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SS14 - Analyse numérique / SS14 - Numerical analysis Org: A. Fortin (Laval) et/and J. Blum (Nice)
- HABIB AMMARI, Ecole Polytechnique, Palaiseau
Algorithms for anomaly detection
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We consider an inverse problem arising in anomaly detection with its
mathematical model based on the T-Scan system. We try to detect a
small anomaly from measured data that is available only on a small
portion of the subject. We carry out rigorous estimates to derive a
simple approximation that gives a non-iterative detection algorithm of
finding the anomaly. We also present a multi-frequency approach to
handle the case where the complex conductivity of the background is
not homogeneous and not known a priori. This is a joint work with
O. Kwon, J. K. Seo, and E. Woo.
- TAHAR ZAMENE BOULMEZAOUD, Université de Pau et des Pays de l'Adour, and Université
Paris VI
Inverted Elements: a new method for solving elliptic
problems in unbounded domains
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In this talk we propose a new numerical method for solving elliptic
equations in unbounded regions of space. The method is based on the
mapping of a part of the domain into a bounded region. An appropriate
family of weighted spaces is used as a functionnal framework for
describing the behavior of functions at large distances. After
exposing the main ideas of the method, we analyse carefully its
convergence. Some 3D computational results are displayed to
demonstrate the performance of the method.
- GEORGES-HENRI COTTET, LMC-IMAG, Université Joseph Fourier, Grenoble
Level set methods for fluid-structure interaction problems in
2D and 3D
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Level set methods have been designed to capture interfaces with some
flexibility in Eulerian formulations of incompressible fluid
mechanics. They have been used by Chang et al. with some success for
variable density flows with surface tension.
In this talk we propose a level-set method motivated by applications
in cell biology which handles elasticity effects on the interface
(either curve or surface). This model satisfies energy equalities
which is a good sign for consistency. It can also be viewed as an
Eulerian alternative of Peskin's immersed boundary methods.
In passing, we also address some technical problems related to
interface distortions in the implementation of level-set methods. We
propose a simple alternative to the usual reinitialization strategies
in general used to overcome these problems.
- YVES DEMAY, Nice
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- ANDRÉ FORTIN, Université Laval, Québec, Canada, G1K 7P4
Remaillage anisotrope et applications au calcul des surfaces
libres
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Dans cet exposé, nous présenterons une stratégie d'adaptation de
maillage pour la résolution de problèmes de mécanique des
fluides avec surfaces libres instationnaires. Les maillages obtenus
sont caractérisés par une forte anisotropie c.-à-d. les
éléments peuvent être fortement allongés dans certaines
directions privilégiées de l'écoulement. L'estimation d'erreur
est basée sur l'introduction d'une métrique dépendant de la
solution.
Les surfaces libres sont calculées par une méthode de surfaces de
niveau (level sets) modifiée de manière à assurer une parfaite
conservation de la masse. Un algorithme de réinitialisation de la
fonction distance associée sera aussi discuté.
Nous présenterons enfin des applications bi et tridimensionnelles
à la déformation et éventuellement au bri de gouttelettes de
fluides en cisaillement dans un autre fluide.
- THIERRY GALLOUET, CMI, 39 rue Joliot-Curie, F-13453 Marseille Cedex 13
Elliptic equations with measure data and numerical schemes
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In order to show the existence of solutions for linear elliptic
equation, the classical method, due to Stampacchia, is to use a
duality argument (and a regularity result for elliptic equations with
regular data). Another classical method is to pass to the limit on
approximate solutions obtained with regular data (converging towards
the measure data). We will present here a third method which consists
in passing to the limit on approximate solutions obtained with
numerical schemes such as finite element schemes or finite volume
schemes. This method is not easier than the previous ones but is
interesting since it yields a way to compute approximate solutions. We
will also present this method for convection-diffusion equations which
lead to noncoercive elliptic equations with measure data.
- MARTIN GANDER, McGill University, 805 Sherbrooke Street West, Montreal,
H3A 2K6
The Parareal Algorithm in the Context of Classical Methods
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A few years ago, Lions, Maday and Turinici introduced a new algorithm
for evolution problems which is parallel in time: the parareal
algorithm. This algorithm decomposes the time domain into subdomains
in time, and then uses fine grid approximations on the time
subdomains, and a coarse grid correction in time to iteratively
construct a better and better approximation in time of the evolution
problem.
I will show in this presentation that the parareal algorithm can be
put into the context of more classical methods. In particular, I will
show that the parareal algorithm is equivalent to a multigrid waveform
relaxation algorithm with a certain choice of smoother, restriction
and extension operators, and it is also equivalent to a multiple
shooting algorithm with a particular coarse approximation of the
Jacobian matrix in the nonlinear Newton solver.
- DANIEL LE ROUX, Université Laval
Modes numériques dans les équations de Saint-Venant,
l'élément PNC1 - P1
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La plupart des méthodes numériques utilisées pour résoudre les
équations de Saint-Venant génèrent des modes parasites en
approximant les ondes de type inertie-gravité. Les modes parasites
les plus dangereux sont les modes stationnaires appartenant aux noyaux
des opérateurs gradient, Coriolis et divergence discrets. Dans un
premier temps nous passerons ces problèmes en revue en choisissant
quelques exemples parmi les grilles de différences finies et
d'éléments finis les plus populaires. Dans un deuxième temps,
nous présenterons la paire d'éléments finis PNC1 -P1, nous analyserons sa relation de dispersion et nous
montrerons que cette paire approxime raisonnablement bien les ondes de
type inertie-gravité. Nous la comparerons notamment à la grille
C-D en resolvant le probleme de Stommel, c'est à dire le cas d'une
circulation forcée par le vent dans un bassin océanique.
- YVON MADAY, Labo. J.-L. Lions, Univ. P. et M. Curie
Some variations on the reduced basis method with
certification
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Reduced basis methods are discretization strategies for the solution
of partial differential equations that use "optimal" ad hoc
bases constructed in a preliminary stage by standard approximation
methods. The number of degrees of freedom required to solve a
particular problem is then extremely small compared to the
corresponding number involved by conventional methods for similar
precision. The range of applications of reduced basis approaches
currently goes well beyond the original application of this framework
to solid mechanics. All aspects of the reduced-basis approach are not
yet mathematically well understood; nevertheless, a posteriori
error estimates and validations are now available that provide for
rigorous certification of accuracy.
In this talk we shall present some new developments of this method
including efficient treatments of nonlinear problems.
- BERTRAND MAURY, Laboratoire de Mathématique, Université Paris-Sud,
Bâtiment 425, 91405 Orsay, France
Pression granulaire et pression fluide
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Nous proposons de donner un aperçu des problèmes posés par
l'estimation des multiplicateurs de Lagrange associés à des
contraintes unilatérales pour certains problèmes d'évolution du
type écoulement granulaire. Nous tâcherons notamment de préciser
les liens qui existent entre le calcul des forces de réactions qui
empêchent l'interpénétration d'une multitude de corps rigides et
le calcul du champ de pression dans la résolution des équations de
Stokes ou de Navier-Stokes incompressibles.
- NILIMA NIGAM, McGill University, 805 Sherbrooke St. W, Montreal, QC
Perturbative Stekhlov-Poincare maps for general domains
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The truncation of an infinite computational domain by an artificial
boundary, which arises in the study of exterior scattering problems,
requires an accurate and efficient implementation of a
Stekhlov-Poincaré map. In this talk, we present a perturbative
technique for computing Stekhlov-Poincaré (Dirichlet to Neumann) maps
for use in exterior acoustic scattering problems. This work extends
the idea of using exact Dirichlet-to-Neumann maps on separable
geometries. Computational experiments as presented. An error analysis
for a finite element method used in conjunction with these
perturbative Stekhlov-Poincaré maps is provided.
This is joint work with Prof. D. P. Nicholls, U. Notre Dame.
- FRANCESCA RAPETTI, Univ. of Nice and Sophia-Antipolis, Parc Valrose, 06108 Nice
Cedex 02, France
Coupling scalar and vector potentials on nonmatching grids
for eddy currents in a moving conductor
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The T-W formulation of the magnetic field has been introduced
in many papers for the approximation of the magnetic quantities
modeled by the eddy current equations. This decomposition allows to
use a scalar function in the main part of the computational domain,
reducing the use of vector quantities to the conducting parts. We
propose to approximate these two quantities on nonmatching grids so
as to be able to tackle a problem where the conducting part can move
in the global domain. The connection between the two grids is managed
with mortar element techniques.
- JACQUES RAPPAZ, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne,
Switzerland
Finite element approximation of multi-scale elliptic problems
using patches of elements
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In this talk we will present a numerical method for solving elliptic
problems with multi-scale data using multiple levels of not
necessarily nested grids. The method consists in calculating
successive corrections to the solution in patches whose discretizations
are not necessarily conforming with the initial mesh. New results
including the spectral analysis of the iteration operator and a
numerical method to evaluate the CBS constant will be presented.
- ERIC SONNENDRUCKER, Université Louis Pasteur, 67084 Strasbourg Cedex, France
Méthodes numériques adaptatives pour l'équation de Vlasov
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La modélisation de nombreux problèmes mettant en jeu des
particules chargées est basée sur l'équation de Vlasov qui est
une équation de transport non linéaire dans l'espace des phases
couplée aux équations de Maxwell ou de Poisson. Le fait que
l'équation soit posée dans l'espace des phases double le nombre de
dimensions et rend donc sa résolution numérique d'autant plus
lourde. Une méthode largement utilisée pour sa résolution
numérique est la méthode semi-Lagrangienne. Les physiciens
l'utilisent de manière courante en 4D de l'espace des phases sur des
maillages uniformes. Mais l'utilisation de maillages uniformes est
loin d'être optimale et augmente de manière importante le coût
de calcul. Nous présenterons dans cet exposé des méthodes
adaptatives basées sur des approximations multi-résolutions
permettant de manière automatique d'utiliser à chaque pas de temps
un maillage parfaitement adapté à la fonction de distribution des
particules qui est la quantité calculée.
- JEAN-PAUL VILA, Toulouse
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