A fundamental and enduring challenge in science and technology is the quantitative prediction of time-dependent nonlinear phenomena. While dynamical simulation (for ballistic trajectories) was one of the first applications of the digital computer, the problems treated, the methods used, and their implementation have all changed a great deal over the years. Astronomers use simulation to study long term evolution of the solar system. Molecular simulations are essential for the design of new materials and for drug discovery. Simulation can replace or guide experiment, which often is difficult or even impossible to carry out as our ability to fabricate the necessary devices is limited.

During the last decades, we have seen dramatic increases in computing power, bringing to the fore an ever widening spectrum of applications for dynamical simulation. At the boundaries of different modeling regimes, it is found that computations based on the fundamental laws of physics are under-resolved in the textbook sense of numerical methods. Because of the vast range of scales involved in modeling even relatively simple biological or material functions, this limitation will not be overcome by simply requiring more computing power within any realistic time. One therefore has to develop numerical methods which capture crucial structures even if the method is far from “converging" in the mathematical sense. In this context, we are forced increasingly to think of the numerical algorithm as a part of the modeling process itself. A major step forward in this area has been the development of structure-preserving or “geometric" integrators which maintain conservation laws, dissipation rates, or other key features of the continuous dynamical model. Conservation of energy and momentum are fundamental for many physical models; more complicated invariants are maintained in applications such as molecular dynamics and play a key role in determining the long term stability of methods. In mechanical models (biodynamics, vehicle simulation, astrodynamics) the available structure may include constraint dynamics, actuator or thruster geometry, dissipation rates and properties determined by nonlinear forms of damping.

In recent years the growth of geometric integration has been very noticeable. Features such as
*symplecticity*or
*time-reversibility*are now widely recognized as essential properties to preserve, owing to their physical significance. This has motivated a lot of research
,
,
and led to many significant theoretical achievements (symplectic and
symmetric methods, volume-preserving integrators, Lie-group methods, ...). In practice, a few simple schemes such as the Verlet method or the Störmer method have been used for years with great
success in molecular dynamics or astronomy. However, they now need to be further improved in order to fit the tremendous increase of complexity and size of the models.

To become more specific, the project
*IPSO*aims at finding and implementing new structure-preserving schemes and at understanding the behavior of existing ones for the following type of problems:

systems of differential equations posed on a manifold.

systems of differential-algebraic equations of index 2 or 3, where the constraints are part of the equations.

Hamiltonian systems and constrained Hamiltonian systems (which are special cases of the first two items though with some additional structure).

highly-oscillatory systems (with a special focus of those resulting from the Schrödinger equation).

Although the field of application of the ideas contained in geometric integration is extremely wide (e.g. robotics, astronomy, simulation of vehicle dynamics, biomechanical modeling,
biomolecular dynamics, geodynamics, chemistry...),
*IPSO*will mainly concentrate on applications for
*molecular dynamics simulation*and
*laser simulation*:

There is a large demand in biomolecular modeling for models that integrate microscopic molecular dynamics simulation into statistical macroscopic quantities. These simulations involve huge systems of ordinary differential equations over very long time intervals. This is a typical situation where the determination of accurate trajectories is out of reach and where one has to rely on the good qualitative behavior of structure-preserving integrators. Due to the complexity of the problem, more efficient numerical schemes need to be developed.

The demand for new models and/or new structure-preserving schemes is also quite large in laser simulations. The propagation of lasers induces, in most practical cases,
several well-separated scales: the intrinsically highly-oscillatory
*waves*travel over long distances. In this situation, filtering the oscillations in order to capture the long-term trend is what is required by physicists and engineers.

ERC Grant awarded to Erwan Faou for his project GEOPARDI

Nicolas Crouseilles has defended his 'Habilitation à diriger les recherches' in january (14th january 2011).

In many physical situations, the time-evolution of certain quantities may be written as a Cauchy problem for a differential equation of the form

For a given
*flow*of (
). From this point of view, a numerical scheme with step size
*geometric integration*is whether
*intrinsic*properties of

This question can be more specifically addressed in the following situations:

The system (
) is said to be

It is then natural to require that
*symmetric*. Symmetric methods for reversible systems of ODEs are just as much important as
*symplectic*methods for Hamiltonian systems and offer an interesting alternative to symplectic methods.

