Function approximation via the subsampled Poincaré inequality

Yifan Chen; Thomas Y. Hou

doi:10.3934/dcds.2020296

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2021, Volume 41, Issue 1: 169-199. Doi: 10.3934/dcds.2020296

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Function approximation via the subsampled Poincaré inequality

Yifan Chen and
Thomas Y. Hou^,

Applied and Computational Mathematics, Caltech, 91106

^* Corresponding author: Thomas Y. Hou
^* Corresponding author: Thomas Y. Hou

Received: December 2019

Revised: June 2020

Early access: August 2020

Published: January 2021

The research was in part supported by NSF Grants DMS-1912654 and DMS-1907977. Y. Chen is supported by the Kortschak Scholars Program

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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.

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Mathematics Subject Classification: Primary: 35A23, 41A44; Secondary: 65D05, 65D07, 62G05.

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Figure 1. Domain $ \Omega = [0, 1]^2 $; the local cube $ \omega_i^{H} $ and the subsampled cube $ \omega_i^{h, H} $

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