On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition

TL;DR

This paper establishes the importance of difference quotients in achieving optimal pointwise in time error bounds for POD reduced order models of the heat equation, resolving longstanding issues in error analysis.

Contribution

It proves that using difference quotients in POD error analysis yields optimal error bounds, addressing a long-standing problem in reduced order modeling.

Findings

Difference quotients lead to optimal error bounds.

Without DQs, errors are suboptimal.

Numerical results confirm theoretical predictions.

Abstract

In this paper, we resolve several long standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In particular, we study the role played by difference quotients (DQs) in obtaining reduced order model (ROM) error bounds that are optimal with respect to both the time discretization error and the ROM discretization error. When the DQs are not used, we prove that both the ROM projection error and the ROM error are suboptimal. When the DQs are used, we prove that both the ROM projection error and the ROM error are optimal. The numerical results for the heat equation support the theoretical results.