The system (
) is said to have an invariant manifold

is kept
*globally*invariant by

As an example, we mention Lie-group equations, for which the manifold has an additional group structure. This could possibly be exploited for the space-discretisation.
Numerical methods amenable to this sort of problems have been reviewed in a recent paper
and divided into two classes, according to whether they use

Hamiltonian problems are ordinary differential equations of the form:

with some prescribed initial values

Besides the Hamiltonian function, there might exist other invariants for such systems: when there exist
*integrable*. Consider now the parallelogram
*oriented*areas of the projections over the planes

where
*canonical symplectic*matrix

A continuously differentiable map

A fundamental property of Hamiltonian systems is that their exact flow is symplectic. Integrable Hamiltonian systems behave in a very remarkable way: as a matter of fact,
their invariants persist under small perturbations, as shown in the celebrated theory of Kolmogorov, Arnold and Moser. This behavior motivates the introduction of
*symplectic*numerical flows that share most of the properties of the exact flow. For practical simulations of Hamiltonian systems, symplectic methods possess an important advantage: the
error-growth as a function of time is indeed linear, whereas it would typically be quadratic for non-symplectic methods.

Whenever the number of differential equations is insufficient to determine the solution of the system, it may become necessary to solve the differential part and the constraint part altogether. Systems of this sort are called differential-algebraic systems. They can be classified according to their index, yet for the purpose of this expository section, it is enough to present the so-called index-2 systems

where initial values

and of the so-called hidden manifold

This manifold

There exists a whole set of schemes which provide a numerical approximation lying on

In applications to molecular dynamics or quantum dynamics for instance, the right-hand side of (
) involves
*fast*forces (short-range interactions) and
*slow*forces (long-range interactions). Since
*fast*forces are much cheaper to evaluate than
*slow*forces, it seems highly desirable to design numerical methods for which the number of evaluations of slow forces is not (at least not too much) affected by the presence of fast
forces.

A typical model of highly-oscillatory systems is the second-order differential equations

where the potential

where
*fast*forces deriving from
*slow*forces deriving from

Another prominent example of highly-oscillatory systems is encountered in quantum dynamics where the Schrödinger equation is the model to be used. Assuming that the Laplacian has been
discretized in space, one indeed gets the
*time*-dependent Schrödinger equation:

where

Given the Hamiltonian structure of the Schrödinger equation, we are led to consider the question of energy preservation for time-discretization schemes.

At a higher level, the Schrödinger equation is a partial differential equation which may exhibit Hamiltonian structures. This is the case of the time-dependent Schrödinger equation, which we may write as

where

with the kinetic and potential energy operators

where

The multiplication by

The numerical approximation of ( ) can be obtained using projections onto submanifolds of the phase space, leading to various PDEs or ODEs: see , for reviews. However the long-time behavior of these approximated solutions is well understood only in this latter case, where the dynamics turns out to be finite dimensional. In the general case, it is very difficult to prove the preservation of qualitative properties of ( ) such as energy conservation or growth in time of Sobolev norms. The reason for this is that backward error analysis is not directly applicable for PDEs. Overwhelming these difficulties is thus a very interesting challenge.

A particularly interesting case of study is given by symmetric splitting methods, such as the Strang splitting:

where

The Helmholtz equation models the propagation of waves in a medium with variable refraction index. It is a simplified version of the Maxwell system for electro-magnetic waves.

The high-frequency regime is characterized by the fact that the typical wavelength of the signals under consideration is much smaller than the typical distance of observation of those signals. Hence, in the high-frequency regime, the Helmholtz equation at once involves highly oscillatory phenomena that are to be described in some asymptotic way. Quantitatively, the Helmholtz equation reads

Here,

One important scientific objective typically is to describe the high-frequency regime in terms of
*rays*propagating in the medium, that are possibly refracted at interfaces, or bounce on boundaries, etc. Ultimately, one would like to replace the true numerical resolution of the
Helmholtz equation by that of a simpler, asymptotic model, formulated in terms of rays.

