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Seminars (HOP)

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Event When Speaker Title Presentation Material
HOP 17th January 2007
14:30 to 15:30
An overview of oscillatory integrals and integral operators in high frequency scattering

In this talk we review recent research on integral equation methods for high frequency time-harmonic scattering. A number of interesting questions arise in this context, including:

    * The kernels of the boundary integral operators become increasingly oscillatory as the frequency increases. What is the dependence on frequency of the condition numbers of these operators? What can be said about this after discretisation?
  • * Is it possible, by use of an ansatz based on high frequency asymptotics, to represent oscillatory solutions with a number of degrees of freedom which is fixed as the frequency increases?
  • * Using such an ansatz, the entries of an approximating linear system are oscillatory integrals. What form do these integrals take and how can we evaluate them?
  • * For which classes of problems might we hope to achieve computational cost which is O(1) in its dependence on the frequency as the frequency increases?

We will indicate the (exciting but limited) progress that has been made internationally in addressing these questions in the last few years, and give some idea of the methods that have been used, focussing particularly on the simplest problem of scattering of time-harmonic acoustic waves by a bounded sound soft obstacle in two and three dimensions. We will also list some of the many open, difficult problems in this area.

HOP 24th January 2007
14:30 to 15:30
Circlets, vectorands and real linear operators
HOP 30th January 2007
14:30 to 15:30
What can happen to the spectrum in low-rank perturbations?
HOP 30th January 2007
16:00 to 17:00
K Sankaran Finite-volume time-domain modelling of the Maxwell system and domain-truncation techniques
HOP 31st January 2007
14:30 to 15:30
Numerical simulation of Bose--Einstein condensation
HOP 1st February 2007
16:00 to 17:00
S Kamvissis On nonlinear steepest descent
HOP 7th February 2007
14:30 to 15:30
A Abdulle Macro-to-micro methods for highly oscillatory partial differential equations
HOP 13th February 2007
14:30 to 15:30
Filon-type methods for highly oscillatory integrals
HOP 13th February 2007
16:00 to 17:00
A Iserles Computing modified Fourier approximations in cubes
HOP 14th February 2007
11:30 to 12:30
D Levin Infinite oscillatory integrals
HOP 14th February 2007
14:30 to 15:30
Computational high frequency wave propagation

The numerical approximation of high frequency wave propagation is important in many applications, including geophysics, electromagnetics and acoustics. When the wavelength is short compared to the overall size of the computational domain direct simulation using standard wave equations is very expensive. Fortunately, there are computationally much less costly models, that are good approximations of many wave equations precisely for very high frequencies. In this talk I will focus on the geometrical optics approximation, which is the infinite frequency limit of wave equations. Geometrical optics has traditionally been simulated using ray tracing. More recently numerical procedures based on various partial differential equations have been introduced. In this talk I will survey such numerical methods for geometrical optics and also briefly discuss finite frequency corrections and some other models

HOP 15th February 2007
11:30 to 12:30
Moment-free numerical integration of highly oscillatory functions
HOP 15th February 2007
14:30 to 15:30
A numerical approach to the method of steepest descent
HOP 21st February 2007
11:30 to 12:30
Some criteria for convergence of Fourier series in different function spaces
HOP 21st February 2007
14:30 to 15:30
Regularized combined field integral equations
HOP 26th February 2007
14:30 to 15:30
O Runborg A fast phase space method for computing creeping waves
HOP 27th February 2007
16:00 to 17:00
S Fulton Highly accurate computation of the Dirichlet-Neumann map
HOP 28th February 2007
11:30 to 12:30
Exponential operator splitting for time-dependent Schroedinger equation
HOP 28th February 2007
14:30 to 15:30
P Monk The use of plane waves to approximate wave problems
HOP 1st March 2007
14:30 to 15:30
Algorithms for sparse approximation and applications
HOP 2nd March 2007
14:30 to 15:30
V Smyshlyaev High frequency asymptotics in scattering by smooth and non-smooth obstacles: analytical and numerical aspects
HOP 6th March 2007
10:00 to 11:00
Robust boundary integral methods in high frequency acoustic scattering
HOP 7th March 2007
11:30 to 12:30
Analysis of plane wave discontinuous Galerkin methods
HOP 7th March 2007
14:30 to 15:30
Integrating exponentials over a simplex: a surprising link with exponential Brownian motion
HOP 12th March 2007
14:30 to 15:30
High frequency scattering by a collection of convex bodies
HOP 13th March 2007
11:30 to 12:30
Nonlinear diffusions as limits of BGK-type kinetic equations
HOP 14th March 2007
11:30 to 12:30
R Melnik Cayley Transform Techniques in PDEs and their Numerical Approximations (on the example of convection-diffusion equations)
HOP 14th March 2007
14:30 to 15:30
A fast multipole method for oscillatory kernels
HOP 15th March 2007
16:00 to 17:00
On asymptotic regimes for the Maxwell-Dirac system
HOP 20th March 2007
14:30 to 15:30
Backward error analysis and its limitations
HOP 20th March 2007
16:00 to 17:00
Molecular dynamics, thermostats and the accuracy of averages
HOP 21st March 2007
14:30 to 15:30
Long-time-step methods for highly oscillatory Hamiltonian problems
HOP 21st March 2007
16:00 to 17:00
S Reich Fluids, parcel dynamics and oscillations
HOP 22nd March 2007
16:00 to 17:00
Analysis of splitting methods for reaction-diffusion equations in the light of stochastic calculus
HOPW02 26th March 2007
10:00 to 11:00
Three recent results on asymptotics of oscillations

The results are separate, and apparently paradoxical, and have implications for physics. First, when two exponentials compete, their interference can be dominated by the contribution with smaller exponent. Second, repeated differentiation of almost all functions in a wide class generates trigonometric oscillations (‘almost all functions tend to cosx’). Third, it is possible to find band-limited functions that oscillate arbitrarily faster than their fastest Fourier component (‘superoscillations’).

HOPW02 26th March 2007
11:30 to 12:15
Heterogeneous multi-scale methods

Continuum simulations of solids or fluids for which some atomistic information is needed are typical example of multi-scale problems with very large ranges of scales. For such problems it is necessary to restrict the simulations on the micro-scale to a smaller subset of the full computational domain. The heterogeneous multi-scale method is a framework for developing and analyzing numerical methods that couple computations from very different scales. Local micro-scale simulations on small domains supply missing data to a macro-scale simulation on the full domain. Examples are local molecular dynamics computations that produce data to a continuum macro-scale model, or a highly oscillatory dynamical system for which a local estimate of resonances is enough to supply data for a smoother evolution of averages. We will focus on dynamical systems.

HOPW02 26th March 2007
14:15 to 15:00
Hamiltonian slow manifolds with internal oscillation

Many Hamiltonian systems possess special families of solutions which can be described approximately as slowly drifting periodic orbits. Examples include the gravitational three-body problem, the interaction of two identical charged particles in a magnetic field, and the propagation and interaction of discrete breathers (time-periodic spatially localised motions in Hamiltonian networks of units). Theory will be presented for how to improve a zeroth order manifold of approximate solutions to r-th order for any r > 0, meaning one that contains all nearby periodic orbits of nearby period and has error of order the r-th power of the drift field (even with a small constant for r=1). In the normally hyperbolic case an exactly invariant nearby submanifold can be constructed. If there are normally elliptic directions, however, this is impossible in general but the above r-th order approximations can be achieved provided that the normal frequencies avoid all multiples of that of the approximately periodic motion and are fast compared with the drift. An effective Hamitlonian is derived to describe the drift of the orbits. Applications to the above fields will be given. An introduction has been published as R.S.MacKay, Slow Manifolds, in Energy localisation and transfer, eds T.Dauxois, A.Litvak-Hinenzon, R.S.MacKay, A.Spanoudaki (World Sci Publ Co, 2004), 149-192.

HOPW02 26th March 2007
15:30 to 16:15
Dispersive and dissipative behaviour of Galerkin approximation using high order polynomials

We consider the dispersive properties of Galerkin finite element methods for wave propagation. The dispersive properties of conforming finite element scheme are analysed in the setting of the Helmholtz equation and an explicit form the discrete dispersion relation is obtained for elements of arbitrary order. It is shown that the numerical dispersion displays three different types of behaviour depending on the order of the polynomials used relative to the mesh-size and the wave number. Quantitative estimates are obtained for the behaviour and rates of decay of the dispersion error in the differing regimes.

HOPW02 26th March 2007
16:15 to 17:00
S Dobrokhotov Generalization of the Maslov Theory for Localized Asymptotics and Tsunami waves

We suggest a new asymptotic representation for the solutions to the multidimensional wave equations with variable velocity with localized initial data. This representation is the generalization of the Maslov canonical operator based also on a simple relationship between fast decaying and fast oscillating solutions, and on boundary layer ideas. It establishes the connection between initial localized perturbations and wave profiles near the wave fronts including the neighborhood of backtracking (focal or turning) and self intersection points. We show that wave profiles are related with a form of initial sources and also with the Lagrangian manifolds organized by the rays and wavefronts. In particular we discuss the influence of such topological characteristics like the Maslov and Morse indices to metamorphosis of the profiles after crossing the focal points. We apply these formulas to the problem of a propagation of tsunami waves in the frame of so-called ``piston model''. Finally we suggest a fast asymptotically-numerical algorithm for simulation of tsunami wave over nonuniform bottom. Different scenarios of the distribution of the waves are considered, the wave profiles of the front are obtained in connection with the different shapes of the source and with the diverse rays generating the fronts. It is possible to use suggested algorithm to predict in real time the zones of the beaches where the amplitude of the tsunami wave has dangerous high values. In this connection we also discuss the following questions: the problems of the regularization of the wave field near focal points; ill-possed problems appearing in the geometry of the wavefronts; the inverse problem connected with the possibility of reconstruction of the source via the measurement of the tsunami wave profile on the shelf etc. This work was done together with S.Sekerzh-Zenkovich, B.Tirozzi, B.Volkov and was partially supported by RFBR grant N 05-01-00968 and Agreement Between University "La Sapienza", Rome and Institute for Problems in Mechanics RAS, Moscow.

