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.

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
y_{0}, the solution
y(
t)at time
tis denoted
. For fixed
t,
becomes a function of
y_{0}called the
*flow*of (
). From this point of view, a numerical scheme with step size
hfor solving (
) may be regarded as an approximation
_{h}of
. One of the main questions of
*geometric integration*is whether
*intrinsic*properties of
may be passed on to
_{h}.

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

The system ( ) is said to be -reversible if there exists an involutive linear map such that

It is then natural to require that
_{h}satisfies the same relation. If this is so,
_{h}is said to be
*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
gwhenever

is kept
*globally*invariant by
. In terms of derivatives and for sufficiently differentiable functions
fand
g, this means that

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
gexplicitly or through a projection step. In both cases, the numerical solution is forced to live on the manifold at the expense of some Newton's iterations.

Hamiltonian problems are ordinary differential equations of the form:

with some prescribed initial values
(
p(0),
q(0)) = (
p
_{0},
q
_{0})and for some scalar function
H, called the Hamiltonian. In this situation,
His an invariant of the problem. The evolution equation (
) can thus be regarded as a differential equation on the manifold

Besides the Hamiltonian function, there might exist other invariants for such systems: when there exist
dinvariants in involution, the system (
) is said to be
*integrable*. Consider now the parallelogram
Poriginating from the point
and spanned by the two vectors
and
, and let
(
,
)be the sum of the
*oriented*areas of the projections over the planes
(
p
_{i},
q
_{i})of
P,

where
Jis the
*canonical symplectic*matrix

A continuously differentiable map
gfrom
to itself is called symplectic if it preserves
, i.e. if

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
(
y(0),
z(0)) = (
y
_{0},
z
_{0})are given and assumed to be consistent with the constraint manifold. By constraint manifold, we imply the intersection of the manifold

and of the so-called hidden manifold

This manifold
is the manifold on which the exact solution
(
y(
t),
z(
t))of (
) lives.

There exists a whole set of schemes which provide a numerical approximation lying on . Furthermore, this solution can be projected on the manifold by standard projection techniques. However, it it worth mentioning that a projection destroys the symmetry of the underlying scheme, so that the construction of a symmetric numerical scheme preserving requires a more sophisticated approach.

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
V(
q)is a sum of potentials
V=
W+
Uacting on different time-scales, with
^{2}Wpositive definite and
. In order to get a bounded error propagation in the linearized equations for an explicit numerical method, the step size must be restricted according to

where
Cis a constant depending on the numerical method and where
is the highest frequency of the problem, i.e. in this situation the square root of the largest eigenvalue of
^{2}W. In applications to molecular dynamics for instance,
*fast*forces deriving from
W(short-range interactions) are much cheaper to evaluate than
*slow*forces deriving from
U(long-range interactions). In this case, it thus 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.

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
H(
t)is finite-dimensional matrix and where
typically is the square-root of a mass-ratio (say electron/ion for instance) and is small (
or smaller). Through the coupling with classical mechanics (
H(
t)is obtained by solving some equations from classical mechanics), we are confronted once again to two different time-scales, 1 and
. In this situation also, it is thus desirable to devise a numerical method able to advance the solution by a time-step
h>
.

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
=
(
x,
t)is the wave function depending on the spatial variables
with
(e.g., with
d= 1or 3 in the partition) and the time
. Here,
is a (small) positive number representing the scaled Planck constant and
iis the complex imaginary unit. The Hamiltonian operator
His written

H=
T+
V

with the kinetic and potential energy operators

where
m_{k}>0is a particle mass and
_{xk}the Laplacian in the variable
, and where the real-valued potential
Vacts as a multiplication operator on
.

The multiplication by
iin (
) plays the role of the multiplication by
Jin classical mechanics, and the energy
is conserved along the solution of (
), using the physicists' notations
where
denotes the Hermitian
L^{2}-product over the phase space. In quantum mechanics, the number
Nof particles is very large making the direct approximation of (
) very difficult.

