Sobolev gradient and conjugate-gradient methods for solving optimization problems for fluid or superfluid systems
Time: 2017-06-05
Published By: Xiaoni Tan
Speaker(s): Ionut Danaila, University of Rouen Normandy, France
Time: 09:30-10:30 June 9, 2017
Venue: 理科一号楼1114教室
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 surroundingvelocity 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 type 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).
This is a joint work with B. Protas, McMaster University, Canada.
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 surroundingvelocity 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 type 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).
This is a joint work with B. Protas, McMaster University, Canada.