Bibliography

[1] S.Yu. Dobrokhotov, S.Ya Sekerzh-Zenkovich, B. Tirozzi, T.Ya. Tudorovskiy, The description of tsunami waves propagation based on the Maslov canonical operator, Doklady Mathematics, 2006, v.74, N 1, pp. 592-596

[2] S.Yu. Dobrokhotov, S.Ya Sekerzh-Zenkovich, B. Tirozzi, T.Ya. Tudorovskiy, Asymptotic theory of tsunami waves: geometrical aspects and the generalized Maslov representation, Publications of Kyoto Research Mathematical Institute, Vol.4, page 118-153, 2006, ISSN 1880-2818. [3] S.Dobrokhotov, S.Sekerzh-Zenkovich, B.Tirozzi, B.Volkov Explicit asymptotics for tsunami waves in framework of the piston model, Russ. Journ. Earth Sciences, 2006, v.8, ES403, pp.1-12

[4] S.Dobrokhotov, S.Sinitsyn, B.Tirozzi, Asymptotics of Localized Solutions of the One-Dimensional Wave Equation with Variable Velocity. I. The Cauchy Problem, Russ. Jour.Math.Phys., v.14, N1, 2007, pp.28-56

HOPW02 26th March 2007
17:00 to 17:30
Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation

The aim of this talk is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.

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HOPW02 27th March 2007
09:00 to 09:45
Numerical oscillations over long times
HOPW02 27th March 2007
09:45 to 10:30
Conservation of energy and actions in numerical discretizations of nonlinear wave equations

For numerical discretizations of nonlinearly perturbed wave equations the long-time near-conservation of energy, momentum, and harmonic actions is studied. The proofs are based on the technique of modulated Fourier expansions in time. Rigorous statements on the long-time conservation properties are shown under suitable numerical non-resonance conditions and under a CFL condition. The time step need not be small compared to the inverse of the largest frequency in the space-discretized system.

This is joint work with Christian Lubich and David Cohen.

HOPW02 27th March 2007
10:30 to 11:00
Highly oscillatory Hamiltonian systems with non-constant mass matrix

We will present a class of numerical methods (based on the trigonometric methods) for such Hamiltonian problems. We will then present a frequency expansion of the numerical solution: the modulated Fourier expansion. The system that determines the coefficients of this expansion has two formal invariants which are related to the total energy and the oscillatory energy of the original system. This allows us to prove the near-conservation of the total and the oscillatory energy for the numerical schemes over very long time intervals.

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HOPW02 27th March 2007
11:30 to 12:15
S Reich Highly oscillatory PDEs, slow manifolds and regularized PDE formations

The main motivation of my talk is provided by geophysical fluid dynamics. The underlying Euler or Navier-Stokes equations display oscillatory wave dynamics on a wide range of temporal and spatial scales. Simplified models are often based on the idea of balance and the concept of a slow manifold. Examples are provided by hydrostatic and geostrophic balance. One would also like to exploit these concepts on a computational level. However, slow manifolds are idealized objects that do not fully characterize the complex fluid behavior. I will describe a novel regularization technique that makes use of balance and slow manifolds in an adaptive manner. The regularization approach is based on a reinterpretation of the (linearly) implicit midpoint rule as an explicit time-stepping method applied to a regularized set of Euler equations. Adaptivity can be achieved by means of a predictor-corrector interpretation of the regularization.

HOPW02 27th March 2007
14:15 to 15:00
Tunnel effect for Kramers-Fokker-Planck type operators

For a large class of Kramers-Fokker-Planck type operators, we determine in the semiclassical (here the low temperature) limit the full asymptotic expansion of the splitting between the lowest eigenvalue (0) and the next one. In a previous work with Herau and C. Stolk we did so in the case when a certain potential (or exponent of a Maxwellian) has preciely one local minimum and then the splitting is "large". In this new work we treat the case when the potential has two local minima. Then the splitting is exponentially small and related to a tunnel effect between the minima. Our most direct source of inspiration has been works by Herau-F.Nier, B.Helffer-Nier, but our methods are quite different.

Joint work with F Herau and M Hitrik.

HOPW02 27th March 2007
15:30 to 16:15
Fast singular oscillating limits of the 3D Navier-Stokes and Euler equations: Global regularity and three-dimensional Euler dynamics

In the first part of the talk, we study highly oscillatory problems for the incompressible 3D Navier-Stokes and prove existence on infinite time intervals of regular strong solutions; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold. The global existence is proven using techniques of fast singular oscillating limits, Lemmas on restricted convolutions and the Littlewood-Paley dyadic decomposition. In the second part of the talk, we analyze regularity and dynamics of highly oscillatory problems for the 3D Euler equations.

Detailed proofs can be found in the following references:

Asymptotic Analysis, vol. 15, No. 2, p. 103-150, 1997 (with A. Babin and B. Nicolaenko).

Indiana University Mathematics Journal, vol. 48, No. 3, p. 1133-1176, 1999 (with A. Babin and B. Nicolaenko).

Indiana University Mathematics Journal, vol. 50, p. 1-35, 2001 (with A. Babin and B. Nicolaenko).

Russ. Mathematical Surveys, vol. 58, No. 2 (350), p. 287-318, 2003 (with B. Nicolaenko).

Methods and Applications of Analysis, vol. 11, No. 4, p. 605-633, 2004 (with B. Nicolaenko, C. Bardos and F. Golse). Hokkaido Mathematical Journal, vol. 35, No. 2, p. 321-364, 2006 (with Y. Giga, K. Inui and S. Matsui).

Annals of Math. Studies, Princeton University Press (Editors: J. Bourgain and C. Kenig), In Press (with Y. Giga and B. Nicolaenko).

HOPW02 27th March 2007
16:15 to 17:00
Some remarks concerning exponential integrators

TBA

HOPW02 27th March 2007
17:00 to 17:30
Spectral results for membrane with perturbed stiffness and mass density

We study the spectrum of a nonhomogeneous membrane that consists of two parts with strongly different stiffness and mass density. The small parameter describes the quotient of stiffness coefficients. The M-th power of the parameter is comparable to the ratio of densities. We show that the asymptotic behaviour of eigenvalues and eigenfunctions depends on rate M. We distinguish three cases M<1, M=1 and M>1.

The strong resolvent convergence of perturbed operators leads to loss of the completeness of limit eigenfunction system in both cases when M is different from 1. Therefore the limit operators describe only a part of the prelimit membrane vibrations. With an eye to close this gap we use the WKB-asymptotic expansions with a quantized small parameter to prove the existence of other kind of eigenvibrations, namely high frequency vibrations.

In the critical case M=1 the limit operator is a nonself-adjoint one, nevertheless the perturbed operators are self-adjoint in a certain topology. The multiplicity of spectrum and structure of root spaces are investigated.

Complete asymptotic expansions for eigenelements are constructed and justified in each case of the perturbations.

HOPW02 28th March 2007
09:00 to 09:45
Nanomechanics of biomolecules: whether oscillatory details are important or not
HOPW02 28th March 2007
09:45 to 10:30
M Hochbruck Exponential integrators for oscillatory second-order differential equations

In this talk, we analyse a family of exponential integrators for second-order differential equations in which high-frequency oscillations in the solution are generated by a linear part. We characterise methods which allow second-order error bounds by presenting a unified error analysis for the whole family of methods. A major advantage of our analysis is that it does not require bounds for point-wise products of matrices and therefore, generalises to abstract differential equations, where the linear part is an unbounded operator with infinitely many large eigenvalues directly.

This is joint work with Volker Grimm

HOPW02 28th March 2007
10:30 to 11:00
Small dispersion limit of the KdV and semiclassical limit of focusing NLS

We compare numerically the small dispersion limit of the Korteweg de Vries equation with the asymptotic formulas obtained in the works of Lax-Levermore and Deif-Venakides and Zhou. We compare numerically the semiclassical limit of the focusing nonlinear Schrodinger equation with the asymptotic formula obtained in the works of Kamvissis-McLaughlin-Miller and Tovbis-Venakides-Zhou. As a results we outline the regions of the (x,t) where the above asymptotic formulas give a satisfactory description of the respective equations.

HOPW02 28th March 2007
11:30 to 12:15
Beyond the adiabatic approximation: exponentially small coupling terms

For multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. We explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two-state systems with real-symmetric Hamiltonian. The superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function. First order perturbation theory for the Hamiltonian in the superadiabatic representation then allows to describe the time-development of exponentially small adiabatic transitions and thus to rigorously confirm Michael Berry’s predictions on the universal form of adiabatic transition histories (Joint work with V. Betz from Warwick).