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:

_{1}= exp(-
i(
t)
V/2)exp(
i(
t)
)exp(-
i(
t)
V/2)
_{0}

where
tis the time increment (we have set all the parameters to 1 in the equation). As the Laplace operator is unbounded, we cannot apply the standard methods used in ODEs to derive long-time
properties of these schemes. However, its projection onto finite dimensional submanifolds (such as Gaussian wave packets space or FEM finite dimensional space of functions in
x) may exhibit Hamiltonian or Poisson structure, whose long-time properties turn out to be more tractable.

The Helmholtz equation modelizes 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,
is the small adimensional parameter that measures the typical wavelength of the signal,
n(
x)is the space-dependent refraction index, and
is a given (possibly dependent on
) source term. The unknown is
. One may think of an antenna emitting waves in the whole space (this is the
), thus creating at any point
xthe signal
along the propagation. The small
term takes into account damping of the waves as they propagate.

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 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
V(
x)is the potential and
(
t,
x)is the time- and space-dependent wave function. However, the size of real devices makes it important to derive simplified models that are valid at a larger scale.
Typically, one wishes to have kinetic transport equations. As is well-known, this requirement needs one to be able to describe “collisions” between electrons in these devices, a concept that
makes sense at the macroscopic level, while it does not at the microscopic (electronic) level. Quantitatively, the question is the following: can one obtain the Boltzmann equation (an equation
that describes collisional phenomena) as an asymptotic model for the Schrödinger equation, along the physically relevant micro-macro asymptotics? From the point of view of modelling, one wishes
here to understand what are the “good objects”, or, in more technical words, what are the relevant “cross-sections”, that describe the elementary collisional phenomena. Quantitatively, the
Boltzmann equation reads, in a simplified, linearized, form :

Here, the unknown is
f(
x,
v,
t), the probability that a particle sits at position
x, with a velocity
v, at time
t. Also,
(
v,
v^{'})is called the cross-section, and it describes the probability that a particle “jumps” from velocity
vto velocity
v^{'}(or the converse) after a collision process.

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
, where
and
approximate
fand
gin a sense that remains to be defined. The idea behind this approximation is to replace the differential manifold
by a suitable approximation
for which an atlas consisting of
*easily computable*charts exists. If this is the case, one can reformulate the vector field
on each domain of the atlas in an
*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
. Nevertheless, it never drifts off the exact manifold considerably, if
and
are chosen appropriately
*close*to each other.

An obvious prerequisite for this idea to make sense is the existence of a neighborhood
of
containing the approximate manifold
and on which the vector field
fis well-defined. In contrast, if this assumption is fulfilled, then it is possible to construct a new admissible vector field
given
. By admissible, we mean tangent to the manifold
, i.e. such that

where, for convenience, we have denoted . For any , we can indeed define

where is the projection along .

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
10
^{-6}
m, over distances of the order
10
^{-2}
mto
10
^{4}
m. In these situations, the propagation produces both a short-scale oscillation and exhibits a long term trend (drift, dispersion, nonlinear interaction with the
medium, or so), which contains the physically important feature. For this reason, one needs to develop ways of filtering the irrelevant high-oscillations, and to build up models and/or
numerical schemes that do give information on the long-term behavior. In other terms, one needs to develop high-frequency models and/or high-frequency schemes.

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's 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).

The Fast Multipole Method (FMM) has been widely developed and studied for the evaluation of Coulomb energy and Coulomb forces. A major problem occurs when the FMM is applied to approximate the Coulomb energy and Coulomb energy gradient within geometric numerical integrations of Hamiltonian systems considered for solving astronomy or molecular-dynamics problems: The FMM approximation involves an approximated potential which is not regular. Its lack of regularity implies a loss of the preservation of the Hamiltonian of the system. In , we contributed to a significant improvement of the FMM with regard to this problem : we investigated a regularization of the Fast Multipole Method in order to recover Hamiltonian preservation. Numerical results obtained on a toy problem confirm the gain of such a regularization of the fast method.

