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Function approximation via the subsampled Poincaré inequality

Chen, Yifan and Hou, Thomas Y. (2021) Function approximation via the subsampled Poincaré inequality. Discrete and Continuous Dynamical Systems - Series A, 41 (1). pp. 169-199. ISSN 1553-5231. https://resolver.caltech.edu/CaltechAUTHORS:20200122-075925100

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Abstract

Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincaré inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincaré inequality, where the measurement function is of subsampled type, with a small but non-zero lengthscale that will be made precise. Our analysis identifies this inequality as a basic tool for function recovery problems. We discuss and demonstrate the optimality of the inequality concerning the subsampled lengthscale, connecting it to existing results in the literature. In application to function approximation problems, the approximation accuracy using different basis functions and under different regularity assumptions is established by using the subsampled Poincaré inequality. We observe that the error bound blows up as the subsampled lengthscale approaches zero, due to the fact that the underlying function is not regular enough to have well-defined pointwise values. A weighted version of the Poincaré inequality is proposed to address this problem; its optimality is also discussed.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.3934/dcds.2020296DOIArticle
https://arxiv.org/abs/1912.08173arXivDiscussion Paper
Alternate Title:The subsampled Poincaré inequality for functional recovery
Additional Information:© 2020 American Institute of Mathematical Sciences. Received December 2019; revised June 2020. The research was in part supported by NSF Grants DMS-1912654 and DMS-1907977. Y. Chen is supported by the Kortschak Scholars Program. We want to thank Professor Henri Berestycki and Jinchao Xu for their interest in our work and for bringing to our attention some of the relevant references. Y. Chen would like to thank Yousuf Soliman for many insightful discussions on the subsampled Poincaré inequality. We thank the anonymous reviewer for the helpful comments that improve this work.
Funders:
Funding AgencyGrant Number
NSFDMS-1912654
NSFDMS-1907977
Kortschak Scholars ProgramUNSPECIFIED
Subject Keywords:Poincaré inequality, subsampled data, function approximation and recovery, degeneracy, weighted inequality
Issue or Number:1
Classification Code:2020 Mathematics Subject Classification. Primary: 35A23, 41A44; Secondary: 65D05, 65D07, 62G05
Record Number:CaltechAUTHORS:20200122-075925100
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200122-075925100
Official Citation:Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100823
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:22 Jan 2020 17:11
Last Modified:16 Dec 2020 18:34

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