Highly oscillatory problems: computation, theory and application

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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.

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering Authors: Ivan Graham, V Dominguez, VP Smyshlyaev Download PDF	2006	HOP
From high oscillation to rapid approximation III: Multivariate expansions Authors: Arieh Iserles, Syvert Norsett Download PDF	2006	HOP
Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions Authors: Ernst Hairer, Christian Lubich, D Cohen Download PDF	2006	HOP
Stabilized FEM-BEM coupling for Maxwell transmission problems Authors: P Meury, R Hiptmair Download PDF	2006	HOP
Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves Authors: Enrique Zuazua, Sylvain Ervedoza Download PDF	2006	HOP
Spectral semi-discretisations of weakly nonlinear wave equations over long times Authors: Christian Lubich, Ernst Hairer Download PDF	2006	HOP
Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations Authors: Christian Lubich, D Cohen, Ernst Hairer Download PDF	2006	HOP
Convergence of a two-grid algorithm for the control of the wave equation Authors: Liviu Ignat, Enrique Zuazua Download PDF	2006	HOP
Stability of the periodic Toda lattice under short range perturbations Authors: Spyridon Kamvissis, G Teschl Download PDF	2006	HOP
Finite-order integration weights can be dangerous Authors: Ian Sloan Download PDF	2006	HOP
On multisymplecticity of partitioned Runge-Kutta methods Authors: Robert McLachlan, BN Ryland Download PDF	2006	HOP
An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks Authors: F Palacios, Enrique Zuazua, C Castro Download PDF	2006	HOP
Wave-number-explicit bounds in time-harmonic scattering Authors: Peter Monk, Simon Chandler-Wilde Download PDF	2006	HOP
Achieving Brouwer's law with implicit Runge-Kutta methods Authors: Robert McLachlan, A Razakarivony, Ernst Hairer Download PDF	2006	HOP
A multiscale method for highly oscillatory ordinary differential equations with resonance Authors: Bjorn Engquist, Yen-Hsi Tsai, Gil Ariel Download PDF	2006	HOP
Finite element heterogeneous multiscale methods with near optimal computational complexity Authors: Bjorn Engquist, Assyr Abdulle Download PDF	2006	HOP
Effectiveness of implicit methods for stiff stochastic differential equations Authors: Assyr Abdulle, W E, Wencheng Li Download PDF	2006	HOP
Multi-symplectic integration of the Camassa-Holm equation Authors: X Raynaud, D Cohen, Brynjulf Owren Download PDF	2006	HOP
Comparison of integrators for the Fermi-Pasta-Ulam problem Authors: Robert McLachlan, Dion O'Neale Download PDF	2006	HOP
A convergent adaptive method for elliptic eigenvalue problems Authors: S Giani, Ivan Graham Download PDF	2006	HOP
Homogenization of spectral problems in bounded domains with doubly high contrasts Authors: Vladimir Kamotski, VP Smyshlyaev, Natalia Babych Download PDF	2006	HOP
Sufficiency of Favard's condition for a class of band-dominated operators on the axis Authors: M Lindner, Simon Chandler-Wilde Download PDF	2006	HOP
From stationary phase to steepest descent Authors: Spyridon Kamvissis Download PDF	2006	HOP
Condition number estimates for combined potential boundary integral operators in acoustic scattering Authors: Ivan Graham, Stephen Langdon, M Lindner, Simon Chandler-Wilde Download PDF	2006	HOP
The focusing NLS equation on the half-line with periodic boundary conditions Authors: Spyridon Kamvissis, AS Fokas Download PDF	2006	HOP
Plane wave discontinuous Galerkin methods Authors: R Hiptmair, Ilaria Perugia, CJ Gittelson Download PDF	2006	HOP
The PML for rough surface scattering Authors: Peter Monk, Simon Chandler-Wilde Download PDF	2006	HOP
High frequency scattering by convex curvilinear polygons Authors: M Mokgolele, Simon Chandler-Wilde, Stephen Langdon Download PDF	2006	HOP
Boundary integral methods in high frequency scattering Authors: Ivan Graham, Simon Chandler-Wilde Download PDF	2006	HOP
Limit operators, collective compactness, and the spectral theory of infinite matrices Authors: M Lindner, Simon Chandler-Wilde Download PDF	2006	HOP
A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods Authors: AS Fokas, V Michel Download PDF	2006	HOP
Spurious solutions in the multiband effective mass theory applied to low dimensional nanostructures Authors: Roderick Melnik, M Willatzen, B Lassen Download PDF	2006	HOP

Workshops

HOPW02

Seminars

Wednesday 17th January 2007
14:30 to 15:30	Simon Chandler-Wilde University of Reading An overview of oscillatory integrals and integral operators in high frequency scattering	Room 1