This is a joint work with E. Hairer, from the University of Geneva.

Following the pioneering work of Butcher , in the study of order conditions for Runge-Kutta methods applied to ordinary differential equations

Hairer and Wanner
introduced the concept of B-series. A B-series
B(
f,
a)(
y)is a formal expression of the form

where the index set
Tis a set of rooted trees. B-series and extensions thereof are now exposed in various textbooks and lie at the core of several recent theoretical developments. B-Series owe their success
to their ability to represent most numerical integrators, e.g. Runge-Kutta methods, splitting and composition methods, underlying one-step method of linear multistep formulae, as well as
*modified*vector fields, i.e. vector fields built on derivatives of a given function. In some applications, B-series naturally combine with each other, according to two different laws. The
composition law of Butcher and the substitution law of Chartier, Hairer and Vilmart.

The aim of the paper is to explain the fundamental role in numerical analysis of these two laws and to explore their common algebraic structure and relationships. It complements, from a numerical analyst perspective, the work of Calaque, Ebraihimi-Fard & Manchon , where more sophisticated algebra is used. We introduce into details the composition and substitution laws, as considered in the context of numerical analysis and relate each law to a Hopf algebra. Then we explore various relations between the two laws and consider a specific map related to the logarithm. Eventually, we mention the extension of the substitution law to P-series, which are of great use for partitionned or split systems of ordinary differential equations.

It is well known that the Euler scheme is of strong order 1/2and weak order 1 in the case of a stochastic differential equation. Two methods are available to prove this result. The first one uses the Kolmogorov equation associated to the stochastic equation and was first used by D. Talay. A second one has been recently discovered by A. Kohatsu-Higa and is based on Malliavin calculus.

In this article, we generalize such results to the infinite dimensional case. We show how to adapt Talay's method. The main difficulty is due to the presence of unbounded operators in the Kolmogorov equation. A tricky change of unknown allows to treat the case of a linear equation. It also works for an equation whose linear part defines a group, the nonlinear Schrödinger equation for instance. The case of a semilinear equations of parabolic type treated here is more difficult and we use Malliavin calculus, but not in the same way as in Kohatsu-Higa's method. We prove for instance that, in the case of a nonlinear heat equation in dimension one with a space time white noise, the Euler scheme has weak order 1/2, it is well known that the strong order is 1/4.

In this paper we analyze the asymptotic dynamics of a system of
Nquantum particles, in a weak coupling regime. Particles are assumed statistically independent at the initial time.

Our approach follows the strategy introduced by the authors in a previous work : we compute the time evolution of the Wigner transform of the one-particle reduced density matrix; it is represented by means of a perturbation series, whose expansion is obtained upon iterating the Duhamel formula; this approach allows us to follow the arguments developed by Lanford for classical interacting particles evolving in a low density regime.

We prove, under suitable assumptions on the interaction potential, that the complete perturbation series converges term-by-term, for all times, towards the solution of a Boltzmann equation.

The present paper completes the previous work: it is proved there that a subseries of the complete perturbation expansion converges uniformly, for short times, towards the solution to the nonlinear quantum Boltzmann equation. This previous result holds for (smooth) potentials having possibly non-zero mean value. The present text establishes that the terms neglected at once previously, on a purely heuristic basis, indeed go term-by-term to zero along the weak coupling limit, at least for potentials having zero mean.

Our analysis combines stationary phase arguments, with considerations on the nature of the various Feynman graphs entering the expansion.

We study the limiting behavior of a nonlinear Schrödinger equation describing a 3 dimensional gas that is strongly confined along the vertical,
zdirection. The confinement induces fast oscillations in time, that need to be averaged out. Since the Hamiltonian in the
zdirection is merely assumed confining, without any further specification, the associated spectrum is discrete but arbitrary, and the fast oscillations induced by the nonlinear equation
entail countably many frequencies that are arbitrarily distributed. For that reason, averaging can not rely on small denominator estimates or like.

