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B03
Numerics of Riemann-Hilbert Problems and Operator Determinants

The Search for the Optimal Contour

Riemann-Hilbert Problems (RHP) are another way of expressing equations satisfying a special property and have some advantages over the traditional forms. Take for example an equation describing the motion of a water wave and its current state: Both the traditional form and the RHP form of the equation enables us to calculate the state of the wave at any point in time. But with the RHP form we can accomplish this without knowing or calculating anything about the state of the wave in between.

Scientific Details+

Though Riemann-Hilbert Problems (RHP) offer some advantages over traditional forms of equations they also introduce new problems. In general, solving a RHP with a computer does not yield good results. As computers can only store finitely many digits of a number, storing a number in a computer always causes a small error. These small errors can add up to a substantial error during a calculation and in the end lead to a result which is far away from the correct solution. In the case of RHPs this happens regularly, but fortunately it is possible to avoid this situation.

A RHP depends on a set of lines in the plane, which is also called a contour. This contour can be modified in specific ways without changing the solution of the RHP. One can think of these modifications as moving the lines around in the plane while honouring some restrictions like ``no line may cross another line during the movement''. Although the solution of a RHP is not changed by these modifications they do have an effect on the calculation error that occurs when the RHP is solved with a computer. Some modifications will increase the error while others will decrease it. So the mission of this project is to develop an algorithm which is capable of determining for a given starting contour a modification of it which minimizes this calculation error. This modified contour can then be used to actually compute the correct solution of the RHP.

Publications+

Papers
  • Folkmar Bornemann, Peter J. Forrester, and Anthony Mays.
    Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros.
    Studies in Applied Mathematics, 138(4):401–437, 2017.
    doi:10.1111/sapm.12160.
  • Folkmar Bornemann.
    A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge.
    The Annals of Applied Probability, 26(3):1942–1946, 2016.
    arXiv:1504.00235, doi:10.1214/15-AAP1121.
  • F. Bornemann, A. Its, S. Olver, and G. Wechslberger.
    Numerical Methods for the Discrete Map $Z^a$.
    In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
    arXiv:1507.06805.
  • Folkmar Bornemann and Peter J Forrester.
    Singular values and evenness symmetry in random matrix theory.
    In Forum Mathematicum, volume 28, 873–891. 2016.
    arXiv:1503.07383, doi:10.1515/forum-2015-0055.
  • Folkmar Bornemann and Michael La Croix.
    The Singular Values of the GOE.
    Random Matrices: Theory Appl. 04, 1550009 (2015) (32 pages), June 2015.
    arXiv:1502.05946, doi:10.1142/S2010326315500094.
  • Benedict Dingfelder and J.A.C. Weideman.
    An improved Talbot method for numerical Laplace transform inversion.
    Numerical Algorithms, 68(1):167–183, 2015.
    arXiv:1304.2505, doi:10.1007/s11075-014-9895-z.
  • G. Wechslberger and F. Bornemann.
    Automatic deformation of Riemann-Hilbert problems with applications to the Painlevé II transcendents.
    Constr. Approx., 39(1):151–171, 2014.
    doi:10.1007/s00365-013-9199-x.
  • Georg Wechslberger and Folkmar Bornemann.
    Automatic Deformation of Riemann–Hilbert Problems with Applications to the Painlevé II Transcendents.
    Constructive Approximation, pages 1–21, June 2013.
    arXiv:1206.2446, doi:10.1007/s00365-013-9199-x.
  • N.S. Witte, F. Bornemann, and P.J. Forrester.
    Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles.
    Nonlinearity, Volume 26, Number 6, pp. 1799-1822, June 2013.
    arXiv:1209.2190, doi:10.1088/0951-7715/26/6/1799.

Books
  • Dominic Volland.
    A Discrete Hilbert Transform with Circle Packings.
    Springer Spektrum, Weisbaden, 2017. ISBN 978-3-658-20456-3/pbk; 978-3-658-20457-0/ebook.
    doi:10.1007/978-3-658-20457-0.

PhD thesis
  • G. Wechslberger.
    Automatic Contour Deformation of Riemann-Hilbert Problems.
    Dissertation, TU Munich, July 2015.
    dgd:192.

Team+

Prof. Dr. Folkmar Bornemann   +

University: TU München
E-Mail: bornemann[at]tum.de
Website: http://www-m3.ma.tum.de/Allgemeines/FolkmarBornemann

Dr. Georg Wechslberger   +

University: TU München
E-Mail: wechslbe[at]ma.tum.de