Pointwise error bounds in POD methods without difference quotients

TL;DR

This paper derives pointwise error bounds for POD methods that exclude difference quotients, showing they remain effective with increasing snapshots if the underlying function is sufficiently smooth, matching the convergence rates of DQ-inclusive methods.

Contribution

It provides new error bounds for POD methods without difference quotients, demonstrating their effectiveness under certain smoothness conditions and matching DQ-based convergence rates.

Findings

Error bounds do not degrade with the number of snapshots for smooth functions.

Convergence rates are close to those of DQ-inclusive methods.

Numerical experiments validate the theoretical estimates.

Abstract

In this paper we consider proper orthogonal decomposition (POD) methods that do not include difference quotients (DQs) of snapshots in the data set. The inclusion of DQs have been shown in the literature to be a key element in obtaining error bounds that do not degrade with the number of snapshots. More recently, the inclusion of DQs has allowed to obtain pointwise (as opposed to averaged) error bounds that decay with the same convergence rate (in terms of the POD singular values) as averaged ones. In the present paper, for POD methods not including DQs in their data set, we obtain error bounds that do not degrade with the number of snapshots if the function from where the snapshots are taken has certain degree of smoothness. Moreover, the rate of convergence is as close as that of methods including DQs as the smoothness of the function providing the snapshots allows. We do this by obtaining discrete counterparts of Agmon and interpolation inequalities in Sobolev spaces. Numerical experiments validating these estimates are also presented.