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HOPW02 28th March 2007
14:00 to 14:30
Approximate energy conservation for symplectic time semidiscretizations of semilinear wave equations

We prove that A-stable symplectic Runge-Kutta time semidiscretizations applied to semilinear wave equations with periodic boundary conditions, analytic nonlinearity and analytic initial data conserve a modified energy p to an exponentially small error. This modified energy is O(h^p)-close to the original energy where p is the order of the method and h the time-stepsize. Standard backward error analysis can not be applied because of the occurrence of unbounded operators in the construction of the modified vectorfield. This loss of regularity in the construction can be taken care of by projecting the PDE to a finite-dimensional space and by coupling the number of excited modes as well as the number of terms in the expansion of the vectorfield with the stepsize. This way we obtain exponential estimates of the form O(\exp(-C/ h^(1/2) )). A similar technique has been used for averaging of rapidly forced Hamiltonian PDEs by [Matthies and Scheel, 2003]. As a side-product, we also provide a convergence analysis of Runge-Kutta methods in Hilbert spaces.

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HOPW02 28th March 2007
14:30 to 15:00
Numerical approximation of multivariate highly oscillatory integrals

The aim of this talk is to describe several methods for numerically approximating the integral of a multivariate highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a method developed by Levin as a point of departure we will construct a new method that uses the same information as a Filon-type method, and obtains the same asymptotic order, while not requiring moments. This allows us to integrate over nonsimplicial domains, and with complicated oscillators. We also present a method for approximating oscillatory integrals with stationary points.

HOPW02 28th March 2007
15:30 to 16:15
Asymptotics for the Camassa-Holm equation

I will present recent results on asymptotic behaviors for the Ca-massa–Holm (CH) equation ut - utxx + 2?ux + 3uux = 2uxuxx + uuxxx on the line, ? being a nonnegative parameter.

Firstly, I will describe the long-time asymptotic behavior of the solution u? (x, t), ? > 0 of the initial-value problem with fast decaying initial data u0(x). It appears that u? (x, t) behaves differently in different sectors of the (x, t)-half-plane. Then I will analyse the behavior of u? (x, t) as ? 0.

The methods are inverse scattering in a matrix Riemann-Hilbert approach and Deift and Zhou’s nonlinear steepest descent method.

Work in collaboration with Dmitry Shepelsky.

HOPW02 28th March 2007
16:15 to 17:00
Norms and condition numbers of oscillatory integral operators in acoustic scattering

In this talk we discuss domain and boundary integral operators arising in the theory and numerical treatment by integral equation methods of the Helmholtz equation or time harmonic Maxwell equations. These integral operators are increasingly oscillatory as the wave number k increases (k proportional to the frequency of the time harmonic incident field). An interesting theoretical question, also of practical significance, is the dependence of the norms of these integral operators and their inverses on k. We investigate this question, in particular for classical single- and double-layer potential operators for the Helmholtz equation on the boundary of bounded Lipschitz domains. The results and techniques used depend on the domain geometry. In certain 2D cases (for example where the boundary is a starlike polygon) bounds which are sharp in their dependence on k can be obtained, but there are many open problems for more general geometries and higher dimension.

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HOPW02 28th March 2007
17:00 to 17:30
Exponential asymptotics for the primitive equations

Following the work of Matthies (2001), it is shown how Gevrey (exponential) regularity of the solution and a classical method used together to prove an exponentially-accurate approximation result for a singular perturbation problem with a small parameter. The model considered is the viscous primitive equations of the ocean, although the method is applicable more generally.

HOPW02 29th March 2007
09:00 to 09:45
Finite difference approximation of homogenization problems for elliptic equations

In this talk, the problem of the approximation by finite differences of solutions to elliptic problems with rapidly oscillating coefficients and periodic boundary conditions will be discussed.

The mesh-size is denoted by $h$ while $\ee$ denotes the period of the rapidly oscillating coefficient. Using Bloch wave decompositions, we analyze the case where the ratio $h/\ee$ is rational. We show that if $h/\ee$ is kept fixed, being a rational number, even when $h,\ee\to 0$, the limit of the numerical solution does not coincide with the homogenized one obtained when passing to the limit as $\ee\to 0$ in the continuous problem. Explicit error estimates are given showing that, as the ratio $h/\ee$ approximates an irrational number, solutions of the finite difference approximation converge to the solutions of the homogenized elliptic equation. We consider both the 1-d and the multi-dimensional case. Our analysis yields a quantitative version of previous results on numerical homogenization by Avellaneda, Hou and Papanicolaou, among others. This is a joint work with Rafael Orive

HOPW02 29th March 2007
09:45 to 10:30
S Venakides Rigorous leading order asymptotic solutions of the semiclassical focusing NLS equation

Solving an integrable nonlinear differential equation reduces to solving the matrix Riemann-Hilbert problem (RHP) that arises from the invese scattering procedure. Typically, this means finding a matrix m(z) that is analytic in the closed complex plane except on a given oriented contour on which m(z) has a jump discontinuity: the value of m(z) left of the contour equals its value on the right multiplied by a given "jump matrix'' V(z). The space-time variables appear as parameters in the jump matrix; the complex variable z is the spectral variable of the linear differential operator that effects the linearization of the nonlinear system through the Lax pair associated with the system.

We derive an explicit formula for the leading asymptotic (oscillatory) solution of the semiclassical focusing nonlinear Schroedinger equation for a class of initial data and prove an error estimate. We extend the analysis when time is large, the asymptotic formula is then expressed in terms of elementary functions.

We first outline the notion of a Lax pair and how the RHP arises naturally.We then describe the steepest descent procedure that leads to an explicitly solvable asymptotic RHP through jump matrix factorizations and contour deformations. We also outline the g-funciton mechanism and describe how it determines the successful deformations.

HOPW02 29th March 2007
10:30 to 11:00
SJA Malham Numerically evaluating the Evans function by Magnus integration

We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and related partial differential equations. In a typical application scenario, we need to repeatedly sample the solution to a system of linear non-autonomous ordinary differential equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane.

In this situation, a substantial portion of the computational effort---the numerical evaluation of the iterated integrals which appear in the Magnus series---can be performed independent of the parameters and hence needs to be done only once. More importantly, for any given tolerance Magnus integrators possess lower bounds on the step size which are uniform across large regions of parameter space and which can be estimated a priori. We demonstrate, analytically as well as through numerical experiment, that these features render Magnus integrators extremely robust and, depending on the regime of interest, efficient in comparison with standard ODE solvers.

HOPW02 29th March 2007
11:30 to 12:15
Y Efendiev Multiscale numerical methods using limited global information

In this talk, I will describe multiscale numerical methods for flows in heterogeneous porous media. The main idea of these methods is to construct local basis functions that can capture the small scale information when they are coupled via some global formulation. I will discuss the use of local boundary conditions, oversampling methods and the use of global information in constructing basis functions. Applications of these methods to stochastic equations will be also discussed. This is joint work with J. Aarnes, T. Hou, L. Jiang, V. Ginting.

HOPW02 29th March 2007
14:00 to 14:30
Exponential integrators and functions of the matrix exponential

Exponential integrators are the most efficient class of methods for the time-stepping of stiff, semilinear, oscillatory PDEs such as the KdV equation. They solve the stiff, linear part of the PDE exactly. In the case of periodic boundary conditions, a Fourier spectral method can be used, so the linear part is diagonal and the methods can be applied straightforwardly. For other spatial discretizations, functions of the matrix exponential are required, which are susceptible to rounding errors. Several methods for evaluating these functions will be discussed.

HOPW02 29th March 2007
14:30 to 15:00
VH Hoang High dimensional finite element methods for elliptic problems with highly oscillating coefficients

Elliptic homogenization problems in a domain $\Omega \subset \R^d$ with $n+1$ separated scales are reduced to elliptic one-scale problems in dimension $(n+1)d$. These one-scale problems are discretized by a sparse tensor product finite element method (FEM). We prove that this sparse FEM has accuracy, work and memory requirement comparable to standard FEM for single scale problems in $\Omega$ while it gives numerical approximations of the correct homogenized limit as well as of all first order correctors, throughout the physical domain with performance independent of the physical problem's scale parameters. Numerical examples for model diffusion problems with two and three scales confirm our results.

HOPW02 29th March 2007
15:30 to 16:15
A multiscale method for stiff ordinary differential equations with resonance

We introduce a multiscale method to compute the effective behavior of a class of stiff and highly oscillatory ODEs. The oscillations may be in resonance with one another and thereby generate some hidden slow dynamics. Our method relies on correctly tracking a set of slow variables whose dynamics is effectively closed, and is sufficient to approximate the effective behavior of the ODEs. This set of variables is found by our numerical methods. We demonstrate our algorithms by a few examples that include a commonly studied problem of Fermi, Pasta, and Ulam (FPU).

HOPW02 29th March 2007
16:15 to 17:00
VP Smyshlyaev Localization and propagation in high contrast highly oscillatory media via 'non-classical' homogenization

We discuss fundamental mathematical tools for analysis of localization and propagation effects in highly oscillatory media with high contrasts. With the underlying two small parameters of the oscillations and the contrast, there is a "critical" scaling when the phenomena at the micro and macro scales are coupled in a non-trivial way, with "unusual" effects observed in an asymptotically explicit way. The related mathematical tool is that of a "non-classical" (high-contrast) homogenization, accounting for "high-frequency" oscillations, as opposes to the "classical" homogenization whose scope is limited by dealing in effect with low frequencies only. Those tools include "non-classical" two-scale asymptotic expansions, two-scale operator and spectral convergence, and two-scale compactness (with the latter building on, among others, recent deep ideas of V.V. Zhikov [1]).

We illustrate this on the problem of wave localization in high contrast periodic media with a defect [2] (a problem relevant to photonic crystal fibres). We also discuss the use of these techniques to the problem of "slowing down" of wave packets in high contrast highly-oscillatory media (the so-called "slow light" effect, with relevance to coupled resonances and metastability), and other prospects. Joint work with Ilia V. Kamotski.

[1] V.V. Zhikov, On an extension of the method of two-scale convergence and its applications, (Russian) Mat. Sbornik 191 (2000), 31-72; English translation in Sbornik Math. 191 (2000), 973-1014; V.V. Zhikov, Gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients. (Russian) Algebra i Analiz 16 (2004), 34-58;

[2] I.V. Kamotski and V.P. Smyshlyaev, Localised eigenstates due to defects in high contrast periodic media via homogenisation. BICS preprint 3/06. http://www.bath.ac.uk/math-sci/bics/preprints/BICS06_3.pdf (2006)

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HOPW02 29th March 2007
17:00 to 17:30
Exponential estimates in averaging and homogenisation

Many partial differential equations with rapid spatial or temporal scales have effective descriptions which can be derived by homogenisation or averaging. In this talk we deal with examples, where quantitative estimates of the error is possible for higher order homogenisation and averaging.

In particular, we provide theorems, which allow homogenisation and averaging beyond all orders by giving exponential estimates of appropriately averaged and homogenised descriptions. Methods include iterated averaging transformations, optimal truncation of asymptotic expansions and highly regular solutions (Gevrey regularity). Prototypical examples are reaction-diffusion equations with heterogeneous reaction terms or rapid external forcing, nonlinear Schr\"odinger equations describing dispersion management, and second-order linear elliptic equations.

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HOPW02 30th March 2007
09:00 to 09:45
A Iserles From high oscillation to rapid approximation

In this talk we address our recent work on the theory of modified Fourier expansions and their generalisation for the approximation of smooth functions in one and several variables. In particular, we debate rapid evaluation of expansion coefficients by means of Filon-like and Gauss--Turan-like quadrature techniques and various acceleration techniques.

HOPW02 30th March 2007
09:45 to 10:30
On the zero-mass constraint for KP type equations

For a rather general class of equations of Kadomtsev-Petviashvili (KP) type, we prove that the zero-mass constraint (in x) is satisfied at any non zero time even if it is not satisfied at initial time zero. Our results are based on a precise analysis of the fundamental solution of the linear part and of its anti x-derivative.

HOPW02 30th March 2007
10:30 to 11:00
High frequency scattering by convex polygons

Standard boundary integral methods for high frequency scattering problems, with piecewise polynomial approximation spaces, suffer from the debilitating restriction that the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency. For problems of acoustic scattering by sound soft convex polygons, it has recently been demonstrated by Chandler-Wilde and Langdon that by including plane wave basis functions supported on a graded mesh in the approximation space, with smaller elements closer to the corners of the polygon, convergence rates that depend only logarithmically on the frequency can be achieved. In this talk we review this approach, and consider some improvements and extensions to more complicated scattering problems.

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HOPW02 30th March 2007
11:30 to 12:00
JM Sanz-Serna Mollified impulse methods revisited

We introduce a family of impulse-like methods for the integration of highly oscillatory second-order differential equations whose forces can be split into a fast and a slow part. Methods of this family are specified by two weight functions: one is used to average positions and the other to mollify the force. In cases where the fast forces are conservative, the new family inlcudes as particular cases the mollified impulse methods introduced by García-Archilla, Skeel and the present author. On the other hand the methods here extend to nonlinear fast forces a well-known class of exponential integrators introduced by Hairer and Lubich. A convergence analysis will be presented that provides insight into the role played by the processes of averaging and mollification. A simple condition on the weight functions is shown to be both necessary and sufficient to avoid order reduction.

HOPW02 30th March 2007
14:00 to 14:30
On the localisation principle of high frequency scattering problems in integral equation discretisations

The numerical simulation of scattering problems at moderate to high frequencies is a challenging problem. The discretization of any suitable mathematical model usually requires resolving the oscillations, which naturally leads to large and, in the context of integral equations, dense discretization matrices. Yet, high frequency scattering problems have a very local nature. The localization principle states that the reflection, refraction or diffraction of a wave that hits an obstacle is governed mainly by the geometry of the scattering object which is local to the point of contact. This principle is exploited by asymptotic methods, such as geometrical optics and the geometrical theory of diffraction. Suddenly, higher frequencies are desirable and lower frequencies become problematic.

In this talk we examine how the localization principle can be exploited numerically in a more classical finite element setting. In particular, we may arrive at a sparse discretization matrix for integral equations by the use of Filon-type quadrature rules for oscillatory integrals. We discuss the advantages and limitations of this approach and we examine the asymptotic nature of this sparse representation.

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HOP 10th April 2007
14:30 to 15:30
G Dujardin Normal form and long time analysis of splitting schemes for the linear Schroedinger equation
HOP 11th April 2007
14:30 to 15:30
Symplectic integrators for highly oscillatory Hamiltonian systems
HOP 12th April 2007
14:30 to 15:30
Stochastic Lie-group integrators
HOP 17th April 2007
14:30 to 15:30
Symplectic Runge - Kutta Methods
HOP 18th April 2007
16:00 to 17:00
E Zuazua Wave propagation on planar graphs
HOP 25th April 2007
14:30 to 15:30
A construction of mixed finite element spaces on polyhedral meshes
HOP 26th April 2007
11:30 to 12:30
An efficient numerical solution of Sturn--Liouville problems
HOP 1st May 2007
14:30 to 15:30
R Melnik Time-dependent models of phase transformations and their numerical approximations
HOP 2nd May 2007
14:30 to 15:30
Improving Poincar\'e: exponential asymptotics for ODEs and integrals with saddles
HOP 3rd May 2007
14:30 to 15:30
The higher order Stokes phenomenon: when is a Stokes line not a Stokes line?
HOP 8th May 2007
11:30 to 12:30
Oscillations and rough paths
HOP 8th May 2007
14:30 to 15:30
Dispersive numerical schemes for linear and nonlinear Schr\"odinger equations
HOP 9th May 2007
14:30 to 15:30
Numerical computation of space-time patterns of PDEs with symmetry
HOP 16th May 2007
14:30 to 15:30
Lie-group variational integrators and their applications to geometric optimal control theory
HOP 17th May 2007
16:00 to 17:00
A Its Generalised Fisher-Hartwig and Szeg\"o-Widom asymptotics
HOP 18th May 2007
14:30 to 15:30
J Vanneste Inertia-gravity-wave generation by slow motion in the atmosphere and oceans
HOP 22nd May 2007
16:30 to 17:30
Geometric integrators for the Camassa-Holm equation
HOP 23rd May 2007
14:30 to 15:30
Lifting the curse of dimensionality - numerical integrations in very high dimensions
HOP 24th May 2007
11:30 to 12:30
A Ostermann Exponential integrators for advection and diffusion problems
HOP 24th May 2007
14:30 to 15:30
Approximate energy conservation for symplectic time semidiscretizations of semilinear wave equations
HOP 29th May 2007
14:30 to 15:30
Fast multiscale denoising of surface meshes in 3D with intrinsic texture
HOP 30th May 2007
14:30 to 15:30
Weak convergence, realisation of holonomic constraints, and the quantum adiabatic theorem
HOP 31st May 2007
14:30 to 15:30
E Celledoni Simulation of rigid body dynamics using Jacobi elliptic functions and integrals
HOP 7th June 2007
16:00 to 17:00
A mathematical study of the Car-Parrinello method
HOP 11th June 2007
14:30 to 15:30
P Olver Invariantisation of numerical schemes
HOP 13th June 2007
11:30 to 12:30
On the approximation of spectra and pseudospectra of non-self-adjoint Hilbert space operators
HOP 13th June 2007
14:30 to 15:30
P Deift Riemann Hilbert problems: theory and applications
HOP 14th June 2007
14:30 to 15:30
M Ganesh A spectrally accurate algorithm for electromagnetic scattering in three dimensions
HOPW07 19th June 2007
10:30 to 11:00
Efficient evaluation of high-frequency propagation and scattering with applications to propagation in non-spherically-symmetric atmospheres
HOPW07 19th June 2007
11:00 to 11:30
High frequency scattering by simple ice crystal shapes
HOPW07 19th June 2007
11:30 to 12:00
M Ganesh An integral equation algorithm for surface scattering
HOPW07 19th June 2007
12:00 to 12:30
P Childs Aspects of seismic inversion
HOPW07 19th June 2007
14:00 to 14:30
Asymptotics for high-frequency multiple scattering
HOPW07 19th June 2007
14:30 to 15:00
Strategies for the evaluation of boundary integrals arising in 3D PUBEM for Helmholtz problems
HOPW07 19th June 2007
15:00 to 15:30
Modelling of high intensity focused ultrasound propagation for cancer therapy
HOPW07 19th June 2007
16:00 to 16:30
High frequency scattering problems from industrial applications
HOP 20th June 2007
14:30 to 15:00
Reconsidering trigonometric integrators
HOP 20th June 2007
15:00 to 15:30
M Khanamirian Numerical solution to highly oscillatory systems of ODEs
HOP 20th June 2007
15:30 to 16:00
From the wave packet estimate to exact observability
HOP 21st June 2007
16:00 to 17:00
The fast evaluation of matrix functions for exponential integrators
HOP 22nd June 2007
15:30 to 16:30
Homogenisation: Analysis and Computation
HOP 27th June 2007
14:30 to 15:00
Interpolation in special orthogonal groups
HOP 27th June 2007
15:00 to 15:30
Fourier analysis of the 1-d discontinuous Galerkin methods
HOP 27th June 2007
15:30 to 16:00
Spectral methods and modified Fourier series
HOP 28th June 2007
14:30 to 15:30
VH Hoang Sparse finite element method for nonlinear monotone problems with multiple scales
HOPW05 2nd July 2007
10:00 to 11:00
Fast high-order high-frequency solvers in computational acoustics and electromagnetics

The numerical solution of highly oscillatory wave-propagation and scattering problems presents a variety of significant challenges: these problems require high discretization densities and often give rise to poorly conditioned numerics; realistic engineering configurations, further, usually require consideration of geometries of great complexity and large extent. In this talk we will consider a number of methodologies that were introduced recently to address these difficulties. We will thus discuss algorithms that can solve, with high-order accuracy, problems of scattering for complex three-dimensional geometries---including, possibly, singular elements such as wires, corners, edges and open screens. In particular, we will describe solutions achieved for two realistic three-dimensional problems of very high frequency---surface scattering and atmospheric GPS propagation---which previous three-dimensional solvers could not address adequately.

For added efficiency, these solvers, which are based on integral equations, high-order integration and fast Fourier transforms, can be used in conjunction with new regularized combined field equations---which require much smaller numbers of iterations in a iterative linear algebra solver than combined field equations available previously. We will also describe a new class of high-order surface representation methods which, starting from point clouds or CAD data, can produce high-order-accurate surface parametrizations of complex engineering surfaces, as required by high-order solvers. Time permitting, applications of these methodologies to solution of time-domain problems and fast evaluation of fully-nonlocal and convergent computational boundary conditions for time-domain problems will be mentioned. In all cases these algorithms exhibit high-order convergence, they run on low memories and reduced operation counts, and they can produce solutions with a high degree of accuracy.

HOPW05 2nd July 2007
11:30 to 12:00
Domain decomposition for multiscale elliptic PDEs

In the Monte-Carlo simulation of fluid flow in stochastic media one typically has to solve elliptic PDEs with highly oscillatory coefficients. In practical applications in hydrogeology, these (random) oscillatory coefficients can have wavelength of order 10^{-3} or smaller and amplitude of the order of 10^{8}. Accurate finite element approximation in 2D requires of the order of 10^{8} degrees of freedom and the resulting linear systems can have condition number close to 10^{16}.

In this talk we will discuss domain decomposition preconditioning for such linear systems. Our overall aim is to solve such problems in a time close to the time required for solving a discretisation of a standard Poisson problem with constant coefficient with the same number of degrees of freedom.

The essential step in constructing two- or multi-level preconditioners is to replace the discretisation on the finest grid with a suitable discretisation on a coarser grid or grids. In the present application, standard (piecewise polynomial based) coarsening fails because, even if the fine mesh resolves the oscillations in the coefficient, coarser meshes typically fail to do so.

By extending the classical domain decomposition theory to this case, we show that a suitable coarsening strategy for heterogeneous media involves the construction of low energy coarse space basis functions. This naturally suggests that multiscale finite element methods can provide good coarse spaces, and leads to a new class of domain decomposition preconditioners. Recent results of Scheichl and Vainikko used the same theoretical technique to explain the robustness of certain algebraically defined preconditioners.

The theoretical results are illustrated by numerical examples on deterministic and random problems.

Reference:

I.G. Graham, P. Lechner and R. Scheichl, Domain Decomposition for Multiscale PDEs, Numer. Math. DOI 10.1007/s00211-007-0074-1 (2007) .

HOPW05 2nd July 2007
14:00 to 15:00
E Zuazua An alternate direction descent method for the control of flows in the presence of shocks

We present a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1-d inviscid Burgers equation. We first prove the existence of minimizers and, by a Gamma-convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes satisfying the so called one-sided Lipschitz condition (OSLC). Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches. The first one, the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a remedy we propose a new method that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions. We develop a new descent stratagey, that we shall call "alternating descent method", distinguishing descent directions that move the shock and those that perturb the profile of the solution away of it. As we shall see, a suitable alternating combination of these two classes of descent directions allows building very efficient and fast descent algorithms.

HOPW05 2nd July 2007
15:30 to 16:00
Condition number estimates for oscillatory integral operators

We present results obtained during the programme on the conditioning of the standard combined potential boundary integral operators in time-harmonic acoustic scattering (cf. the talk by Oscar Bruno), in particular addressing behaviour as the wavenumber k tends to infinity, when the integral operator becomes increasingly oscillatory. While study of this topic goes back to Kress and Spassov (Numer. Math. 1983), the focus previously has been on the canonical case of a circle/sphere for which spherical harmonics are the eigenfunctions and the singular values are known explicitly. However, even for this case, it is only recently (Dominguez, Graham, Smyshlyaev, Preprint NI07004-HOP and Numer. Math. 2007) that rigorous upper bounds have been obtained on the operator and its inverse as a function of k. For non-spherical scatterers the only result is a recent upper bound on the inverse operator for piecewise smooth starlike domains (Chandler-Wilde & Monk, to appear SIAM J. Math. Anal.). In this talk we derive a range of lower bounds on the operator and its inverse, which show that the behaviour for large k depends subtly on the geometry. The sharpness of these lower bounds is demonstrated by numerical simulation and, in some instances, by provable upper bounds. The main computational message is that, while the condition number grows only mildly (like k^{1/3}) for a circle/sphere, behaviour can be much worse (like k^{5/4}) for non-starlike domains.

HOPW05 2nd July 2007
16:00 to 16:30
New Galerkin methods for high-frequency scattering simulations

Recently developed integral equation methods for surface scattering simulations ---that combine the advantages of rigorous solvers (error controllability) with those of asymptotic methods (frequency-independent discretizations)--- have displayed the potential of delivering scattering returns within a prescribed error-tolerance in times that do not depend on the wavenumber k. The near-optimal characteristics of these novel schemes, pioneered by Bruno, Geuzaine, Monro and Reitich, have rapidly generated great interest and significant number of work in recent years.

In single-scattering configurations, an actual proof that provides a rigorous upper bound for the operation count of O(k^{1/9}) in the case of circular/spherical boundaries was recently established by Dominguez, Graham and Smyshlyaev for a p-version boundary element implementation of a similar approach. As in the original algorithm, they take profound advantage of the exponential decay (with increasing wavenumber k) of the corresponding physical density in the deep shadow region, and approximate this quantity by zero there.

In this talk, we present two improved Galerkin schemes for the solution of single-scattering problems, and we show that, within the framework of our first scheme, the error in best approximation of the surface current grows at most at O(k^a) (for any a>0) over the "entire" boundary. Moreover, as we show, our second approach based on a novel change of variables around the "transition regions" reduces this dependency to O(log(k)).

HOPW05 2nd July 2007
16:30 to 17:00
Plane wave discontinuous Galerkin methods

Standard low order Lagrangian finite element discretization of boundary value problems for the Helmholtz equation $-\Delta u - \omega^{2}u=f$ are afflicted with the so-called pollution phenomenon: though for sufficiently small $h\omega$ an accurate approximation of $u$ is possible, the Galerkin procedure fails to provide it. Attempts to remedy this have focused on incorporating extra information in the form of plane wave functions $\boldsymbol{x}\mapsto \exp(i\omega\boldsymbol{d}\cdot\boldsymbol{x})$, $|\boldsymbol{d}|=1$, into the trial spaces. Prominent examples of such methods are the plane wave partition of unity finite element method of Babuska and Melenk, and the ultra-weak Galerkin discretization due to Cessenat and Despres. Both perform well in computations.

It turns out that the latter method can be recast as a special so-called discontinuous Galerkin (DG) method employing local trial spaces spanned by a few plane waves. This perspective paves the way for marrying plane wave approximation with many of the various DG methods developed for 2nd-order elliptic boundary value problems. We have pursued this for a generic mixed DG method and a primal DG method which generalized the ultra-weak scheme.

For these methods we have developed a convergence analysis for the $h$-version, which achieves convergence through mesh refinement. Key elements are approximation estimates for plane waves and sophisticated duality techniques. The latter entail estimating how well local plane waves can approximate the solution of a dual problem. Unfortunately, we could not help invoking general polynomial estimates in Sobolev spaces for this purpose. This incurs unsatisfactory pollution-affected final error bounds $O(\omega^{2}h)$. On the other hand, a more detailed 1D analysis confirms that the plane wave DG method does not suffer from pollution. To bridge the gap is an open theoretical challenge.

HOPW05 3rd July 2007
09:00 to 10:00
Integrators for Highly oscillatory Hamiltonian systems: an homogenisation approach

We introduce a class of symplectic (and in fact also non symplectic) schemes for the numerical integration of highly oscillatory Hamiltonian systems. The bottom line for the approach is to exploit the Hamilton-Jacobi form of the equations of motion. Because we perform a two-scale expansion of the solution of the Hamilton-Jacobi equations itself, we readily obtain, after an appropriate discretization, symplectic integration schemes. Adequate modifications also provide non symplectic schemes. The efficiency of the approach is demonstrated using several variants.

This is joint work with F. Legoll (LAMI-ENPC, France)

HOPW05 3rd July 2007
10:00 to 10:30
Generalised polynomial chaos for analysing the stability

If a dynamical system exhibits a periodic response, analysing the stability of this state yields crucial information. We consider oscillators modelled by systems of ordinary differential equations or differential algebraic equations. Hence we focus on stability properties of periodic solutions with respect to perturbations in corresponding initial values. Floquet theory represents a local concept for analysing the stability. Alternatively, we consider a stochastic perturbation following some probability distribution to obtain global information on stability. This strategy yields a system with stochastic input parameters. Thus results concerning the expected value and the variance of the corresponding solution are of interest. Monte Carlo methods can be used to compute these key figures approximately, where often a huge number of realisations is required. We apply the alternative approach of generalised polynomial chaos to obtain according approximations. Numerical simulations of oscillators using this strategy are presented.

HOPW05 3rd July 2007
11:00 to 11:30
Adaptive parameterization in molecular dynamics

I will describe work on formulation and numerical methods for molecular dynamics. Increasingly, in tackling complex models, it is necessary to manipulate the model "on-the-fly" during simulation, examples being quantum-mechanical/molecular mechanical models (QM/MM) and coarse-graining in molecular dynamics. I will discuss a new technique for adaptive parameterization of molecular landscapes.

HOPW05 3rd July 2007
11:30 to 12:00
Splitting methods for oscillatory non-autonomous linear systems

Several relevant physical phenomena are modelled (totally or partially) by linear PDEs that, once spatially discretized, give rise to systems of coupled harmonic oscillators. To diagonalise these systems is usually prohibitively expensive. Then, to use splitting methods (involving matrix-vector products and possibly FFTs) is a valid alternative. A theoretical analysis about the stability and accuracy of splitting methods on the harmonic oscillator allows us to build new methods which outperform the existing methods from the literature.

Non-autonomous problems are also of great interest but, the most efficient methods for the autonomous case are not valid in this setting. From the Magnus series expansion (as a formal solution to the non-autonomous problem) we show how to adapt these methods by treating the "time" separately from the coordinates. This technique allows us to build new methods whose performance is tested on the Schrödinger equation with time-dependent potentials.

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HOPW05 3rd July 2007
14:00 to 15:00
Treatment of oscillations in classical molecular dynamics

Atomic oscillations present in classical molecular dynamics (MD) restrict the step size that can be used. These oscillations are decidedly nonlinear, and there is only a modest separation of temporal scales. For reasons to be explained, amplitudes of the highest frequency oscillations are small, and, hence, MD is an example of a (mildly) stiff oscillatory problem. Actually, getting accurate trajectories is not usually the aim of MD; instead, time averages or time correlation functions are sought, and the implications of this are examined. Two techniques for lengthening the step size have found general use: multiple time stepping and bond-length constraints. Accuracy and stability of these techniques is discussed.

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HOPW05 3rd July 2007
15:30 to 16:00
The Magnus expansion in the adiabatic picture

In this talk we review the main aspects of the Magnus expansion for treating time-dependent systems evolving in a near-adiabatic regime. We discuss its applicability and the convergence domain. We will show, in particular, how this procedure alllows us to obtain very accurate approximations to the transition probability for time-dependent two-state quantum systems even far from the adiabatic limit.

HOPW05 3rd July 2007
16:00 to 16:30
The separable shadow hybrid Monte Carlo (S2HMC) method for improved performence over hybrid Monte Carlo

Hybrid Monte Carlo (HMC) is a rigorous sampling method that uses molecular dynamics (MD) as a global Monte Carlo move. The acceptance rate of HMC decays exponentially with system size. The Shadow Hybrid Monte Carlo (SHMC) was previously introduced to overcome this performance degradation by sampling instead from the shadow Hamiltonian defined for MD when using a symplectic integrator. However SHMC's performance is limited by the need to generate momenta for the MD step from a non-separable shadow Hamiltonian. The Separable Shadow Hybrid Monte Carlo (S2HMC) method, based on a separable formulation of the shadow Hamiltonian, allows allows efficient generation of momenta and retains the advantage of SHMC.

HOPW05 3rd July 2007
16:45 to 17:15
Level set methods for capturing semiclassical dynamics of Schr\"odinger equations with different potentials

In this talk we present newly developed level methods for capturing semiclassical dynamics of Schr\"{o}dinger equations with different potentials. We discuss the essential ideas behind the techniques, the coupling of these techniques to handle several canonical potentials, including the phase space based level set method for given smooth potentials; the field space based level set method for self-consistent potentials governed by the Poisson equation; as well as the Bloch-band based level set method for periodic potentials. The relations between computed multi-valued solutions and desirable physical observables are established. Numerical examples are presented to validate the numerical methods.

HOPW05 3rd July 2007
17:15 to 17:45
Adiabatic invariance and geometric phase in slowly deforming domains

We consider the evolution of a 2d perfect fluid as its domain is deformed slowly in a prescribed fashion. Subject to some assumptions, the leading-order Eulerian flow is found to be steady and depend only on the instantaneous form of the boundary; it is thus an adiabatic invariant of the system. Also to leading order, the Lagrangian particle trajectory is found to consist of a dynamical and a geometric component, in the fashion of the Hannay-Berry phase.

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HOPW05 4th July 2007
09:00 to 10:00
MCMC methods for multiscale measure

In many application domains there is growing interest in the fitting of stochastic differential equations (SDEs, diffusions) to data. The data may come from experimental observations or from large scale computer simulations. In many case the data has a multiscale character which is incompatible with a (or the desired) diffusion process at small scales. However it may be compatible at intermediate scales. In order to understand this situation I will study the fitting of SDEs to data generated by multiscale diffusion processes, in situations where averaging and homogenization apply. The parametric model will be the averaged or homogenized equation; the data, however, will be chosen from the multiscale model. Understanding this mismatch between data and model will shed light on the original problem of incompatibility between model and data at small scales.

HOPW05 4th July 2007
10:00 to 10:30
Multiscale numerical methods using limited global information and applications

In this talk, I will describe multiscale numerical methods for flows in heterogeneous porous media. The main idea of these methods is to construct local basis functions that can capture the small scale information when they are coupled via some global formulation. I will discuss the use of local boundary conditions, oversampling methods and the use of global information in constructing basis functions. Applications of these methods to stochastic equations will be also discussed. This is joint work with J. Aarnes, T. Hou, L. Jiang, V. Ginting.

HOPW05 4th July 2007
11:00 to 11:30
A Boutet de Monvel Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition

We consider the focusing NLS equation on the quarter plane. Initial data are vanishing at infinity while boundary date are time-periodic (ae^[2i\omega t]). The main tool is the asymptotic analysis of the associated matrix Riemann-Hilbert problem. We will show that we obtain 4 different asymptotics in different regions:

region 1: a Zakharov-Manakov vanishing asymptotics region 2: a train of asymptotics solitons region 3: a modulated elliptic wave region 4: a plane wave.

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HOPW05 4th July 2007
11:30 to 12:00
S Reich A Mach-uniform algorithm based on regularised fluid equations

A Mach-uniform algorithm is an algorithm suitable for simulations at any level of the Mach number. Semi-implicit time discretization methods and pressure correction algorithms are among the popular choices to obtain Mach-uniform algorithms with applications to meteorology and combustion simulations. Here we suggest an alternative (although related) approach based on a regularized pressure formulation of the Euler equations. The equations are integrated in time by a staggered time-stepping method which avoids artificial numerical damping of large scale accoustic waves. The regularization is motivated by a simple spatial multi-scale analysis.

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HOPW05 4th July 2007
14:00 to 15:00
A Abdulle S-ROCK: explicit methods for stiff stochastic problems

Abstract:

In this talk, we discuss a new class of methods for the solution of stiff stochastic differential equations. (This is a joint work with S. Cirilli).

HOPW05 4th July 2007
15:30 to 16:00
S Smitheman The Dirichlet to Neumann map for the modified Helmholtz and Helmholtz equations with complex boundary data

We present a spectral collocation type method for computing the Dirichlet to Neumann map for the modified Helmholtz equation. For regular and irregular polygons, we demonstrate quadratic convergence for sine basis functions and exponential convergence for Chebyshev basis functions.

We go on to outline how our method can be extended to the Helmholtz equation, for which we also present numerical results.

Our work is an extension of previous results of Prof. Fokas and collaborators for the Laplace equation (J. of Comput. and Appl. Maths. 167, 465-483 (2004)).

HOPW05 4th July 2007
16:00 to 16:30
Mathematical analysis and numerical simulation of Bose-Einstein condensation

In this talk, I review the mathematical results of the dynamcis of Bose-Einstein condensate (BEC) and present some efficient and stable numerical methods to compute ground states and dynamics of BEC. As preparatory steps, we take the 3D Gross-Pitaevskii equation (GPE) with an angular momentum rotation, scale it to obtain a four-parameter model and show how to reduce it to 2D GPE in certain limiting regimes. Then we study numerically and asymptotically the ground states, excited states and quantized vortex states as well as their energy and chemical potential diagram in rotating BEC. Some very interesting numerical results are observed. Finally, we study numerically stability and interaction of quantized vortices in rotating BEC. Some interesting interaction patterns will be reported.

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HOPW05 4th July 2007
16:45 to 17:15
Radial basis function collocation for Schr\"odinger's equation

Gaussian radial basis functions may provide an efficient alternative to standard function spaces used in computational quantum mechanics. Combining them with a collocation spatial discretization may give simple and efficient integrators for the time dependent Schrodinger equation. My talk will describe an exploration of these possibilities.

HOPW05 4th July 2007
17:15 to 17:45
A Zanna Exact and approximate methods for the free rigid body

We consider both exact and approximate methods for the free rigid body equations. In particular, we are interested in understanding whether and when exact algorithms can be competitive with approximate ones. Both the exact and the approximate methods are extensively tested for several problems, including satellite and molecular dynamics simulations.

HOPW05 5th July 2007
09:00 to 10:00
S Vandewalle Numerical integration of highly oscillatory functions based on analytic continuation

We consider the integration of highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived for a fairly general class of oscillatory integrals with an analytic integrand. The accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency, and for the case of an increasing number of points and fixed frequency. These results are then used to obtain quadrature rules for more general oscillatory integrals, i.e., for functions that exhibit some smoothness but that are not analytic.

