Snapshot Location in Proper Orthogonal Decomposition for Linear and Semi-linear Parabolic Partial Differential Equations

It is well-known that the performance of POD and POD-DEIM methods depends on the selection of the snapshot locations. In this work, we consider the selections of the locations for POD and POD-DEIM snapshots for spatially semi-discretized linear or semi-linear parabolic PDEs. We present an approach that for a fixed number of snapshots the optimal locations may be selected such that the global discretization error is approximately the same in each associated sub-interval. The global discretization error is assessed by a hierarchicaltype a posteriori error estimator developed from automatic time-stepping for systems of ODEs. We compare the global discretization error of this snapshot selection on error equilibration for the full order model (FOM) with that for the reduced order model (ROM) to study its impact. This contribution also shows that the equilibration of the global discretization error for the FOM is preserved by its corresponding POD and POD-DEIM based ROM. The numerical examples illustrating the performance of this approach are provided.