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

Koc, Birgul; Rubino, Samuele; Schneier, Michael; Singler, John R.; Iliescu, Traian

doi:10.48550/arxiv.2010.03750

Cited by 1 publication

(15 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have the following result about the pointwise error bounds for this POD case. A similar result with a different set of constant weights can be found in [28,Theorem 3.7]. The proof of the result below is similar so we omit it here.…”

Section: Pod With Difference Quotientssupporting

confidence: 65%

“…Throughout this work, we refer to this method as the standard DQ approach. This approach has been studied by many including [11,17,21,[28][29][30]. In this work we consider backward Euler for the time stepping scheme and the difference quotients.…”

Section: Pod With Difference Quotientsmentioning

confidence: 99%

“…The results in this work can be extended to variable weights or other choices of constant weights. One popular choice of constant weight in the literature is M −1 as in [21,28] where M represents the total number of data snapshots for the standard difference quotient approach to POD. Similarly to the standard POD case, with certain choices of weights, one can approximate time integrals with various quadrature rules.…”

Section: Pod With Difference Quotientsmentioning

confidence: 99%

“…POD numerical analysis papers tend to focus on the time discretization error and the ROM discretization error since the spatial discretization error can typically be handled using existing techniques. For more information on these three types of error and the numerical analysis of POD, see the introduction of the recent work [28]. The current work focuses only on the POD ROM errors and leads to an improved understanding of the numerical analysis of POD ROMs, particularly in regards to difference quotients and pointwise error bounds.…”

Section: Introductionmentioning

confidence: 99%

“…In this work a notion of optimality was introduced and results strongly suggested that DQs were needed to achieve optimal pointwise-in-time convergence rates. Recently in [28], further progress was made. In this work it is shown that a critical assumption often made when using standard POD without DQs is automatically guaranteed to be satisfied when DQs are included with the data.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A new approach to proper orthogonal decomposition with difference quotients

Locke¹,

Singler²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Pod With Difference Quotientssupporting

confidence: 65%