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Pointwise error bounds in POD methods without difference quotients

Bosco García-Archilla, Julia Novo

TL;DR

The paper addresses whether POD methods that omit difference quotients (DQs) from snapshot data can still yield reliable pointwise error bounds as the snapshot count grows. It develops discrete Agmon and interpolation inequalities in Sobolev spaces to bound time-pointwise errors in terms of time-derivative norms and the POD tail, yielding decay rates close to optimal with increased smoothness (e.g., $u\in H^m(0,T;X)$). The theory is applied to heat-equation ROMs using implicit Euler and BDF2 time discretizations, yielding explicit error bounds that separate projection-tail and time-discretization contributions, and it is validated by numerical experiments showing bounded overestimation factors. Overall, the work clarifies when omitting DQs does not degrade pointwise accuracy and provides a pathway to near-optimal convergence in POD-based reduced-order models for time-dependent PDEs.

Abstract

In this paper we consider proper orthogonal decomposition (POD) methods that do not include difference quotients (DQs) of snapshots in the data set. The inclusion of DQs have been shown in the literature to be a key element in obtaining error bounds that do not degrade with the number of snapshots. More recently, the inclusion of DQs has allowed to obtain pointwise (as opposed to averaged) error bounds that decay with the same convergence rate (in terms of the POD singular values) as averaged ones. In the present paper, for POD methods not including DQs in their data set, we obtain error bounds that do not degrade with the number of snapshots if the function from where the snapshots are taken has certain degree of smoothness. Moreover, the rate of convergence is as close as that of methods including DQs as the smoothness of the function providing the snapshots allows. We do this by obtaining discrete counterparts of Agmon and interpolation inequalities in Sobolev spaces. Numerical experiments validating these estimates are also presented.

Pointwise error bounds in POD methods without difference quotients

TL;DR

The paper addresses whether POD methods that omit difference quotients (DQs) from snapshot data can still yield reliable pointwise error bounds as the snapshot count grows. It develops discrete Agmon and interpolation inequalities in Sobolev spaces to bound time-pointwise errors in terms of time-derivative norms and the POD tail, yielding decay rates close to optimal with increased smoothness (e.g., ). The theory is applied to heat-equation ROMs using implicit Euler and BDF2 time discretizations, yielding explicit error bounds that separate projection-tail and time-discretization contributions, and it is validated by numerical experiments showing bounded overestimation factors. Overall, the work clarifies when omitting DQs does not degrade pointwise accuracy and provides a pathway to near-optimal convergence in POD-based reduced-order models for time-dependent PDEs.

Abstract

In this paper we consider proper orthogonal decomposition (POD) methods that do not include difference quotients (DQs) of snapshots in the data set. The inclusion of DQs have been shown in the literature to be a key element in obtaining error bounds that do not degrade with the number of snapshots. More recently, the inclusion of DQs has allowed to obtain pointwise (as opposed to averaged) error bounds that decay with the same convergence rate (in terms of the POD singular values) as averaged ones. In the present paper, for POD methods not including DQs in their data set, we obtain error bounds that do not degrade with the number of snapshots if the function from where the snapshots are taken has certain degree of smoothness. Moreover, the rate of convergence is as close as that of methods including DQs as the smoothness of the function providing the snapshots allows. We do this by obtaining discrete counterparts of Agmon and interpolation inequalities in Sobolev spaces. Numerical experiments validating these estimates are also presented.
Paper Structure (9 sections, 14 theorems, 125 equations, 2 figures, 5 tables)

This paper contains 9 sections, 14 theorems, 125 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Application to a POD method.
  4. Numerical experiments
  5. Abstract results
  6. The periodic case
  7. The general case
  8. Application to sequences of funtion values
  9. Summary and conclusions

Key Result

Theorem 2.1

Let $u$ be bounded in $H^1(0,T,X)$. Then, for the constant $c_A$ defined in c_A below, the following bound holds: If $u_0+\cdots+u_M=0$, then the second term on the right-hand side above can be omitted.

Figures (2)

  • Figure 1: $L^2$ norms of the time derivatives of the finite element approximation $\boldsymbol u_h$ to the periodic orbit solution of \ref{['bruss']}.
  • Figure 2: $L^2$ norms of the elements $\varphi^1,\ldots,\varphi^J$ of the POD basis for $X=H^1_0$ and constant $C_P$ in \ref{['Poincare']}.

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 4.1
  • Remark 4.2
  • ...and 9 more