Parametric reduced order models based on a Riemannian barycentric interpolation

A new strategy for constructing parametric Galerkin reduced order models is presented in this article. This strategy is achieved thanks to the Riemannian manifold, quotient of the set of full-rank matrices by the orthogonal group. Starting from a set of training parametrized subspaces of the same dimension, namely obtained by the proper orthogonal decomposition, the projection subspace for a new untrained parameter value is sought as the Karcher barycenter of the data points. The principal advantage of this strategy is that it leads to parametric reduced order models that are naturally flexible with respect to parameter variations. This property is a result of the simple expressions of the geodesic exponential and logarithmic mappings involved in the calculations of the approximated projection subspace. Confrontation with the usual interpolation in the tangent space to the Grassmann manifold is carried out for the flow problem past a circular cylinder and the flow in a lid-driven cavity. Comparable results in terms of accuracy are obtained, while the proposed approach is shown to be computationally cheaper by allowing real-time update of Galerkin projections.