To overcome these difficulties, we prove that the fast oscillations are
*almost periodic*in time, with values in a
*Sobolev-like*space that we completely identify. We then exploit the existence of
*long time averages*for almost periodic function to perform the necessary averaging procedure in our nonlinear problem.

Consider the Schrödinger operator with semiclassical parameter
h, in the limit where
hgoes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator's resolvent at a positive energy
are bounded by
O(
h^{-1})if and only if the associated Hamilton flow is non-trapping at energy
. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis
requires the use of an appropriate regularization.

In doing so, we completely characterize how migrations do modify both the qualitative and quantitative properties of the global demography.

Our analysis relies on a convenient version of the central manifold theorem, in conjunction with a spectral gap estimate on the involved migration operator.

This is a joint work with S. Descombes, from the University of Nice.

Although the numerical simulation of the Heat equation in several space dimension is now well understood, there remain a lot of challenges in the presence of an external
source,
*e.g.*for reaction-diffusion problems, or more generally for the complex Ginzburg-Landau equation. From a mathematical point of view, these belong to the class of semi-linear parabolic
partial differential equations and can be represented in the general form

When one wishes to approximate the solution of the above parabolic non-linear problem, a method of choice is based on operator-splitting: the idea is to split the abstract evolution equation into two parts which can be solved explicitly or at least approximated efficiently.

For a positive step size
h, the most simple numerical integrator is the Lie-Trotter splitting which is an approximation of order 1, while the symmetric version is referred to as the Strang splitting and is an
approximation of order 2. For higher orders, one can consider general splitting methods of the form

However, achieving higher order is not as straightforward as it looks. A disappointing result indeed shows that all splitting methods (or composition methods) with real coefficients must
have negative coefficients
a_{i}and
b_{i}in order to achieve order 3 or more. The existence of at least one negative coefficient was shown in
,
, and the existence of a negative coefficient for both operators was proved in
. An elegant geometric proof can be found in
. As a consequence, such splitting methods
*cannot*be used when one operator, like
, is not time-reversible.

In order to circumvent this order-barrier, there are two possibilities. One can use a linear, convex (see
,
,
for methods of order 3 and 4) or non-convex (see
,
where an extrapolation procedure is exploited), combinations of elementary splitting methods like (
). Another possibility is to consider splitting methods with
*complex*coefficients
a_{i}and
b_{i}with positive real parts (see
in celestrial mechanics). In 1962/1963, Rosenbrock
considered complex coefficients in a similar context.

In , we consider splitting methods, and we derive new high-order methods using composition techniques originally developed for the geometric numerical integration of ordinary differential equations . The main advantages of this approach are the following:

the splitting method inherits the stability property of exponential operators;

we can replace the costly exponentials of the operators by cheap low order approximations without altering the overall order of accuracy;

using complex coefficients allows to reduce the number of compositions needed to achieve any given order;

This a joint work with Ander Murua, from the University of the Basque Country.

When one needs to compute the numerical solution of a differential equation of a specific type (ordinary, differential-algebraic, linear...) with a method of a given class of numerical schemes, a deciding criterion to pick up the right one is its order of convergence: the systematic determination of order conditions thus appears as a pivotal question in the numerical analysis of differential equations. Given a family of vector fields with some specific property (say for instance linear, additively split into a linear and a nonlinear part, scalar...) and a set of numerical schemes (rational aprooximations of the exponential, exponential integrators , Runge-Kutta methods...), a fairly general recipe consists in expanding into series both the exact solution of the problem and its numerical approximation: order conditions are then derived by comparing the two series term by term, once their independence has been established. Depending on the equation and on the numerical method, these series can be indexed by integers or trees, and can expressed in terms of elementary differentials or commutators of Lie-operators. Despite the great variety of situations encountered in practice and of ad-hoc techniques, the problems raised are strikingly similar and can be described as follows:

is it possible to construct a set of algebraically independent order conditions?

what are the order conditions corresponding to a scheme obtained by composition of two given methods?

are there numerical schemes within the class considered of arbitrarily high order for arbitrary vector field?

are there numerical schemes within the set of methods considered that approximate modified fields?

The Butcher group
and its underlying Hopf algebra of rooted trees were originally formulated to address these questions for
Runge-Kutta methods. In the past few years, these concepts turned out to have far-reaching applications in several areas of mathematics and physics: they were rediscovered in noncommutative
geometry by Connes and Moscovici
and they describe the combinatorics of renormalization in quantum field theory as described by Kreimer
. In the present work, we show that the Hopf algebra of rooted trees associated to Butcher's group can be seen as a
particular instance of a more general construction: given a group
Gof integrations schemes (satisfying some natural assumptions), we exhibit a sub-algebra of the algebra of functions acting on
G, which is graded, commutative and turns out to be a Hopf algebra. Within this algebraic framework, we then address the questions listed above and provide answers that are relevant to
many practical situations.

Finally, we describe how our theory can be used to obtain order conditions for composition schemes.

The main goal of this work is to derive and analyze new schemes for the numerical approximation of least-squares problems set on high dimensional spaces. This work
,originates from the Statistical Analysis of Distributed Multipoles (SADM) algorithm introduced by Chipot
*et al.*in 1998 for the derivation of atomic multipoles from the quantum mechanical electrostatic potential mapped on a grid of points surrounding a molecule of interest. The main idea is
to draw subsystems of the original large least-square problem and compute the average of the corresponding distribution of solutions as an approximation of the original solution. Moreover, this
methods not only provides a numerical approximation of the solution, but a global statistical distribution reflecting the accuracy of the physical model used.

Strikingly, it turns out that this kind of approach can be extended to many situations arising in computational mathematics and physics. The principle of the SADM algorithm is in fact very general, and can be adapted to derive efficient algorithms that are robust with the dimension of the underlying space of approximation. This provides new numerical methods that are of practical interest for high dimensional least-squares problems where traditional methods are impossible to implement.

The goal of this paper is twofold:

Give a general mathematical framework, and analyze the consistency, convergence and cost of these new algorithms in an abstract setting and in specific situations where calculations can be made explicit (Wishart or subgaussian distribution). The main outcome is that the subsystems drawn from the original system have to be chosen rectangular and not square (as initially proposed in the SADM method) to obtain convergent and efficient algorithms.

Apply these results to revisit and improve the SADM method. This is mainly done in Section 5 by considering the three-point charge model of water.

The time dependent linear Schrödinger equation for nuclei on the whole space is semi-discretised using Hermite and Gauss-Hermite basis functions. These are well suited on the one hand for the conservation properties of the numerical solution and, on the other hand, for their remarkable approximation properties. In Ref. , we investigate theoretically and numerically the convergence of the spectral and pseudospectral Gauss-Hermite semi-discretisation schemes.

Consider a Hamiltonian system

where
, and with a separable Hamiltonian
Hof the form

where
V(
q)is the potential function. In many applications, such as for instance molecular dynamics, it is of importance that the numerical flow used to compute the solution of
preserves the volume form and the Hamiltonian. However, it is generally admitted that no standard method can satisfy both
requirements, apart from exceptional situations such as for instance a quadratic Hamiltonian. A possible approach could be to solve in sequence the
dHamiltonian systems with Hamiltonians

obtained by freezing all components (denoted with a bar) except the two conjugate coordinates
q_{i}and
p_{i}. If each subsystem can be solved exactly and the same step-size is used for all, the resulting “numerical" method preserves the desired quantities, since each sub-step is symplectic and
preserves
H^{[
i]}(and thus
H). Considering that each subsystem is of dimension 2 and thus integrable, it can be hoped that an exact solution is indeed obtainable in some specific situations. Nevertheless, such
situations are rather non-generic, though it is important to mention at this stage the special case of
*multi-quadratic*potentials, i.e. potentials such that for all
i= 1, ...,
dand all
,
V^{[
i]}is
*quadratic*in
q_{i}. In this context, the method described above has been introduced in by R. Quispel and R.I. McLachlan in
.

In order to retain the possibility of solving exactly each sub-system and at the same time to cover more general problems, we give up the requirement of exact Hamiltonian preservation and we
consider a multi-quadratic piecewise approximation of
H. If instead of
we now solve

where
is a
C^{1, 1}multi-quadratic approximation of
H, the aforementioned procedure applied with exact solution of the sub-systems gives a first-order method which preserves
exactly as well as the volume form. If
for a compact subset
Kof
containing the numerical solution, then
His conserved up to an error of size
over arbitrarily long intervals of integration (including infinite ones).

Note that this approach remains valid for more general Hamiltonians (non-separable for instance), provided an exact solution can be computed, so that all theoretical results concerning the conservation of energy and volume will be stated for general Hamiltonians. In contrast, we will describe the implementation of the method with quadratic B-splines only for the case of separable Hamiltonians.

An adiabatic invariant is a property of a physical system which stays constant when changes are made slowly. In mechanics, an adiabatic change is a small perturbation of the Hamiltonian where the change of the energy is much slower than the orbital frequency (see for instance , ). The area enclosed by the different motions in phase space are then the adiabatic invariants. In the case of a perturbed Hamiltonian of the form

with , the classical procedure for deriving the invariants of motion is to look for a change of variables, close to the identity, in powers of

in order to eliminate the angle variables of the Hamiltonian. This method, that goes back to Poincaré, was refined in the 20th century by Birkhoff , Kolmogorov/Arnold/Moser (KAM) , , Nekhoroshev , and forms now the classical perturbation theory.

Using this coordinate transform method, the classical conclusion is that the series, though divergent, are asymptotic in the sense that, for instance,

for exponentially large time
t. Hence,
I(
t)is an adiabatic invariant for system (
), in the sense that its variation is small for a long time interval.

In the paper
, we consider perturbed
*reversible*systems for which the classical method can be applied (see for instance
,
,
). The systems we consider are of the following form:

where
is a small parameter,
sis an odd function of
and
an even function of

For such systems, we propose an alternative construction of the invariants. It stems from the expansion of
Iitself and involves no change of variables in
(
a,
): the procedure thus remains extremely basic. We assume here that
is a constant vector, independent of
a. This simplifies further some of the proofs while still covering most cases of interest

Although the form of equations ( ) seems very specific, a lot of systems in classical mechanics (reversible integrable ones to be precise) can be transformed into action-angle variables (see for instance Chapter XI in ). A prominent example of such a mechanical system is the Fermi-Pasta-Ulam model which nicely illustrates the persistence of adiabatic quantities (in this model, an adiabatic invariant is built up from the oscillatory energies of the stiff springs).

Results derived in this paper apply to the Fermi-Pasta-Ulam equations as much as to many other systems in celestial mechnanics for instance. Moreover, they might be helpful to analyse geometric properties of numerical methods or to obtain stability results of a more theoretical nature such as those proved in or Chapter XI.

In several recent publications, numerical integrators based on Jacobi elliptic functions are proposed for solving the equations of motion of the rigid body. Although this approach yields theoretically the exact solution, a standard implementation shows an unexpected linear propagation of round-off errors. In Ref. , we explain how deterministic error contribution can be avoided, so that round-off behaves like a random walk. Key Words. rigid body integrator, Jacobi elliptic functions, probabilistic error propagation, long-time integration, compensated summation, quaternion, Discrete MoserÐVeselov algorithm.

This is a joint work with M. Chyba and E. Hairer.

For general optimal control problems, PontryaginÕs maximum principle gives necessary optimality conditions which are in the form of a Hamiltonian differential equation. For its numerical integration, symplectic methods are a natural choice. The article investigates to which extent the excellent performance of symplectic integrators for long-time integrations in astronomy and molecular dynamics carries over to problems in optimal control. Numerical experiments supported by a backward error analysis show that, for problems in low dimension close to a critical value of the Hamiltonian, symplectic integrators have a clear advantage. This is illustrated using the Martinet case in sub-Riemannian geometry. For problems like the orbital transfer of a spacecraft or the control of a submerged rigid body such an advantage cannot be observed. The Hamiltonian system is a boundary value problem and the time interval is in general not large enough so that symplectic integrators could benefit from their structure preservation of the flow. Key Words. symplectic integrator, backward error analysis, sub-Riemannian geometry, Martinet, abnormal geodesic, orbital transfer, submerged rigid body.

The INGEMOL project is concerned with the numerical simulation of differential equations by so-called geometric methods, i.e. methods preserving some of the qualitative features of the exact solution. Conserving the energy or the symmetry is often physically relevant and of paramount importance in some applications such as molecular simulation or propagation of laser waves in fibers (these are the main applications considered within the project, though several others are possible: robotics, celestial mechanics…). Though a lot has been achieved by numerical analysts in the domain of numerical integration during the last two decades, with most significantly the introduction of symplectic schemes and their analysis through backward error techniques, a lot remains to be done in situations where the existing theory fails to give a useful answer; the goal of the INGEMOL project is to help solving these difficulties in some well-identified cases : 1. whenever symmetric multi-step methods are used for Hamiltonian systems, 2. whenever splitting methods are used for the Schrödinger equation, 3. whenever the system under consideration has highly-oscillating solutions.

Taking into account in the theory the unboundedness of the operators or the high oscillations of the solutions allows for the construction, in a second step, of more appropriate numerical schemes with fewer or none of the present restrictions.

Eventually, it is planned to implement the new schemes with in view their application to the simulation of laser waves and to molecular simulation.

P. Chartier is coordinator of the project. INGEMOL associates the following persons and teams:

F. Castella, P. Chartier, M. Crouzeix, G. Dujardin, A. Debussche, E. Faou, G. Vilmart: IPSO

Ch. Chipot: Structure et réactivité des systèmes moléculaire complexes, CNRS, Nancy.

S. Descombes: ENS LYON.

E. Cancès, C. Le Bris, F. Legoll, T. Lelièvre, G. Stoltz: CERMICS, ENPC, Marne-la-Vallée.

This is an exchange program between the ipso team and the numerical analysis groups in Tübingen, headed by C. Lubich and in the University of the Basque Country headed by A. Murua. E. Faou is the coordinator of the french part of this project. In 2008, this program financed the following one-week visits:

L. Gauckler from Tübingen

E. Faou (1 time), G. Dujardin from IPSO.

P. Chartier from IPSO.

A. Murua from the Basque Country.

This program is valid for two years (2008 and 2009).

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

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

P. Chartier is guest editor-in-chief of a special issue of M2AN devoted to numerical methods for the integration of ODEs.

E. Faou is the leader of the INRIA associated team MIMOL (2008–2010) grouping members of:

The IPSO team (INRIA Rennes, France, head: P. Chartier),

The numerical analysis group of the University of Tübingen, (Germany, head: C. Lubich),

The computer science department of the University of the Basque country, (Spain, San-Sebastian, head: A. Murua).

F. Castella is
**Co-organizer**, with R. Illner, of a session "Kinetic Methods in PDE's", in the framework of the second Canada-France mathematical congress,
**Montreal**.

F. Castella is
**Co-organizer**, with D. Bresch, B. Desjardins and M. Peybernes, of the
**summer school**of the GdR “CHANT”,
**Roscoff**(Finistère), 70 participants.