In some sense, and in comparison with, say, the wave equation, the specificity of the Helmholtz equation is the following. While the wave equation typically describes the evolution of waves
between some initial time and some given observation time, the Helmholtz equation takes into account at once the propagation of waves over
*infinitely long*time intervals. Qualitatively, in order to have a good understanding of the signal observed in some bounded region of space, one readily needs to be able to describe the
propagative phenomena in the whole space, up to infinity. In other words, the “rays” we refer to above need to be understood from the initial time up to infinity. This is a central difficulty
in the analysis of the high-frequency behaviour of the Helmholtz equation.

The Schrödinger equation is the appropriate way to describe transport phenomena at the scale of electrons. However, for real devices, it is important to derive models valid at a larger scale.

In semi-conductors, the Schrödinger equation is the ultimate model that allows to obtain quantitative information about electronic transport in crystals. It reads, in convenient adimensional units,

where

Here, the unknown is

The technique consists in solving an approximate initial value problem on an approximate invariant manifold for which an atlas consisting of
*easily computable*charts exists. The numerical solution obtained is this way never drifts off the exact manifold considerably even for long-time integration.

Instead of solving the initial Cauchy problem, the technique consists in solving an approximate initial value problem of the form:

on an invariant manifold
*easily computable*charts exists. If this is the case, one can reformulate the vector field
*easy*way. The main obstacle of
*parametrization*methods
or of
*Lie-methods*
is then overcome.

The numerical solution obtained is this way obviously does not lie on the exact manifold: it lives on the approximate manifold
*close*to each other.

An obvious prerequisite for this idea to make sense is the existence of a neighborhood

where, for convenience, we have denoted

where

Laser physics considers the propagation over long space (or time) scales of high frequency waves. Typically, one has to deal with the propagation of a wave having a wavelength of the order
of

This task has been partially performed in the context of a contract with Alcatel, in that we developed a new numerical scheme to discretize directly the high-frequency model derived from physical laws.

Generally speaking, the demand in developing such models or schemes in the context of laser physics, or laser/matter interaction, is large. It involves both modeling and numerics (description of oscillations, structure preserving algorithms to capture the long-time behaviour, etc).

In a very similar spirit, but at a different level of modelling, one would like to understand the very coupling between a laser propagating in, say, a fiber, and the atoms that build up the fiber itself.

The standard, quantum, model in this direction is called the Bloch model: it is a Schrödinger like equation that describes the evolution of the atoms, when coupled to the laser field. Here the laser field induces a potential that acts directly on the atom, and the link bewteeen this potential and the laser itself is given by the so-called dipolar matrix, a matrix made up of physical coefficients that describe the polarization of the atom under the applied field.

The scientific objective here is twofold. First, one wishes to obtain tractable asymptotic models that average out the high oscillations of the atomic system and of the laser field. A
typical phenomenon here is the
*resonance*between the field and the energy levels of the atomic system. Second, one wishes to obtain good numerical schemes in order to solve the Bloch equation, beyond the oscillatory
phenomena entailed by this model.

In classical molecular dynamics, the equations describe the evolution of atoms or molecules under the action of forces deriving from several interaction potentials. These potentials may be short-range or long-range and are treated differently in most molecular simulation codes. In fact, long-range potentials are computed at only a fraction of the number of steps. By doing so, one replaces the vector field by an approximate one and alternates steps with the exact field and steps with the approximate one. Although such methods have been known and used with success for years, very little is known on how the “space" approximation (of the vector field) and the time discretization should be combined in order to optimize the convergence. Also, the fraction of steps where the exact field is used for the computation is mainly determined by heuristic reasons and a more precise analysis seems necessary. Finally, let us mention that similar questions arise when dealing with constrained differential equations, which are a by-product of many simplified models in molecular dynamics (this is the case for instance if one replaces the highly-oscillatory components by constraints).

In reference
, different parallel algorithms are proposed for the numerical
resolution of the quasi-neutrality equation in the GYSELA code. A set of benchmarks on a parallel machine has permitted to evaluate the performance of the different versions of the
quasi-neutrality solver. In particular, in
, these improvements are combined with memory optimization which enable
a scalability of the GYSELA code up to

The CEMRACS is an annual summer research session promoted by the SMAI. The 15th edition of 2010 has been organized by N. Crouseilles, H. Guillard, B. Nkonga and E. Sonnendrücker around "Numerical modeling of fusion plasmas". The volume gathers artless resulting from research projects initiated during the CEMRACS 2010.

