAM Journal on Numerical Analysis, 47(5), 3359-3373. [More Information]
  • Olver, S. (2009). On the convergence rate of a modified Fourier series. Mathematics of Computation, 78(267), 1629-1645.

2007

  • Olver, S. (2007). Moment-free numerical approximation of highly oscillatory integrals with stationary points. European Journal of Applied Mathematics, 18(4), 435-447. [More Information]
  • Olver, S. (2007). Numerical approximation of vector-valued highly oscillatory integrals. Bit (Lisse): numerical mathematics, 47(3), 637-655. [More Information]

2006

  • Olver, S. (2006). Moment-free numerical integration of highly oscillatory functions. IMA Journal of Numerical Analysis, 26, 213-227. [More Information]
  • Olver, S. (2006). On the quadrature of multivariate highly oscillatory integrals over non-polytope domains. Numerische Mathematik, 103, 643-665. [More Information]

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