F. Castella is the director of the GdR CNRS 'CHANT' (équations Cinétiques et Hyperboliques : Aspects Numériques, Théoriques, et de modélisation'). [budget=15000 Euros per year, approximately 300 persons, and about 4 events organized per year].

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.

P. Chartier is member of the Commission d'Evaluation at INRIA.

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

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

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

E. Faou is oral examiner at ENS Cachan Bruz (“agrégation”).

E. Faou is lecturer at the Ecole Normale Supérieure de Cachan Bretagne. Course:
*Ordinary differential equations.*

P. Chartier gave a lecture at the Ecole CIMPA, Tlemcem, Algeria, May 2008.

P. Chartier gave a talk at the conference “Splitting Methods in Time Integration” in Innsbruck, October 2008.

P. Chartier gave a talk at the workshop “Numerical methods and Hopf algebras of trees” in Clermont-Ferrand, October 2008.

P. Chartier was invited to give a talk at at Basel University, November 2008.

P. Chartier was invited to give a talk at the University of Geneva, December 2008.

P. Chartier was invited to give a talk at the University of Nice, December 2008.

E. Faou was invited to give at the Canada-France congress in Montreal, June 2008. (Invitation to the mini-symposium:
*Variational and Numerical Methods in Geometry, Physics and Chemistry*, organized by M.J. Esteban, L. Bronsard and E. Cancés).

E. Faou attended the Workshop in Berder on Hamiltonian PDEs, organized by the university of Nantes.

E. Faou gave a Seminar in the University of Pau (France), November 2008.

E. Faou gave aSeminar in the Observatoire de Paris (Astronomy and dynamical systems team), October 2008.

E. Faou gave a Seminar in the University of Lille (France), October 2008.

E. Faou gave a Seminar in the University of Tübingen (Germany), June 2008.

E. Faou gave a Seminar in the University of Mulhouse (France), March 2008.

F. Castella gave a six hours lecture at
*'Ecole de Physique des Houches'*on interacting particles systems, les Houches, France.

F. Castella attented the Workshop "Mathematical Models for Transport in Macroscopic and Mesoscopic Systems", Berlin,Germany.

A. Debussche gave a talk in the workshop Stochastic Partial Differential Equations and Applications - VIIIÓ, Levico Terme (Trento), 6-12 janvier2008

A. Debussche gave a talk in the workshop Numerical Analysis of Stochastic PDEs 2008 (NASPDE08), ETH Zurich, 16-17 mai 2008 New Perspectives on Malliavin Calculus, CRM, Barcelona, 25 juin 2008

A. Debussche gave a talk in the Journees MAS 2008, Rennes , 25-27 aout 2008.

A. Debussche gave a talk in the workshop Stochastic Partial Differential Equations Computations & Applications, ICMS Edinburgh, 29 sept.-1er oct. 1008.

G. Vilmart gave a talk at the workshop “Numerical methods and Hopf algebras of trees” in Clermont-Ferrand, October 2008.

G. Vilmart gave a talk at the Fall Metting of the Swiss Mathematical Society, Berne (Switzerland) Oct. 2008

G. Vilmart gave a talk at the Séminaire Mulhousien de mathématiques, Mulhouse (France), Oct. 2008

G. Vilmart gave a talk at Fourth Graduate Colloquium, Swiss Doctoral Program in mathematics, Neuchâtel (Switzerland), Sep. 2008

G. Vilmart attended the II International Summer School on Geometry, Mechanics, and Control, La Palma, Canary Islands (Spain), June 2008.

G. Vilmart gave a talk at Colloque Numérique Suisse, Fribourg (Switzerland), Apr. 2008

P. Chartier visited the University of the Basque Country for three weeks.

E. Faou visited the University of Tübingen in june 2008 .

The team has invited the following persons :

L. Gauckler on a one-week visit.

A. Murua on a a two-week visit.