Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyper-reduction

Projection-based model reduction techniques rely on the definition of a small dimensional subspace in which the solution is approximated. Using local subspaces reduces the dimensionality of each subspace and enables larger speedups. Transitions between local subspaces require special care and updating the reduced bases associated with each subspace increases the accuracy of the reduced-order model. In the present work, local reduced basis updates are considered in the case of hyper-reduction, for which only the components of state vectors and reduced bases defined at specific grid points are available. To enable local reduced basis updates, two comprehensive approaches are proposed. The first one is based on an offline/online decomposition. The second approach relies on an approximated metric acting only on those components where the state vector is defined. This metric is computed offline and used online to update the local bases. An analysis of the error associated with this approximated metric is then conducted and it is shown that the metric has a kernel interpretation. Finally, the application of the proposed approaches to the model reduction of two nonlinear physical systems illustrates their potential for achieving large speedups and good accuracy.