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In a recent work [B. Koc et al., arXiv:2010.03750, SIAM J. Numer. Anal., toappear], the authors showed that including difference quotients (DQs) isnecessary in order to prove optimal pointwise in time error bounds for properorthogonal decomposition (POD) reduced order models of the heat equation. Inthis work, we introduce a new approach to including DQs in the POD procedure.Instead of computing the POD modes using all of the snapshot data and DQs, weonly use the first snapshot along with all of the DQs and special POD weights.We show that this approach retains all of the numerical analysis benefits ofthe standard POD DQ approach, while using a POD data set that has half thenumber of snapshots as the standard POD DQ approach, i.e., the new approach ismore computationally efficient. We illustrate our theoretical results withnumerical experiments.

A new approach to proper orthogonal decomposition with difference quotients