Tuesday 23rd January 2007
10:00 to 11:00	Discussion of highly oscillatory quadrature	Discussion Room

Wednesday 24th January 2007
14:30 to 15:30	Olavi Nevanlinna Circlets, vectorands and real linear operators	Room 1

Tuesday 30th January 2007
14:30 to 15:30	Olavi Nevanlinna What can happen to the spectrum in low-rank perturbations?	Room 1
16:00 to 17:00	Krishnaswamy Sankaran ETH Zürich Finite-volume time-domain modelling of the Maxwell system and domain-truncation techniques	Room 1

Wednesday 31st January 2007
14:30 to 15:30	Numerical simulation of Bose--Einstein condensation	Room 1

Thursday 1st February 2007
16:00 to 17:00	Spyridon Kamvissis Technical University of Crete On nonlinear steepest descent	Room 1

Tuesday 6th February 2007
10:00 to 11:00	Discussion of highly oscillatory quadrature	Discussion Room

Wednesday 7th February 2007
14:30 to 15:30	Macro-to-micro methods for highly oscillatory partial differential equations	Room 1

Tuesday 13th February 2007
14:30 to 15:30	Syvert Norsett Filon-type methods for highly oscillatory integrals	Room 1
16:00 to 17:00	Arieh Iserles University of Cambridge Computing modified Fourier approximations in cubes	Room 1

Wednesday 14th February 2007
11:30 to 12:30	David Levin Infinite oscillatory integrals	Room 1
14:30 to 15:30	Olof Runborg Computational high frequency wave propagation	Room 1

Thursday 15th February 2007
11:30 to 12:30	Peter Olver University of Minnesota Moment-free numerical integration of highly oscillatory functions	Room 1
14:30 to 15:30	Stefan Vandewalle KU Leuven A numerical approach to the method of steepest descent	Room 1

Wednesday 21st February 2007
10:00 to 11:00	Discussion meeting on highly oscillatory quadrature	Discussion Room
11:30 to 12:30	Sergey Tikhonov Scuola Normale Superiore di Pisa Some criteria for convergence of Fourier series in different function spaces	Room 1
14:30 to 15:30	Regularized combined field integral equations	Room 1

Monday 26th February 2007
14:30 to 15:30	Olof Runborg A fast phase space method for computing creeping waves	Room 1

Tuesday 27th February 2007
16:00 to 17:00	Scott Fulton Highly accurate computation of the Dirichlet-Neumann map	Room 1

Wednesday 28th February 2007
11:30 to 12:30	Mechthild Thalhammer Exponential operator splitting for time-dependent Schroedinger equation	Room 1
14:30 to 15:30	Peter Monk University of Delaware The use of plane waves to approximate wave problems	Room 1

Thursday 1st March 2007
14:30 to 15:30	Anna Gilbert University of Michigan Algorithms for sparse approximation and applications	Room 1

Friday 2nd March 2007
14:30 to 15:30	High frequency asymptotics in scattering by smooth and non-smooth obstacles: analytical and numerical aspects	Room 1

Tuesday 6th March 2007
10:00 to 11:00	Ivan Graham University of Bath Robust boundary integral methods in high frequency acoustic scattering	Room 1

Wednesday 7th March 2007
11:30 to 12:30	Analysis of plane wave discontinuous Galerkin methods	Room 1
14:30 to 15:30	Brad Baxter Integrating exponentials over a simplex: a surprising link with exponential Brownian motion	Room 1

Monday 12th March 2007
14:30 to 15:30	Fatih Ecevit High frequency scattering by a collection of convex bodies	Room 1

Tuesday 13th March 2007
11:30 to 12:30	Nonlinear diffusions as limits of BGK-type kinetic equations	Room 1

Wednesday 14th March 2007
11:30 to 12:30	Roderick Melnik Cayley Transform Techniques in PDEs and their Numerical Approximations (on the example of convection-diffusion equations)	Room 1
14:30 to 15:30	Bjorn Engquist A fast multipole method for oscillatory kernels	Room 1

Thursday 15th March 2007
16:00 to 17:00	On asymptotic regimes for the Maxwell-Dirac system	Room 1

Tuesday 20th March 2007
14:30 to 15:30	Ernst Hairer Observatoire Astronomique de l'Universite de Geneve Backward error analysis and its limitations	Room 2
16:00 to 17:00	Molecular dynamics, thermostats and the accuracy of averages	Room 2

Wednesday 21st March 2007
14:30 to 15:30	Christian Lubich Eberhard Karls Universität Tübingen Long-time-step methods for highly oscillatory Hamiltonian problems	Room 1
16:00 to 17:00	Sebastian Reich Universität Potsdam Fluids, parcel dynamics and oscillations	Room 1

Thursday 22nd March 2007
16:00 to 17:00	Analysis of splitting methods for reaction-diffusion equations in the light of stochastic calculus	Room 1

