Erwan Faou



INRIA
Projet Ipso
ENS Cachan Bretagne
Avenue Robert Schumann
F-35170 Bruz

Phone: (33) 299 05 52 81
Fax: (33) 2 99 84 71 71
email: Erwan.Faou@inria.fr
















Fotógrafo: B. Grébert, UNAM Cuernavaca, Mexico
(March 2009....)
Other pictures: the AMS on the way to Iztaccihuatl


non resonant caseresonant case
Figures: Numerical resonances in the approximation of NLS by splitting methods.
Left: The evolution of the actions associated with the solution of the Non Linear Schrödinger equation are represented in logarithmic scale: Illustration of Birkhoff normal form results!
Right: This is not the EEG monitoring of a trader in september 2008, but the same representation as before with a resonant time step very close (0.1%) to the one used in the left figure...

Explanations may be found in the slides of the talks I gave in Beijing (AMSS, Chinese Academy of Sciences, May 2009):
Normal form and geometric numerical integration of Hamiltonian PDE. Part I: Linear equations, and Part II: Non linear equations.




Publications:

P. Chartier and E. Faou, Equations différentielles ordinaires: analyse numérique géométrique
Book to be submitted to the collection Mathématiques appliquées pour le Master/SMAI.

Papers

Erwan Faou and Benoît Grébert
Hamiltonian interpolation of splitting approximations for nonlinear PDEs
Submitted
pdf
Monique Dauge, Erwan Faou and Victor Péron
Comportement asymptotique à haute conductivité de l'épaisseur de peau en électromagnétisme
To appear in C. R. Acad. Sci. Paris, Sér. I.
pdf
Erwan Faou and Benoît Grébert
Resonances in long time integration of semi linear Hamiltonian PDEs
Submitted
pdf
Nicolas Champagnat, Christophe Chipot and Erwan Faou
A probabilistic approach to high-dimensional least-squares approximations.
Submitted
pdf
Erwan Faou and Benoît Grébert
Quasi invariant modified Sobolev norms for semi linear reversible PDEs
Nonlinearity 23 (2010) 429-443
pdf
Philippe Chartier, Eric Darrigrand and Erwan Faou
A Regular Fast Multipole Method for geometric numerical integration of Hamiltonian systems
To appear in BIT
pdf
Erwan Faou, Benoît Grébert and Eric Paturel
Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization.
Numer. Math. 114 (2010) 429-458
pdf
Erwan Faou, Benoît Grébert and Eric Paturel
Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part II: Abstract splitting.
Numer. Math. 114 (2010) 459-490
pdf
Monique Dauge and Erwan Faou
Koiter Estimate Revisited
Math. Models Methods Appl. Sci. (M3AS) 20 No 1 (2010)  1-42
pdf
Arnaud Debussche and Erwan Faou
Modified energy for split-step methods applied to the linear Schrödinger equation
SIAM J. Numer. Anal. 47 (2009) 3705-3719
pdf
Erwan Faou and Tony Lelièvre
Conservative stochastic differential equations: Mathematical and numerical analysis
Math. Comp. 78 (2009) 2047-2074
pdf
François Castella, Philippe Chartier and Erwan Faou
An averaging technique for highly-oscillatory Hamiltonian problems
SIAM J. Numer. Anal. 47 No 4 (2009) 2808-2837
pdf
Erwan Faou, Vasile Gradinaru and Christian Lubich
Computing semi-classical quantum dynamics with Hagedorn wavepackets
SIAM J. Sci. Comp. 31 No 4 (2009) 3027-3041
pdf
Erwan Faou
Analysis of splitting methods for reaction-diffusion problems using stochastic calculus
Math. Comp. 78 (2009) 1467-1483.
pdf
Erwan Faou and Vasile Gradinaru
Gauss-Hermite wavepacket dynamics: convergence of the spectral and pseudo-spectral approximation.
IMA J. Numer. Anal. 29 (2009) 1023-1045.
pdf
Philippe Chartier and Erwan Faou
A simple proof of the existence of adiabatic invariants for perturbed reversible problems
J. Phys. A: Math. Theor. 41 No 47 (2008) 475204
pdf
Philippe Chartier and Erwan Faou
Geometric integrators for piecewise smooth Hamiltonian systems.
Math. Model. Numer. Anal. (M2AN) 42 No 2 (2008) 223-241
pdf
Guillaume Dujardin and Erwan Faou
Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential.
Numer. Math. 106 No 2 (2007) 223-262
pdf
Guillaume Dujardin and Erwan Faou
Long time behavior of splitting methods applied to the linear Schrödinger equation.
C. R. Acad. Sci. Paris, Sér. I. 344 (2007) 89-92

Erwan Faou
Nosé-Hoover dynamics in a shaker.

J. Chem. Phys. 124 184104 (2006)
ps, pdf
Philippe Chartier, Erwan Faou, Ander Murua
An algebraic approach to invariant preserving integrators: The case of quadratic and Hamiltonian invariants.
Numer. Math. 103 No 4 (2006) 575-590
pdf
Erwan Faou and Christian Lubich
A Poisson integrator for Gaussian wavepacket dynamics.
Comput. Vis. Sci. 9 No 2 (2006) 45-55
ps 
Eric Cancès, François Castella, Philippe Chartier, Erwan Faou, Claude Le Bris, Frédéric Legoll and Gabriel Turinici
Long-time averaging for integrable Hamiltonian dynamics.
Numer. Math. 100 No 2 (2005) 211-232
ps, pdf
Erwan Faou, Ernst Hairer and Truong-Linh Pham
Energy conservation with non-symplectic methods: Examples and counter-examples 
BIT 44 No 4 (2004) 699-709
ps
F. Leplingard, C. Martinelli, S. Borne, L. Lorcy, T. Lopez, D. Bayart, F. Castella, P. Chartier and E. Faou
Modeling of multi-wavelength Raman fiber lasers using a new and fast algorithm.
IEEE Photonics Technology Letters 16 No 12 (2004) 2601-2603
pdf
Erwan Faou
Multiscale Expansions for Linear Clamped Elliptic Shells.
Comm. P.D.E. 29 Vol 11 & 12 (2004) 1799-1845
ps, pdf 
Eric Cancès, François Castella, Philippe Chartier, Erwan Faou, Claude Le Bris, Frédéric Legoll and Gabriel Turinici
High-order averaging schemes with error bounds for thermodynamical properties calculations by molecular dynamics simulations.
J. Chem. Phys. 121 No 21 (2004) 10346-10355
INRIA Research Report
François Castella, Philippe Chartier, Erwan Faou, Dominique Bayart, Florence Leplingard and Catherine Martinelli
Raman Laser Modeling: Mathematical and Numerical Analysis of a model.
Math. Model. Numer. Anal. (M2AN) 38 No 3 (2004) 457-475
ps, pdf 
Monique Dauge, Erwan Faou and Zohar Yosibash
Plates and shells: Asymptotic expansions and hierarchical models.
Chapter 8
, Vol I of the Encyclopedia for Computational Mechanics.
Edited by Erwin Stein, René de Borst, Thomas J.R. Hughes (2004).
abstract, pdf 
François Castella, Philippe Chartier and Erwan Faou
Analysis of a Poisson system with boundary conditions.
C. R. Acad. Sci. Paris, Sér. I. 336 (2003) 703-708
ps
Erwan Faou
Elasticity on a thin shell: Formal series solution.
Asympt. Anal. 31 (2002) 317-361
ps, pdf
Erwan Faou
asymptotiques dans les coques elliptiques: Equations tridimensionnelles linéarisés
.
C. R. Acad. Sci. Paris, Sér. I. 333 (2001) 389-394
ps, pdf
Erwan Faou
Développements asymptotiques dans les coques elliptiques: Modèle de Koiter
.
C. R. Acad. Sci. Paris, Sér. I. 333 (2001) 139-143
ps, pdf
Georgiana Andreoiu and Erwan Faou
Complete asymptotics for shallow shells.
Asympt. Anal. 25 (2001) 239-270
ps , pdf 
Georgiana Andreoiu, Monique Dauge and Erwan Faou
Développements asymptotiques complets pour des coques faiblement courbées encastrées ou libres.

C. R. Acad. Sci. Paris, Sér. I. 330 (2000) 523-528
ps, ps.g
Erwan Faou
Elasticité linéarisée tridimensionnelle pour une coque mince: Résolution en série formelle en puissances de l'épaisseur.
C. R. Acad. Sci. Paris, Sér. I. 330 (2000) 415-420
ps, ps.gz
Monique Dauge, Ivica Djurdjevic, Erwan Faou and Andreas Rössle
Eigenmode Asymptotics in Thin Elastic Plates.
J. Math. Pures Appl. 78 No 9 (1999) 925-964
ps, ps.gz


Thesis and HDR

Thèse soutenue le 21 juin 2000
Développements asymptotiques dans les coques minces linéairement élastiques

pdf file  --- ps file  --- ps.gz file 

Habilitation degree (obtained october 16, 2007)
Some geometrical aspects in shell theory and in numerical integration of hamiltonian systems.
pdf file of the document. The slides are also available, as well as the video (I cannot endure the preview very long....)



Teaching

You can download here (postscript file, 67 pages) the notes (in french) of a lecture on geometric integration and KAM theory given in june 2003, in collaboration with P.Chartier. These notes are inspired by the book "Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations" written by E. Hairer, C. Lubich and G. Wanner and published by Springer



PhD. Students




Some slides 


Castellon Conference on Geometric Integration 2006 (site). "Normal form for splitting methods applied to the linear Schrödinger equation", joint work with Guillaume Dujardin.  Here are the slides.


Talk on shell theory given in june 2006 in a meeting on asymptotic methods organized at the ENSTA by POEMs. Here are the slides (in french).

Slides of a talk on Gaussian Wave Packets approximation of the Schödinger equation given in SciCADE'05 in Nagoya (japan).


Choosing a resonant step size can lead to dramatic effects but beautiful pictures: This is behavior of the Fourier coefficients corresponding to the solution of the linear Schrödinger equation on a torus computed by a splitting method with a non-resonnant stepsize and an analytic potential.
non resonant case
Now here is the same picture but with a resonant stepsize....

resonant case
You can find here the corresponding movies of the solution of the linear Schrödinger equation using a splitting method with a good stepsize and a bad stepsize.