SEMINAIRE D'ANALYSE NUMÉRIQUE
Année universitaire 2017-2018
Jeudi
17 mai : Ionut
DANALLA
(Laboratoire
de
Mathématiques
Raphaël Salem,
Université de
Rouen
Normandie)
Sobolev
gradient and
conjugate-gradient
methods
for solving
optimization
problems for
fluid or
superfluid
systems
This is a joint work with B. Protas, McMaster University, Canada.
Steepest descent methods using Sobolev gradients proved very effective to
solve minimisation problems in different application fields [1].
We present two original methods to solve minimisation problems using Sobolev
gradients.
The first problem concerns the reconstruction of the velocity field in
a fluid flow dominated by a large scale vortex ring structure.
We reconstruct the vorticity distribution inside the
axisymmetric vortex ring from some incomplete and possibly noisy
measurements of the surrounding
velocity field. The numerical approach inspired from Shape Optimization
theory is described in detail in [2].
The second problem is the minimisation of the constrained
Gross-Pitaevskii energy functional describing superfluid Bose-Einstein
condensates.
We present the new Sobolev gradient method suggested in [3] to efficiently
compute stationary states with quantized vortices and used in [4]
to simulate rotating Bose-Einstein condensates. The method is reformulated
in the framework of the Riemann Optimization theory to derive an efficient
nonlinear conjugate-gradient method (details in [5]).
Both numerical algorithms were implemented using a finite-element method and
programmed using the free software FreeFem++ [6], an easy-to-use and highly
adaptive software offering many advantages for the implementation of
complex algorithms: syntax close to the mathematical formulation, advanced
automatic mesh generator, mesh adaptation, automatic interpolation,
interface with state-of-the-art numerical libraries (PETSC, UMFPACK,
SUPERLU, MUMPS, METIS, IPOPT, etc).
We illustrate the suggested new numerical methods by computing various cases
from fluid mechanics (vortex rings)
and condensed matter physics (Bose-Einstein condensates with quantized
vortices).
[1] J. W. Neuberger, Sobolev Gradients and Differential Equations, Springer,
2010.
[2] I. Danaila and B. Protas, Optimal reconstruction of inviscid vortices,
Proceedings of the Royal Society A, 471: 20150323, 2015.
[3] I. Danaila, P. Kazemi, A new Sobolev gradient method for direct
minimization of
the Gross-Pitaevskii energy with rotation. SIAM J. of Scientific
Computing,
32:2447-2467, 2010.
[4] G. Vergez, I. Danaila, S. Auliac, F.Hecht, A finite-element
toolbox for the stationary Gross-Pitaevskii equation with rotation, Computer
Physics Communications, 209, p. 144-162, 2016.
[5] I. Danaila, B. Protas, Computation of Ground States of the
Gross-Pitaevskii
Functional via Riemannian Optimization, SIAM J. of Scientific Computing, 39,
pp. B1102-B1129, 2017.
[6] www.freefem.org.
