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A new approach to proper orthogonal decomposition with difference quotients

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Abstract

In a recent work (Koc et al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), 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 approximately half the number of snapshots as the standard POD DQ approach, i.e., the new approach requires less computational effort. We illustrate our theoretical results with numerical experiments.

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Funding

J. Singler was supported by the US National Science Foundation (NSF) under grant number 2111421.

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Authors and Affiliations

  1. School of Mathematics and Science, University of Tennessee Southern, 433 W Madison Street, Pulaski, TN, 38478, USA

    Sarah Locke Eskew

  2. Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 W. 12th Street, Rolla, MO, 65401, USA

    John R. Singler

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  1. Sarah Locke Eskew
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Correspondence to Sarah Locke Eskew.

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Communicated by: Stefan Volkwein

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John R. Singler contributed equally to this work.

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Eskew, S.L., Singler, J.R. A new approach to proper orthogonal decomposition with difference quotients. Adv Comput Math 49, 13 (2023). https://doi.org/10.1007/s10444-023-10011-9

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