Hierarchical Shape Optimization : Multi-Level Algorithms and
Unsymmetrical Nash Games

Jean-Antoine Désidéri
INRIA, Sophia Antipolis Méditerranée Center (France)

In the first part of the talk, multi-level strategies for parametric shape optimization will be reviewed. These methods employ nested parameterization supports of either shape, or shape deformation, and the classical process of degree elevation resulting in exact geometrical data transfer from coarse to fine representations. The corresponding algorithms mimic classical multigrid strategies and are found very effective in terms of convergence acceleration.

In the second part, we will focus on the definition and evaluation of unsymmetrical Nash games to handle multi-objective shape optimization problems with a biased split of territory that in effect, introduces a form of hierarchy between the criteria.

Finally, the two parts will be linked by showing how the diagonalization of a reduced Hessian matrix, introduced above to define a split of territory, can also be used to define an algebraic multi-level shape-optimization algorithm, in a manner less dependent on the original setting of the geometrical parameterization.