A new approach to proper orthogonal decomposition with difference quotients

TL;DR

This paper introduces a more computationally efficient method for proper orthogonal decomposition with difference quotients, reducing the number of snapshots needed while maintaining optimal error bounds for heat equation models.

Contribution

The authors propose a new POD DQ approach that uses fewer snapshots, specifically only the first snapshot and all DQs, without losing analysis benefits.

Findings

Retains optimal pointwise error bounds

Uses half the number of snapshots

Maintains numerical analysis advantages

Abstract

In a recent work [B. Koc et al., arXiv:2010.03750, SIAM J. Numer. Anal., to appear], the authors showed that including difference quotients (DQs) is necessary in order to prove optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order models of the heat equation. In this 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, we only 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 of the standard POD DQ approach, while using a POD data set that has half the number of snapshots as the standard POD DQ approach, i.e., the new approach is more computationally efficient. We illustrate our theoretical results with numerical experiments.