Principal angles between subspaces and reduced order modelling accuracy in optimization

The paper considers robust parametric optimization problems using multi-point formulations and addresses the issue of the approximation of the gradient of the functional by reduced order models. The question of interest is the impact of such approximations on the search subspace in the multi-point optimization problem. The mathematical concept used to evaluate these approximations is the principal angles between subspaces and practical ways to evaluate these are provided. An additional indicator is provided when a descent minimization algorithm is used. The approach appears also to be an interesting tool for uncertainty quantification of the design in the presence of models of increasing complexity. The application of these concepts is illustrated in the design of the shape of an aircraft robust over a range of transverse winds.