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Highly Oscillatory Problems: Computation, Theory and Application

15th January 2007 to 6th July 2007

Organisers: Professor B Engquist (Austin), Professor T Fokas (Cambridge), Professor E Hairer (Geneva) and Professor A Iserles (Cambridge)

Scientific Advisory Committee:: Professor S Chandler-Wilde (Reading), Professor T Hou (Caltech), Professor C Lubich (Tübingen) and Professor S Nørsett (Trondheim)

Programme Theme

High oscillation pervades a very wide range of applications: electromagnetics, fluid dynamics, molecular modelling, quantum chemistry, computerised tomography, plasma transport, celestial mechanics, medical imaging, signal processing. . . . It has been addressed by a wide range of mathematical techniques, inter alia from asymptotic theory, harmonic analysis, theory of dynamical systems, theory of integrable systems and differential geometry. The computation of highly oscillatory problems spawned a large number of different numerical approaches and algorithms. The purpose of this programme is to foster research into different aspects of high oscillation – the theoretical, the computational and the applied – from a united standpoint and to promote the synergy implicit in an interdisciplinary activity.

The immediate motivation for this programme originates in a number of recent important advances in theoretical and computational aspects of high oscillation, in particular

  • Great strides in the understanding of homogenization and averaging processes in highly oscillatory partial differential equations and the development of robust computational algorithms exploiting these phenomena.
  • Important new developments in asymptotic theory, e.g. exponential asymptotics.
  • Symplectic algorithms for the discretization of Hamiltonian problems and improved understanding of their long-term behaviour in the presence of oscillations.
  • Integral expansion methods (e.g. Magnus and Neumann) and exponential integrators for highly oscillatory initial-value differential equations.
  • Efficient methods for the computation of highly oscillatory integrals in one or more dimensions.

Each of these developments per se is highly significant, yet it falls short of providing us with a decisive answer to the full range of questions arising in highly oscillatory phenomena. In their totality they represent a genuine revolution in our understanding of high oscillation. This places tremendous added value on bringing all these developments together and weaving the different strands into a unified theory and practice.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons