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Workshop Programme

for period 2 - 6 July 2007

Effective Computational Methods for Highly Oscillatory Solutions

2 - 6 July 2007

Timetable

Monday 2 July
08:30-09:55 Registration
09:55-10:00 David Wallace - Welcome
10:00-11:00 Bruno, O (Caltech)
  Fast high-order high-frequency solvers in computational acoustics and electromagnetics Sem 1
 

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.

 
11:00-11:30 Coffee and Poster session
11:30-12:00 Graham, I (Bath)
  Domain decomposition for multiscale elliptic PDEs Sem 1
 

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) .

 
12:30-13:30 Lunch at Wolfson Court (Residents only)
14:00-15:00 Zuazua, E (Madrid)
  An alternate direction descent method for the control of flows in the presence of shocks Sem 1
 

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.

 
15:00-15:30 Tea and Poster session
15:30-16:00 Chandler-Wilde, S (Reading)
  Condition number estimates for oscillatory integral operators Sem 1
 

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.

 
16:00-16:30 Ecevit, F (MPI Leipzig)
  New Galerkin methods for high-frequency scattering simulations Sem 1
 

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)).

 
16:30-17:00 Hiptmair, R (Zuerich)
  Plane wave discontinuous Galerkin methods Sem 1
 

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.

 
17:00-18:00 Welcome Wine Reception and Poster session
18:45-19:30 Dinner at Wolfson Court (Residents only)
Tuesday 3 July
09:00-10:00 le Bris, C (CERMICS - ENPC)
  Integrators for Highly oscillatory Hamiltonian systems: an homogenisation approach Sem 1
 

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)

 
10:00-10:30 Pulch, R (Wuppertal)
  Generalised polynomial chaos for analysing the stability Sem 1
 

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.

 
10:30-11:00 Coffee and Poster session
11:00-11:30 Leimkuhler, B (Edinburgh)
  Adaptive parameterization in molecular dynamics Sem 1
 

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.

 
11:30-12:00 Blanes, S (Valencia)
  Splitting methods for oscillatory non-autonomous linear systems Sem 1
 

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.

Related Links

 
12:30-13:30 Lunch at Wolfson Court (Residents only)
14:00-15:00 Skeel, R (Purdue)
  Treatment of oscillations in classical molecular dynamics Sem 1
 

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.

Related Links

 
15:00-15:30 Tea and Poster session
15:30-16:00 Casas, F (Jaume)
  The Magnus expansion in the adiabatic picture Sem 1
 

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.

 
16:00-16:30 Sweet, C (Notre Dame)
  The separable shadow hybrid Monte Carlo (S2HMC) method for improved performence over hybrid Monte Carlo Sem 1
 

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.

 
16:45-17:15 Liu, H (Iowa)
  Level set methods for capturing semiclassical dynamics of Schr\"odinger equations with different potentials Sem 1
 

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.

 
17:15-17:45 Wirosoetisno, D (Durham)
  Adiabatic invariance and geometric phase in slowly deforming domains Sem 1
 

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.

Related Links

 
18:45-19:30 Dinner at Wolfson Court (Residents only)
Wednesday 4 July
09:00-10:00 Stuart, A (Warwick)
  MCMC methods for multiscale measure Sem 1
 

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.

 
10:00-10:30 Efendiev, Y (Texas)
  Multiscale numerical methods using limited global information and applications Sem 1
 

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.

 
10:30-11:00 Coffee and Poster session
11:00-11:30 Boutet de Monvel, A (Paris)
  Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition Sem 1
 

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.

Related Links

 
11:30-12:00 Reich, S (Potsdam)
  A Mach-uniform algorithm based on regularised fluid equations Sem 1
 

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.

Related Links

 
12:30-13:30 Lunch at Wolfson Court (Residents only)
14:00-15:00 Abdulle, A (Edinburgh)
  S-ROCK: explicit methods for stiff stochastic problems Sem 1
 

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).

 
15:00-15:30 Tea and Poster session
15:30-16:00 Smitheman, S (Cambridge)
  The Dirichlet to Neumann map for the modified Helmholtz and Helmholtz equations with complex boundary data Sem 1
 

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)).

 
16:00-16:30 Bao, W (Singapore)
  Mathematical analysis and numerical simulation of Bose-Einstein condensation Sem 1
 

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.

Related Links

 
16:45-17:15 Degani, I (Bergen)
  Radial basis function collocation for Schr\"odinger's equation Sem 1
 

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.

 
17:15-17:45 Zanna, A (Bergen)
  Exact and approximate methods for the free rigid body Sem 1
 

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.

 
18:00-19:30 CUP Reception (1 Trinity Street, Cambridge)
Thursday 5 July
09:00-10:00 Vandewalle, S (Leuven)
  Numerical integration of highly oscillatory functions based on analytic continuation Sem 1
 

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.

 
10:00-10:30 Brunner, H (Newfoundland)
  Open problems in the computational solution of volterra functional equations with highly oscillatory solutions Sem 1
 

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.

 
10:30-11:00 Coffee and Poster session
11:00-11:30 Olver, S (Cambridge)
  Asymptotic least squares approximation for highly oscillatory differential equations Sem 1
 

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.

 
11:30-12:00 Iserles, A (Cambridge)
  Modified fourier expansions and spectral problems for highly oscillatory Fredholm operators Sem 1
 

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.

 
12:30-13:30 Lunch at Wolfson Court (Residents only)
14:00-15:00 Lasser, C (Berlin)
  Approximation of non-adiabatic quantum dynamics by surface hopping Sem 1
 

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.

 
15:00-15:30 Tea and Poster session
15:30-16:00 Sparber, C (Vienna)
  Asymptotics for linear and nonlinear Schr\"odinger equations with periodic potentials Sem 1
 

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.

 
16:00-16:30 Lubich, C (Tubingen)
  Computational quantum dynamics using Hagedorn wavepackets Sem 1
 

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.

 
16:45-17:15 Engquist, B (Texas at Austin)
  On macroscale variables for multiple time scale problems Sem 1
17:15-17:45 Tsai, Y (Texas at Austin)
  A multiscale method for computing highly oscillatory ODEs Sem 1
 

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.

 
19:30-23:00 Conference Garden Party at Clare Hall
Friday 6 July
09:00-10:00 McLachlan, R (Massey)
  Partitioned Runge-Kutta methods in space and time Sem 1
 

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.

 
10:00-10:30 Hairer, E (Geneve)
  High oscillations versus parasitic solutions Sem 1
 

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

 
10:30-11:00 Coffee and Poster session
11:00-11:30 Munthe-Kaas, H (Bergen)
  Towards symplectic Lie group integrators Sem 1
 

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.

 
11:30-12:00 Celledoni, E (NTNU)
  Eulerian and semi-Lagrangian exponential integrators for convection dominated problems Sem 1
 

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.

 
12:30-13:30 Lunch at Wolfson Court (Residents only)
14:00-14:30 Ledoux, V (Ghent)
  High order coefficient approximations methods for the Sturm-Liouville boundary value problem Sem 1
 

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).

 
14:30-15:00 Fass\'o, F (Padova)
  Numerical investigation of the conjugate locus for the Euler top Sem 1
 

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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15:00-15:30 Tea and Poster session
15:30-16:00 Leok, M (Purdue)
  Lie group and homogeneous variational integrators and their applications to geometric optimal control theory Sem 1
 

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.

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16:00-16:30 Melnik, R (Wilfrid Laurier)
  Spurious solutions in applications of effective envelope theory: roots of the problem and methodologies for its remedy Sem 1
 

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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16:30-17:00 Fradkin, L (London)
  Backscatter diffraction coefficients for surface-breaking cracks: Comparison of two semi-analytical approaches Sem 1
 

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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18:45-19:30 Dinner at Wolfson Court (Residents only)

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