The approach described in this paper is related to the steepest descent or saddle point method from complex analysis. However, it does not employ asymptotic expansions. It can be used for small or moderate frequencies as well as for very high frequencies. We consider both the one-dimensional case, and the multi-dimensional case. Finally, we briefly elaborate on the use of the new integration rules in the context of solving highly oscillatory integral equations.

HOPW05 5th July 2007
10:00 to 10:30
H Brunner Open problems in the computational solution of volterra functional equations with highly oscillatory solutions

The approximation of solutions to Volterra integral equations by collocation or discontinuous Galerkin methods leads to a set of `semi-discretised' equations that in general are not amenable to numerical computation: an additional discretisation process that is able efficiently and accurately to cope with the highly oscillatory nature of the kernel of the given Volterra integral operator is needed. Here, the use of Filon-type quadrature is an obvious possibility; however, it is not yet clear how best to do this when the kernel is weakly singular.

In this talk I will describe current work related to the above problem, and especially to collocation methods for various types of Volterra functional equations, including equations with variable (and possibly vanishing) delay arguments. It will also be shown that in the case of Volterra integral equations of the first kind, the choice of the quadrature scheme in discontinuous Galerkin methods can have a major effect on the convergence properties of the approximate solution.

HOPW05 5th July 2007
11:00 to 11:30
Asymptotic least squares approximation for highly oscillatory differential equations

This talk presents a new approach for approximating highly oscillatory ordinary differential equations. By using the asymptotic expansion in a least squares system, we are able to obtain a result that preserves the asymptotic accuracy of the expansion, while converging rapidly to the exact solution. We are thus able to accurately approximate such differential equations by solving a very small linear system. We apply this method to the computation of highly oscillatory integrals, as well as second order oscillatory differential equations.

HOPW05 5th July 2007
11:30 to 12:00
A Iserles Modified fourier expansions and spectral problems for highly oscillatory Fredholm operators

Although highly oscillatory Fredholm operators are compact and have point spectrum, their calculation by standard means, e.g. the finite section method, is notoriously difficult. As an alternative, we propose expanding the underlying eigenfunctions in modified Fourier series. This leads to infinite-dimensional algebraic eigenvalue problems that exhibit intriguing structure and rapid decay of coefficients. This is exploited in an effective numerical algorithm.

HOPW05 5th July 2007
14:00 to 15:00
Approximation of non-adiabatic quantum dynamics by surface hopping

The time-dependent Schroedinger equation provides the fundamental description of quantum mechanical molecular dynamics. Its multiscale character suggests a splitting in two coupled subproblems, the so called Born-Oppenheimer approximation: One solves a family of stationary Schroedinger equations in the electronic degrees of freedom (one equation for each nucelonic configuration) and subsequently a time-dependent Schroedinger equation in the nucleonic degrees of freedom, whose potential has been determined by the electronic problem. This splitting fails to provide an approximation, if different electronic eigenvalues are not uniformly separated for all nucleonic configurations. The talk explains a microlocal point of view on this non-adiabatic coupling between electronic and nucleonic degrees of freedom and derives an associated deterministic surface hopping algorithm. Its numerical realization crucially relies on the sampling of highly oscillatory initial data on high-dimensional configuration spaces, which can be tackled by a Monte Carlo approach. The presented results are joint work with C. Fermanian, S. Kube, and M. Weber.

HOPW05 5th July 2007
15:30 to 16:00
C Sparber Asymptotics for linear and nonlinear Schr\"odinger equations with periodic potentials

We consider semiclassically scaled linear and nonlinear Schrödinger equation with highly oscillatory periodic potentials. We will discuss rigorous asymptotic descriptions for such problems as well as a recently developed numerical approach based on Bloch decomposition.

HOPW05 5th July 2007
16:00 to 16:30
Computational quantum dynamics using Hagedorn wavepackets

This talk, which is based on joint work with Erwan Faou and Vasile Gradinaru, reports on work in progress on a newly developed numerical approach to many-body quantum dynamics in the semi-classical regime. We present a symmetric splitting integrator for the propagation of multidimensional extensions of Gauss-Hermite wavepackets appearing in analytical work by Hagedorn. The integrator evolves positions and momenta of the wavepackets according to the Stoermer-Verlet integrator of classical mechanics, and gains its computational feasibility and efficiency for many particles by the possibility of thinning out the moving basis sets according to a hyperbolic cross approximation or a Hartree-type approximation in a moving frame. The algorithm reduces to the Strang splitting of the Schroedinger equation in the limit of the full set of orthonormal basis functions, and it is robust in the semiclassical limit.

HOPW05 5th July 2007
16:45 to 17:15
On macroscale variables for multiple time scale problems
HOPW05 5th July 2007
17:15 to 17:45
A multiscale method for computing highly oscillatory ODEs

We prsent a multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to $\epsilon$ perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam.

HOPW05 6th July 2007
09:00 to 10:00
Partitioned Runge-Kutta methods in space and time

In a partitioned RK (PRK) method, different coefficients are used for different variables. This allows, for example, explicit symplectic integrators for separable Hamiltonians; in fact, these are a special case of splitting methods. There also exist implicit symplectic PRK methods, but these seem to have no clear application, since non- partitioned (Gauss) methods dominate them. Recently, a possible application has emerged for implicit symplectic PRK methods in the spatial discretization of Hamiltonian PDEs. Here they can yield stable multisymplectic methods, while by contrast, explicit PRK methods are unconditionally unstable.

HOPW05 6th July 2007
10:00 to 10:30
High oscillations versus parasitic solutions

The theory of modulated Fourier expansions is a powerful tool for the study of the long-time behaviour of differential equations with highly oscillatory solutions (conservation of energy, momentum, and harmonic actions).

There is a discrete analogue that permits to study the long-time behaviour of linear multistep methods applied to (non-oscillatory) Hamiltonian systems. The parasitic solutions of the difference equations play the role of harmonic oscillations.

In this talk we explain the common ideas of both theories. This is joint-work with Christian Lubich.

Related Links

HOPW05 6th July 2007
11:00 to 11:30
Towards symplectic Lie group integrators

We will discuss recent progress towards analyzing and constructing symplectic Lie group integrators. Potential applications for Lie Poisson systems acted upon by the coadjoint action, and other systems with a symplectic group action.

HOPW05 6th July 2007
11:30 to 12:00
Eulerian and semi-Lagrangian exponential integrators for convection dominated problems

We consider a new class of integration methods particularly suited for convection diffusion problems with dominating convection. These methods are exponential integrators and their peculiarity is that they allow for the computation of exponentials of the linearized convection term.

The main reason for developing this type of methods is that as it turns out they can be applied to the numerical integration of the considered PDEs in a semi-Lagrangian fashion. The main challenge in the numerical approximation of convection dominated phenomena is to avoid the occurrence of spurious oscillations in the numerical solution, (numerical dispersion), without adding diffusion. This task is achieved nicely by semi-Lagrangian methods. In these methods linear convective terms are integrated exactly by computing first the characteristics corresponding to the grid ponts of the adopted discretization, and then producing the numerical approximation via a simple although expensive interpolation procedure.

HOPW05 6th July 2007
14:00 to 14:30
High order coefficient approximations methods for the Sturm-Liouville boundary value problem

The determination of the eigenvalues of Sturm-Liouville problems is of great interest in mathematics and its applications. However most eigenvalue problems cannot be solved analytically, and computationally efficient approximation techniques are of great applicability. An important class of methods obtain eigenvalue approximations by applying an integrator based on coefficient approximation in a shooting process. These coefficient approximation methods replace the coefficient functions of the Sturm-Liouville equation by simpler approximations and then solve the approximating problem. The standard reference in the piecewise constant approximation case is due to S. Pruess [1], and therefore the methods are often referred to as Pruess methods. The Pruess method has some significant advantages. While a naive integrator is forced to make increasingly smaller steps in the search for large eigenvalues (due to the increasingly oscillatory nature of the solution), the stepsize is not restricted by the oscillations in the solution for a Pruess method. A drawback of the Pruess methods is the difficulty in obtaining higher order methods; unless Richardson extrapolation is used the method is only second order.

Higher order methods based on coefficient approximation can be realized using a perturbation technique. This approach leads to the so-called Piecewise Perturbation Methods (PPM) [2]. The PPM add some perturbation corrections to the solution of the approximating problem in order to obtain a more accurate approximation to the solution of the original problem. High order PPM were found to be well suited to be used in a shooting procedure to compute the eigenvalues efficiently and accurately. This resulted in a Matlab software package which can be used to compute the eigenvalues of a Sturm-Liouville or Schr\"odinger problem up to high accuracy ({\sc MATSLISE} [3]).

Recently it was shown that the piecewise perturbation approach may be viewed as the application of a modified Neumann expansion [4]. The excellent performance of piecewise perturbation methods for the Sturm-Liouville problem can thus be seen as a convincing illustration of the power and potential of the Neumann series integrators. Another integral series which has been recognized as a very effective computational tool for problems with highly oscillatory solution, is the Magnus expansion. Also integrators based on this Magnus expansion can be combined with coefficient approximation and form another extension of the Pruess ideas to high order approximations. \\

[1] Pruess, S. Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation. SIAM J. Numer. Anal. 10 (1973).

[2] Ixaru, L. Gr., De Meyer, H. and Vanden Berghe, G. CP methods for the Schr\"odinger equation revisited, J. Comput. Appl. Math. 88 (1997).

