On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition

Abstract

In this paper, we resolve several long-standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In particular, we study the role played by difference quotients (DQs) in obtaining reduced order model (ROM) error bounds that are optimal with respect to both the time discretization error and the ROM discretization error. When the DQs are not used, we prove that both the POD projection error and the ROM error are suboptimal. When the DQs are used, we prove that both the POD projection error and the ROM error are optimal. The numerical results for the heat equation support the theoretical results.

Reduced Order Model Closures: A Brief Tutorial

“…More than a decade later, Singler improved Kunisch and Volkwein's results, by proving sharper error bounds [47]. Recently, optimal pointwise in time error bounds were proved in [30]. These results finally bring the G-ROM numerical analysis to a level comparable to (although not as developed as) the level of the numerical analysis of the FEM.…”

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

“…However, it turns out that these snapshots are needed for obtaining pointwise estimates in time. In fact, we can repeat the arguments of the previous section to get such estimates for the L 2 (Ω) d error, and also for the H 1 (Ω) d error applying the inverse inequality (19), that cannot be obtained, at least with the same arguments, without adding those snapshots.…”

“…It was observed that if the POD basis functions are based on the projection onto the Hilbert space X = L 2 (Ω) d , the difference quotients are not needed to prove optimal error bounds in certain norms, see [4,15,27,23]. However, as pointed out in [19], even in this case, the inclusion of the difference quotients allows to get pointwise estimates in time that generally cannot be proved if there are no difference quotients in the set of snapshots. On the other hand, from the numerical point of view, it is not clear that the difference quotients should be included in the actual simulations with the POD-ROM.…”

“…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 , . .…”

A new reduced order model of imcompressible Stokes equations

“…In recent years, model order reduction (MOR) becomes more popular for solving partial differential equations (PDEs); see [1,6,[10][11][12][13][14][15][16]. Especially, there has been a growing interest in the application of ROMs to modeling incompressible flows [5,9].…”