Under certain scaling the nonlinear Schrödinger equation with random dispersion converges to the nonlinear Schrödinger equation with white noise dispersion. The aim of these works is to
prove that this latter equation is globally well posed in

Motivated by paleoclimatological issues, we determine in asymptotic

first exit times for the Chafee-Infante equation forced by heavy-tailed Levy diffusions from reduced domains of attraction in the limit of small intensity. We show that in contrast to the case of Gaussian diffusion the expected first exit times are polynomial in terms of the intensity.

In the first article , we establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system. The primary application of this abstract framework is to infer the local existence of strong, pathwise solutions to the 3D primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise. In the second article global existence is obtained.

The goal of geometric numerical integration is the simulation of evolution equations by preserving their geometric properties over long times. This question is of particular importance in the case of Hamiltonian partial differential equations typically arising in many application fields such as quantum mechanics or wave propagations phenomena. This implies many important dynamical features such as energy preservation and conservation of adiabatic invariants over long times. In this setting, a natural question is to know how and to which extent the reproduction of such long time qualitative behavior is ensured by numerical schemes.

Starting from numerical examples, these notes try to provide a detailed analysis in the case of the Schrödinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. This text analyzes the possible stability and instability phenomena induced by space and time discretization, and provides rigorous mathematical explanations for them.

In we prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation

where

This text generalizes the previous one in the following sense.

In the situation on the previous article, we analyze the first time at which one of the Markov chains reaches its absorbing state. When the number of Markov chains goes to infinity, we
analyze the asymptotic behavior of the system for an
*arbitrary*probability mass function governing the competition. We give conditions for the existence of the asymptotic distribution and we show how these results apply to cluster-based
distributed storage when the competition is handled using a geometric distribution.

The stroboscopic averaging method (SAM) is a technique for the integration of highly oscillatory differential systems

Other participants (outside IPSO) are D. Bouche, J.P. Braeunig, Ch. Steiner, M. Mehrenberger. The main goal of this contract is to determine equivalent equations for standard numerical schemes dedicated to advection equations. In particular, the first term arising in these equivalent equation concerns the numerical diffusion. These computations enable to quantify in a analytical way the numerical diffusion.

Geometric methods and sampling: application to molecular simulation. The project is financed for 3 years, coordinated by Tony Lelièvre and gathers the following teams and persons:

Team of Eric Cancès at CERMICS

Team IPSO

Mathias Rousset from INRIA Lille

Christophe Chipot, from the CNRS in Nancy.

P. Chartier is the coordinator for IPSO.

The full description is available at
https://

Leader E. Sonnedrücker. The full description is available at
http://

Title: Geometric Partial Differential Equations

Instrument: ERC Starting Grant (Starting)

Duration: September 2011 - August 2016

Coordinator: INRIA (France)

See also:
http://

Abstract: The goal is to develop new numerical methods for the approximation of evolution equations possessing strong geometric properties such as Hamiltonian systems or stochastic differential equations. Use intensive numerical simulations to discover and analyze new nonlinear phenomena.

P. Chartier is member of the editorial board of M2AN.

P. Chartier is member of the editorial board of ESAIM Proceedings.

A. Debussche is member of the editorial board of SINUM.

A. Debussche is member of the editorial board of Differential and Integral Equations.

A. Debussche is Director of the mathematics department of the antenne de Bretagne ENS Cachan.

The team organized a workshop on numerical methods for stiff problems, Saint-Malo (January).

F. Méhats and F. Castella were members of the organization and scientific committees of the Conference in honor of N. Ben Abdallah, Toulouse.

P. Chartier was member of the scientific committee of SciCADE11, Toronto, Canada, July 11-15, 2011.

P. Chartier was member of the Commission d'Evaluation at INRIA until june.

P. Chartier is member of the bureau of the Comité des Projets at INRIA-Rennes.

A. Debussche is member of the CNU, Section 26.

P. Chartier: Seminar INRIA Pau, December 12, 2011.

P. Chartier: Seminar University of Geneva, November 2, 2011.

P. Chartier: FOCM'11, Budapest, Hungary, July 4-6, 2011 (Invited Speaker)

P. Chartier: Meeting on Geometric Numerical Integration, Oberwolfach, Germany, March 20-26, 2011 (Invited Speaker)

P. Chartier: Real Matematica Sociedad Espagnola 2011 conference, special session on “Numerical integrators for Hamiltonian systems and related problem”, Avila, February 1-5, 2011 (Invited Speaker)

E. Faou: June 2011, Workshop :
*KAM theory and geometric integrators*, BIRS, Banff, Canada. Organized with W. Craig and B. Grébert.