[3] Ledoux, V., Van Daele, M., and Vanden Berghe, G. Matslise: A matlab package for the numerical solution of Sturm-Liouville and Schrodinger equations. ACM Trans. Math. Software 31 (2005).

[4] Degani, I., AND Schiff, J. RCMS: Right Correction Magnus Series approach for oscillatory ODEs. J. Comput. Appl. Math. 193 (2006).

HOPW05 6th July 2007
14:30 to 15:00
Numerical investigation of the conjugate locus for the Euler top

Conjugate and cut loci of geodesic flows have a significant interest in Riemannian geometry, but few examples are known especially on manifolds of dimension greater than 2. An interesting example from mechanics is given by the flow of the Euler top, namely, a geodesic flow on SO(3) with a left invariant metric. The (first) conjugate locus can be determined analytically if two of the three moments of inertia of the body are equal. In that case, the conjugate locus is either a segment or circle (if the body is oblate) or a non-injective mapping of an astroid of revolution (if the body is prolate) [Bates and Fasso` 2006, see the link below]. The analytic study of the generic case of distinct moments of inertia is much more difficult, if not even prohibitive. We thus resort to the numerical construction of the conjugate locus, based on accurate numerical integrations of the flow and of its tangent map. The dependency of the conjugate locus on the moments of inertia is studied in a deformation scenario from the symmetric case. (Joint collaboration with L. Bates).

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HOPW05 6th July 2007
15:30 to 16:00
Lie group and homogeneous variational integrators and their applications to geometric optimal control theory

The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.

We will discuss the application of geometric structure-preserving numerical schemes to the optimal control of mechanical systems. In particular, we consider Lie group variational integrators, which are based on a discretization of Hamilton's principle that preserves the Lie group structure of the configuration space. In contrast to traditional Lie group integrators, issues of equivariance and order-of-accuracy are independent of the choice of retraction in the variational formulation. The importance of simultaneously preserving the symplectic and Lie group properties is also demonstrated.

In addition, we will introduce a numerically robust shooting based optimization algorithm that relies on the conservation properties of geometric integrators to accurately compute sensitivity derivatives, thereby yielding an optimization algorithm for the control of mechanical systems that is exceptionally efficient. The role of geometric phases in these control algorithms will also be addressed.

Recent extensions to homogeneous spaces yield intrinsic methods for Hamiltonian flows on the sphere, and have potential applications to the simulation of geometric exact rods, structures and mechanisms.

The research has been supported in part by NSF grant DMS-0726263 and DMS-0504747.

Related Links

HOPW05 6th July 2007
16:00 to 16:30
R Melnik Spurious solutions in applications of effective envelope theory: roots of the problem and methodologies for its remedy

The appearance of spurious solutions is a long standing problem in the application of the envelope function approximation. The subject of this talk is the exact envelope function theory developed by M.G. Burt and further extended by B.A. Foreman. Using this theory, multiband models can be derived for nanoscale heterostructures, similar to the “effective mass” models developed by J.M. Luttinger and W. Kohn that opened the door to modern treatments of important classes of bulk semiconductor materials and nanostructures. In applying this approach, we have to undergo several levels of approximations of the Hamiltonian. In this talk we analyze such approximations critically and show how spurious non-physical solutions in the resulting coupled systems of partial differential equations can be avoided. The discussion is motivated by our research into low-dimensional semiconductor nanostructures, in particular quantum dots.

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HOPW05 6th July 2007
16:30 to 17:00
Backscatter diffraction coefficients for surface-breaking cracks: Comparison of two semi-analytical approaches

Evaluation of diffraction coefficients for surface-breaking cracks has been an outstanding problem in mathematical modelling of ultrasonic inspection for over twenty years. Several groups of applied mathematicians have been involved in developing suitable semi-analytical solutions over an even longer period. Recently two different approaches, one based on the Budaev and Bogy's technique and one based on the Gautesen's method have been developed and coded. We report on their successful cross-validation and validation against experimental data.

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HOPW08 13th September 2010
08:55 to 09:00
Welcome by ESF Rapporteur
HOPW08 13th September 2010
09:00 to 10:00
Non polynomial approximations of wave problems
HOPW08 13th September 2010
10:00 to 11:00
Computing Slowly Advancing Features in Fast-Slow Systems without Scale Separation - A Young Measure Approach
HOPW08 13th September 2010
11:30 to 12:30
Fourier series on triangles and tetrahedra
HOPW08 13th September 2010
14:00 to 14:30
L Gauckler Modulated Fourier expansions for the long-time analysis of Hamiltonian PDEs
HOPW08 13th September 2010
14:30 to 15:00
R Quispel Integral-Preserving Integrators
HOPW08 13th September 2010
15:00 to 15:30
On the numerical solution of first-kind Volterra integral equations with highly oscillatory kernels
HOPW08 13th September 2010
16:00 to 17:00
Multiple integrals under differential constraints: two-scale convergence and homogenization
HOPW08 13th September 2010
17:00 to 18:00
On Wigner and Bohmian Measures
HOPW08 13th September 2010
18:00 to 18:30
E Karatsuba On approximation of special oscillating series of quantum theory
HOPW08 14th September 2010
09:00 to 10:00
A Addulle Numerical integration for numerical homogenizations methods
HOPW08 14th September 2010
10:00 to 11:00
Multi-scale spaces for multiscale high-contrast problems and their applications
HOPW08 14th September 2010
11:30 to 12:30
Computing expectation values for molecular quantum dynamics
HOPW08 14th September 2010
14:00 to 14:30
Variance reduction in stochastic homogenization using antithetic variables
HOPW08 14th September 2010
14:30 to 15:00
Oscillatory systems with three separated timescales
HOPW08 14th September 2010
15:00 to 15:30
Developing Integrators for Highly Oscillatory Hamiltonian Systems Using Homogenization
HOPW08 14th September 2010
16:30 to 17:00
Preconditioning the Helmholtz equation using rapidly oscillating functions
HOPW08 14th September 2010
17:00 to 17:30
Coercivity, Nonnormality and Numerical Range of boundary integral operators in high-frequency scattering
HOPW08 14th September 2010
17:30 to 18:00
Convergent high-frequency algorithms for single and multiple scattering
HOPW08 14th September 2010
18:00 to 18:30
Runga-Kutta convolution quadrature: convergence theory and applications to time-domain BIE of acoustic scattering
HOPW08 15th September 2010
09:00 to 10:00
Modulated Fourier expansions and the Fermi-Pasta-Ulam problem
HOPW08 15th September 2010
10:00 to 11:00
T Lelievre Metastability, rare events and sampling problems in molecular dynamics
HOPW08 15th September 2010
11:30 to 12:00
Approximation by plane waves
HOPW08 15th September 2010
12:00 to 12:30
The generalised eigenfunction method and time-dependent linear water-wave impact on a vertical elastic plate
HOPW08 15th September 2010
14:00 to 14:30
Numerical solution of Riemann–Hilbert problems: Painlevé II
HOPW08 15th September 2010
14:30 to 15:00
J Geier Efficient finite difference schemes for highly oscillatory linear ODE
HOPW08 15th September 2010
15:00 to 15:30
Highly Oscillatory Integrals and their Applications
HOPW08 15th September 2010
16:00 to 17:00
C Sanz-Serna Optimal Tuning of Hybrid Monte Carlo
(This is joint work with A. Beskos, N.S. Pillai, G.O. Roberts and A.M. Stuart).
HOPW08 15th September 2010
17:00 to 18:00
The Hybrid Monte Carlo Algorithm in High Dimensions
HOPW08 15th September 2010
18:00 to 18:30
Error Estimates for Gaussian Beam Superposition
HOPW08 16th September 2010
09:00 to 10:00
H Owhadi Homogenization with non-separated scales and high contrast
HOPW08 16th September 2010
10:00 to 11:00
A new transform method and some of its numerical implementations
HOPW08 16th September 2010
11:30 to 12:00
C-B Schoenlieb Higher-order total variation for oscillating patterns
HOPW08 16th September 2010
12:00 to 12:30
Oscillation Analysis of Some Hybrid Dynamical Systems of Transmission Pipelines
HOPW08 16th September 2010
14:00 to 14:30
Tensor product method for radiative transfer
HOPW08 16th September 2010
14:30 to 15:00
W Hao Bloch Decomposition-Based Gaussian Beam Method for the Schrodinger equation with Periodic Potentials
HOPW08 16th September 2010
15:00 to 15:30
Strichartz estimates for the Schroedinger equation on a tree and applications
HOPW08 16th September 2010
16:00 to 16:30
Upscaling of network models for porous media
HOPW08 16th September 2010
16:30 to 17:00
Thin domains with highly oscillating boundaries
HOPW08 16th September 2010
17:00 to 17:30
Oscillatory diffusion equation
HOPW08 16th September 2010
17:30 to 18:00
An Effective Finite Difference Approach for Optical Beam Propagations
HOPW08 16th September 2010
18:00 to 18:30
Global and Blow-up patterns of the Cauchy problem of a fourth-order thin film equation
HOPW08 17th September 2010
09:00 to 10:00
Applications of High Frequency Methods in Aeronautics: status and needs
HOPW08 17th September 2010
10:10 to 10:40
Nonlinear hyperbolic conservation laws: diffusive-dispersive limits
HOPW08 17th September 2010
10:40 to 11:10
Coercivity of boundary integral operators in high frequency scattering
HOPW08 17th September 2010
11:30 to 12:30
Existence of approximate solitary waves in simplectic algorithms of integration
(co-authors: Erwan Faou and Benoit Grebert)
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons