On the effect of different samplings to the solution of parametric PDE eigenvalue problems

The use of sparse sampling is a consolidated technique for the reduced order modeling of parametric PDEs. In this note we investigate the choice of sampling points within the framework of reduced order techniques for the approximation of eigenvalue problems originating from parametric PDEs. We use the standard proper orthogonal decomposition technique to obtain the basis of the reduced space and Galerkin orthogonal technique to get the reduced problem. We present some numerical results and observe that, as in the case of the source problem, also for eigenvalue problems the use of sparse sampling is a good idea and that, when the number of sampling points is assigned, sparse sampling provides better results than uniform sampling. In the spirit of the journal, we present our results in the form of examples and counterexamples.