Tuesday 10th April 2007
14:30 to 15:30	Normal form and long time analysis of splitting schemes for the linear Schroedinger equation	Room 2

Wednesday 11th April 2007
14:30 to 15:30	Symplectic integrators for highly oscillatory Hamiltonian systems	Room 2

Thursday 12th April 2007
14:30 to 15:30	Simon Malham Heriot-Watt University Stochastic Lie-group integrators	Room 2

Tuesday 17th April 2007
14:30 to 15:30	Robert McLachlan Symplectic Runge - Kutta Methods	Room 1

Wednesday 18th April 2007
16:00 to 17:00	Enrique Zuazua Wave propagation on planar graphs	Room 1

Wednesday 25th April 2007
14:30 to 15:30	Snorre Christiansen University of Oslo A construction of mixed finite element spaces on polyhedral meshes	Room 1

Thursday 26th April 2007
11:30 to 12:30	Veerle Ledoux An efficient numerical solution of Sturn--Liouville problems	Room 1

Tuesday 1st May 2007
14:30 to 15:30	Roderick Melnik Time-dependent models of phase transformations and their numerical approximations	Room 1

Wednesday 2nd May 2007
14:30 to 15:30	Improving Poincar\'e: exponential asymptotics for ODEs and integrals with saddles	Room 1

Thursday 3rd May 2007
14:30 to 15:30	Christopher Howls University of Southampton The higher order Stokes phenomenon: when is a Stokes line not a Stokes line?	Room 1

Tuesday 8th May 2007
11:30 to 12:30	Oscillations and rough paths	Room 1
14:30 to 15:30	Liviu Ignat Universidad Autonoma de Madrid Dispersive numerical schemes for linear and nonlinear Schr\"odinger equations	Room 1

Wednesday 9th May 2007
14:30 to 15:30	Vera Thummler Universität Bielefeld Numerical computation of space-time patterns of PDEs with symmetry	Room 1

Wednesday 16th May 2007
14:30 to 15:30	Melvin Leok Indiana University-Purdue University Indianapolis Lie-group variational integrators and their applications to geometric optimal control theory	Room 1

Thursday 17th May 2007
16:00 to 17:00	Alexander Its Generalised Fisher-Hartwig and Szeg\"o-Widom asymptotics	Room 1

Friday 18th May 2007
14:30 to 15:30	Jacques Vanneste University of Edinburgh Inertia-gravity-wave generation by slow motion in the atmosphere and oceans	Room 1

Tuesday 22nd May 2007
16:30 to 17:30	Brynjulf Owren Geometric integrators for the Camassa-Holm equation	Room 1

Wednesday 23rd May 2007
14:30 to 15:30	Ian Sloan University of New South Wales Lifting the curse of dimensionality - numerical integrations in very high dimensions	Room 1

Thursday 24th May 2007
11:30 to 12:30	Alexander Ostermann Exponential integrators for advection and diffusion problems	Room 1
14:30 to 15:30	Claudia Wulff University of Surrey Approximate energy conservation for symplectic time semidiscretizations of semilinear wave equations	Room 1

Tuesday 29th May 2007
14:30 to 15:30	Michael Ascher Fast multiscale denoising of surface meshes in 3D with intrinsic texture	Room 1

Wednesday 30th May 2007
14:30 to 15:30	Folkmar Bornemann Weak convergence, realisation of holonomic constraints, and the quantum adiabatic theorem	Room 1

Thursday 31st May 2007
14:30 to 15:30	Elena Celledoni Simulation of rigid body dynamics using Jacobi elliptic functions and integrals	Room 1

Thursday 7th June 2007
16:00 to 17:00	Folkmar Bornemann A mathematical study of the Car-Parrinello method	Room 1

Monday 11th June 2007
14:30 to 15:30	Peter Olver University of Minnesota Invariantisation of numerical schemes	Room 1

Wednesday 13th June 2007
11:30 to 12:30	On the approximation of spectra and pseudospectra of non-self-adjoint Hilbert space operators	Room 1
14:30 to 15:30	Percy Deift Courant Institute of Mathematical Sciences Riemann Hilbert problems: theory and applications	Room 1

Thursday 14th June 2007
14:30 to 15:30	Mahadevan Ganesh Colorado School of Mines A spectrally accurate algorithm for electromagnetic scattering in three dimensions	Room 1

Wednesday 20th June 2007
14:30 to 15:00	Dion O'Neale Massey University Reconsidering trigonometric integrators	Room 1
15:00 to 15:30	Numerical solution to highly oscillatory systems of ODEs	Room 1
15:30 to 16:00	Chuang Zheng Universidad Autonoma de Madrid From the wave packet estimate to exact observability	Room 1

Thursday 21st June 2007
16:00 to 17:00	Thomas Schmelzer University of Oxford The fast evaluation of matrix functions for exponential integrators	Room 1

Friday 22nd June 2007
15:30 to 16:30	Bjorn Engquist Homogenisation: Analysis and Computation	Room 1

Wednesday 27th June 2007
14:30 to 15:00	Interpolation in special orthogonal groups	Room 1
15:00 to 15:30	Aurora-Mihaela Marica Universidad Autonoma de Madrid Fourier analysis of the 1-d discontinuous Galerkin methods	Room 1
15:30 to 16:00	Spectral methods and modified Fourier series	Room 1

Thursday 28th June 2007
14:30 to 15:30	Sparse finite element method for nonlinear monotone problems with multiple scales	Room 1

Monday 2nd July 2007
09:00 to 17:00	Poster session	Discussion Room

Highly oscillatory problems: computation, theory and application

Organisers

Bjorn Engquist Princeton University
Ernst Hairer Observatoire Astronomique de l'Universite de Geneve
Arieh Iserles University of Cambridge

Speakers

Michael Ascher
Brad Baxter
Folkmar Bornemann
Elena Celledoni
Simon Chandler-Wilde Brunel University
Snorre Christiansen University of Oslo
Percy Deift Courant Institute of Mathematical Sciences
Fatih Ecevit
Bjorn Engquist Princeton University
Scott Fulton
Mahadevan Ganesh Colorado School of Mines
Anna Gilbert University of Michigan
Ivan Graham University of Bath
Ernst Hairer Observatoire Astronomique de l'Universite de Geneve
Christopher Howls University of Southampton
Liviu Ignat Universidad Autonoma de Madrid
Arieh Iserles University of Cambridge
Alexander Its
Spyridon Kamvissis Technical University of Crete
Veerle Ledoux
Melvin Leok
David Levin
Christian Lubich Eberhard Karls Universität Tübingen
Simon Malham Heriot-Watt University
Aurora-Mihaela Marica Universidad Autonoma de Madrid
Robert McLachlan
Roderick Melnik
Peter Monk University of Delaware
Olavi Nevanlinna
Syvert Norsett
Dion O'Neale Massey University
Peter Olver University of Minnesota
Alexander Ostermann
Brynjulf Owren
Sebastian Reich Universität Potsdam
Olof Runborg
Krishnaswamy Sankaran ETH Zürich
Thomas Schmelzer University of Oxford
Ian Sloan University of New South Wales
Mechthild Thalhammer
Vera Thummler Universität Bielefeld
Sergey Tikhonov Scuola Normale Superiore di Pisa
Stefan Vandewalle KU Leuven
Jacques Vanneste University of Edinburgh
Claudia Wulff University of Surrey
Chuang Zheng Universidad Autonoma de Madrid
Enrique Zuazua BCAM - Basque Center for Applied Mathematics

Show All

Highly oscillatory problems: computation, theory and application

Programme theme

Preprints

A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering

From high oscillation to rapid approximation III: Multivariate expansions

Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions

Stabilized FEM-BEM coupling for Maxwell transmission problems

Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves

Spectral semi-discretisations of weakly nonlinear wave equations over long times

Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations

Convergence of a two-grid algorithm for the control of the wave equation

Stability of the periodic Toda lattice under short range perturbations

Finite-order integration weights can be dangerous

On multisymplecticity of partitioned Runge-Kutta methods

An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks

Wave-number-explicit bounds in time-harmonic scattering

Achieving Brouwer's law with implicit Runge-Kutta methods

A multiscale method for highly oscillatory ordinary differential equations with resonance

Finite element heterogeneous multiscale methods with near optimal computational complexity

Effectiveness of implicit methods for stiff stochastic differential equations

Multi-symplectic integration of the Camassa-Holm equation

Comparison of integrators for the Fermi-Pasta-Ulam problem

A convergent adaptive method for elliptic eigenvalue problems

Homogenization of spectral problems in bounded domains with doubly high contrasts

Sufficiency of Favard's condition for a class of band-dominated operators on the axis

From stationary phase to steepest descent

Condition number estimates for combined potential boundary integral operators in acoustic scattering

The focusing NLS equation on the half-line with periodic boundary conditions

Plane wave discontinuous Galerkin methods

The PML for rough surface scattering

High frequency scattering by convex curvilinear polygons

Boundary integral methods in high frequency scattering

Limit operators, collective compactness, and the spectral theory of infinite matrices

A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods

Spurious solutions in the multiband effective mass theory applied to low dimensional nanostructures

Workshops

The theory of highly oscillatory problems

Applying geometric integrators

Multiresolution and high oscillation

Oscillatory integrals and integral equations in high frequency scattering and wave propagation

Effective computational methods for highly oscillatory solutions

High frequency wave propagation and scattering

Highly oscillatory problems

Seminars

Organisers

Speakers

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