E. Faou: January 2011, Workshop :
*Advanced Numerical Studies in Nonlinear Partial Differential Equations*, University of Edinburgh, UK. Organized with S. Kuksin, B. Leimkuhler and C. Sulem.

E. Faou: November 2011: Seminar at the CNR, Pavia

E. Faou: October 2011: Séminaire du laboratoire Jacques-Louis Lions, University of Paris 6.

E. Faou: September 2011: Colloque “Rencontre Mathématiques-Mécanique, hommage à Paul Germain”, Congrès français de mécanique, Besançon.

E. Faou: June 2011: Conference:
*Nonlinear Dispersive Partial Differential Equations and Related Topics*, Institut Henri Poincaré, Paris.

E. Faou: June 2011: Seminar at Fields Institute, Toronto (Canada)

E. Faou: March 2011: Workshop on Geometric Numerical Integration, Oberwolfach (Germany)

E. Faou: February 2011: Invitation to the university of Tokyo (Japan).

E. Faou: February 2011: Séminaire
*systèmes dynamiques*, Univ. Paris 7.

F. Castella: talk at the conference in honnor of N. Ben Abdallah, Toulouse.

Arnaud Debussche: September 6-11, mini course on “Stochastic Navier-Stokes equations: well posedness and ergodic properties" in the CIME summer school "Topics in mathematical fluid-mechanics” at Cetraro, Italy.

Arnaud Debussche: july 6 - 8 2011, Workshop:
*FoCM 2011, Conference on the Foundations of Computational Mathematics*, Budapest. Organisation de la session
*Stochastic Computations*avec T. Mueller-Gronbach et B. Baxter.

Arnaud Debussche: january 19 - 21 2011, Workshop:
*Maximum principles, fractional diffusion and differential or integral inequalities for deterministic and stochastic partial differential equations*, Université d'Evry-Val d'Essone.
Organised by L. Denis, P.G. Lemarie-Rieusset and J. Matos.

Arnaud Debussche: January 25 2011: Exposé à la journée de l'ANR HANDDY, université de Nantes.

Arnaud Debussche: February 1-5 2011,
*Congreso de la real Sociedad Matematica Espanola 2011, centenaraio de la RSME*, Avila. Mini course (3h)..

Arnaud Debussche: March 25 2011:
*One day on SPDE and applications*, université du Mans, organisez by A. Matoussi.

Arnaud Debussche: April 28-30:
*Stochastics and Dynamics*, Brown Lefschetz Center for Dynamic Systems, organised by K. Ramanan and B. Rozovsky.

Arnaud Debussche: June 2011: Séminaire EDP, IECN, université de Nancy.

Arnaud Debussche: November 2011: Colloquium de Mathématiques, université de Pau.

Arnaud Debussche: December 2011 :
*Workshop: The stochastic Schrödinger equations in selected physics problems*, ENST, CEA, Gif-sur-Yvette.

Nicolas Crouseilles: Numerical methods for stiff problems in Hamiltonian systems and kinetic equations, at Saint-Malo.
http://

Nicolas Crouseilles: : talk at the meeting of ANR E2T2

Florian Méhats: Colloque "Asymptotic dynamics driven by solitons and traveling fronts in nonlinear PDE", Santiago (Chili).

Florian Méhats: Workshop "Asymptotic Regimes for Schrodinger equation", Vienna (Autriche).

Florian Méhats: Workshop "KAM theory and Geometric Integration", Banff (Canada).

Florian Méhats: Workshop du GdR CHANT "Transport et Nanostructures", Grenoble.

Florian Méhats: Seminar MODANT Grenoble.

Florian Méhats: Colloque "Kinetic models of classical and quantum particle systems", Toulouse.

Florian Méhats: Seminars IHP, Orsay, Rennes.

HdR : Nicolas Crouseilles, “Contributions à la simulation numérique des modèles de Vlasov en physique des plasmas”, Université de Strasbourg