New Proper Orthogonal Decomposition Approximation Theory for PDE Solution Data

Locke, Sarah; Singler, John R.

doi:10.1137/19m1297002

Cited by 9 publications

(7 citation statements)

References 30 publications

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“…In the following result, we give POD approximation errors in different norms and using other projections onto X r . Similar results have been proved in multiple works (see, e.g., [28,29,39,48,50]), and our proof relies on techniques from these works. We note that this result can be obtained directly from the general results in the recent reference [39]; however, we include a proof to be complete.…”

Section: Pod Without Difference Quotients (Nodq Case)supporting

confidence: 82%

“…Similar results have been proved in multiple works (see, e.g., [28,29,39,48,50]), and our proof relies on techniques from these works. We note that this result can be obtained directly from the general results in the recent reference [39]; however, we include a proof to be complete. In this work, a bounded linear operator \Pi : Z \rightar Z for a normed space Z is a projection onto Z r \subset Z if \Pi 2 = \Pi and the range of \Pi equals Z r .…”

Section: Pod Without Difference Quotients (Nodq Case)supporting

confidence: 82%

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

Koc¹,

Rubino²,

Schneier³

et al. 2021

SIAM J. Numer. Anal.

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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 POD projection error and the ROM error are suboptimal. When the DQs are used, we prove that both the POD projection error and the ROM error are optimal. The numerical results for the heat equation support the theoretical results.

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Section: Pod Without Difference Quotients (Nodq Case)supporting

confidence: 82%

Section: Pod Without Difference Quotients (Nodq Case)supporting

confidence: 82%

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

Koc¹,

Rubino²,

Schneier³

et al. 2021

SIAM J. Numer. Anal.

Self Cite

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“…• The pointwise ROM error in Equation (36) with the error norm Y = L 2 is optimal-I if there exists a constant C such that…”

Section: Optimalitymentioning

confidence: 99%

“…Because of the widespread use of POD in applications, many researchers have studied POD ROMs from a numerical analysis perspective; see, e.g., [5,21,22,30,31,36,37,40,45]. In order for POD to be beneficial for applications, researchers must understand how the various errors involved behave.…”

Section: Introductionmentioning

confidence: 99%

A new approach to proper orthogonal decomposition with difference quotients

Locke¹,

Singler²

2021

Preprint

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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.

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“…In recent years, model order reduction (MOR) becomes more popular for solving partial differential equations (PDEs); see [1,6,[10][11][12][13][14][15][16]. Especially, there has been a growing interest in the application of ROMs to modeling incompressible flows [5,9].…”

Section: Introductionmentioning

confidence: 99%