An Efficient Output Error Estimation for Model Order Reduction of Parametrized Evolution Equations

Abstract

In this work we present an a posteriori output error bound for model order reduction of parametrized evolution equations. With the help of the dual system and a simple representation of the relationship between the field variable error and the residual of the primal system, a sharp output error bound is derived. Such an error bound successfully avoids the accumulation of the residual over time, which is a common drawback in the existing error estimations for time-stepping schemes. An estimation needs to be performed for practical computation of the error bound, and as a result, the output error bound reduces to an output error estimation. The proposed error estimation is applied to four kinds of problems. The first one is a linear convection-diffusion equation, which is used to compare the performance of the new error estimation and an existing primal-dual error bound. The second one is the unsteady viscous Burgers' equation, an academic benchmark of nonlinear evolution equations in fluid dynamics often used as a first test case to validate nonlinear model order reduction methods. The other two problems arise from chromatographic separation processes. Numerical experiments demonstrate the performance and efficiency of the proposed error estimation. Furthermore, optimization based on the resulting reduced-order models is successful in terms of accuracy and runtime for obtaining the optimal solution. B911for linear or quadratically nonlinear parabolic problems [34,37,38,39]. Notably, these error estimations are all derived in the functional space in the framework of the finite element (FE) discretization except for [17]. In the FE discretization framework, the weak form of the partial differential equation (PDE) is used to derive the error bound, while the error bound in [17] is derived in the framework of the finite volume discretization for error estimation of the field variables.In this paper, we propose an efficient output error estimation for projection-based MOR methods applied to parametrized nonlinear evolution problems. For (nonlinear) evolution problems, time-stepping schemes are often used to solve them [24], and error estimations for projection-based MOR methods have been studied in recent years; see, e.g., [13,17,40]. The error estimator, however, may lose sharpness when a large number of time steps are needed, because the error estimator is actually a summation of the error over the previous time steps. To circumvent this problem, we introduce a suitable dual system at each time instance in the evolution process associated with the primal system, i.e., the original system. The output error for the primal system can thus be estimated sharply and efficiently with the help of the dual system and a simple representation of the relationship between the residual and the error of the field variable.Actually, an a posteriori output error bound for the RB method using the primaldual approach can be found in [16]. However, the derived output error bound is suitable only for linear evolution equations. From the ...

A New Error Estimator for Reduced-Order Modeling of Linear Parametric Systems

“…In every iteration, the s sample corresponding to the maximal error estimator is chosen as the next expansion point s i (Step 22). Steps 5,8,12,16 and Step 20 orthogonalize the vectors in V (s i ) and V du (s i ), V r du (s α i ), V rpr (s α i ), V rrpr (s β i ) against the existing vectors in V and V du , V r du , V rpr , V rrpr , respectively. In Algorithm 1, some steps are only implemented for certain error estimators, depending on which error estimator is being used.…”

“…, (Ã(s α i )) q−1B (s α i )}. 16: = ∆(s i ). 39: end while Algorithm 2 Greedy ROM construction for parametric systems (1) .…”

“…end if 14: if ∆(μ) = ∆ pr 1 (μ), or ∆ pr 2 (μ), or ∆ 3 (μ), or ∆ pr 3 (μ) then 15: compute V rpr µ i α following (44). 16: = ∆(μ i ). 39: end while Algorithm 4 Improving ∆ 1 (μ), ∆ 2 (μ) and ∆ pr 2 (μ) for nearly symmetric and parametric systems (1) .…”

Accelerating optimization and uncertainty quantification of nonlinear SMB chromatography using reduced-order models

“…While the error estimators from [11,36,54] are developed for nonlinear time-dependent problems, they may become inaccurate for systems with switching procedure. In our recent work [55], an efficient output error estimation is derived, and it has been successfully applied to a nonlinear batch chromatographic model and a linear SMB model. In this work, we will use the error estimation proposed in [55] to construct the ROM for the nonlinear SMB chromatography.…”

“…It remains still an active topic in MOR. In Section 4, we will introduce a recently proposed error estimation from [55], which will be used as the error indicator in the POD-Greedy algorithm to construct the RB V for the SMB model.…”