A new approach to proper orthogonal decomposition with difference quotients

Abstract

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.

POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution

“…These temporal derivatives can be easily computed using the righthand side of the mixed finite element Galerkin equation. In the present paper we follow an idea presented in the recent paper [22] in which it is shown that there is no need to include in the set of snapshots other than one approximation to the velocity at a fixed time, instead of the full set u h (t j ) M j=0 as it is usually done in the literature. The numerical analysis in [22] is carried out for the heat equation and with difference quotients, (u h (t j ) − u h (t j−1 )/(t j − t j−1 ) , approaching the time derivative.…”

“…In the present paper we follow an idea presented in the recent paper [22] in which it is shown that there is no need to include in the set of snapshots other than one approximation to the velocity at a fixed time, instead of the full set u h (t j ) M j=0 as it is usually done in the literature. The numerical analysis in [22] is carried out for the heat equation and with difference quotients, (u h (t j ) − u h (t j−1 )/(t j − t j−1 ) , approaching the time derivative. In the present paper we consider instead the Galerkin time derivatives although the analysis for the difference quotients case is essentially the same (or even simpler).…”

“…Several works in the literature have already studied the subject of increasing the set of snapshots with approximations to the time derivatives. However, apart from [22], all of them include as snapshots the set of the approximations at different times, instead of only one snapshot. Also, starting with the pioneering paper [21], most of these papers include a different type of approximation than considered in this paper, namely difference quotients.…”

“…Trying to keep the theoretical advantages of including approximations of the temporal derivative of the velocity in the set of snapshots but relaxing their drawbacks in practical simulations, we study in this paper, both theoretically and numerically, the inclusion of time derivatives of the discrete velocity. As in [22], our approach has half the number of snapshots as in the standard POD finite quotient approach. Also, as in the difference quotient case, we are able to get pointwise in time estimates using as projecting space X = H 1 0 (Ω) d and L 2 (Ω) d .…”

“…Also, as in the difference quotient case, we are able to get pointwise in time estimates using as projecting space X = H 1 0 (Ω) d and L 2 (Ω) d . To this end, we follow [22,Lemma 3.3] (see also [19,Lemma 3.6]) to prove that the X-norm of a function at any point in time (let's say t j ) is bounded in terms of the X-norm of the value at t 0 plus the mean values of its time derivative taken in a time interval (let's say {t 0 , t 1 , . .…”