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 ...

In this work, we show that the reduced basis method accelerates a partial differential equation constrained optimization problem, where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization. Such an optimization problem arises, for example, from batch chromatography. To reduce the computational burden of repeatedly solving the large-scale system under parameter variations, a parametric reduced-order model with a small number of equations is derived by using the reduced basis method. As a result, the small reduced-order model, rather than the full system, is solved at each step of the optimization process. An adaptive technique for selecting the snapshots is proposed, so that the complexity and runtime for generating the reduced basis are largely reduced. An outputoriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically. An early-stop criterion is proposed to circumvent the stagnation of the error and to make the construction of the reduced model more efficient. Numerical examples show that the adaptive technique is very efficient in reducing the offline time. The optimization based on the reduced model is successful in terms of the accuracy and the runtime for acquiring the optimal solution. is often used to derive a ROM, which has been applied to accelerate optimization problems [5,6]. However, a ROM from POD is reliable only in the neighborhood of the input parameter setting at which the ROM is constructed. There is no guarantee for the accuracy of the ROM at a different parameter setting. To circumvent the problem, a trust-region technique was suggested to manage the POD-based ROM in [5]. Here, the ROM is updated according to the quality of the approximation. However, the repeated construction of the ROM reduces the significance of the reduction in computational resources obtained by MOR. In contrast, the technique of parametric model order reduction (PMOR) enables the generation of a parametric ROM with acceptable accuracy over the feasible parameter domain, such that a single ROM is sufficient for the optimization process. Among the various PMOR methods [7][8][9][10][11][12][13], few of them are applicable for nonlinear problems with parameters. The reduced basis method (RBM), however, has been developed for nonlinear parametric systems [14][15][16][17]. Moreover, endowed with a posteriori error estimation, the parametric ROM can be generated automatically.The RBM has been proved to be a powerful tool for rapid and reliable evaluation of the parameterized PDEs [14][15][16][17]. The reduced basis (RB), used to construct the ROM, is computed from snapshots (the solutions of the PDEs at certain selected samples of parameters and/or chosen time steps) through a greedy algorithm. When applied to optimization, the original system resulting from the discretization of PDEs is first replaced by a ROM generated by the RBM, then the related quantities can be evaluated rapidly by solving the...

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Numerical simulation of large-scale complex systems is challenging, especially when such a task needs to be done many times under parameter variations, e.g., in the context of optimization, control, and parameter estimation. Model order reduction (MOR) is a useful technique for constructing a low-cost surrogate, i.e., a reduced order model (ROM), which can reproduce the input-output response of the original large-scale system, with compromise on the accuracy to an acceptable extent. To compute a ROM, an efficient a posteriori error estimation is crucial because it enables the computation to be reliable and automatic.In this work we propose an efficient output error estimation for projection based MOR methods applied to parametrized nonlinear evolution problems. For (nonlinear) evolution equations, time-stepping schemes are often used to solve them, and error estimations for reduced basis methods have been studied in recent years [1,3]. The error estimator may loose 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 dual system at each time instance in the evolution process. With the help of the dual system and under suitable assumptions, the output error of the reduced model can be estimated more sharply. The proposed output error bound is independent of the choice of the projection matrix, and hence is independent of the MOR methods.The proposed error estimation is applied to two kinds of evolution problems arising from chromatographic separation processes. One is batch chromatography with bi-Langmuir isotherm equations, which is a nonlinear parametric evolution problem. The other is a linear parametric periodic switching system, the model of simulated moving bed chromatography with linear isotherm equations. Reduced basis methods are used to construct the reduced order model. More precisely, the POD-Greedy algorithm [2] is employed to generate the reduced basis. Figure 1 illustrates the decay of the error bounds and the corresponding true error during the reduced basis extension process for the batch chromatographic model. It is seen that the proposed output error bound (ErrorBound-2) is much sharper than ErrorBound-1 [1, 3]. The resulting reduced models are employed to solve the underlying optimization problems. Employing the reliable compact reduced models, the optimization problems are solved much faster with sufficient accuracy. The results of the optimization of batch chromatography are summarized in Table 1. It is seen that the optimal solution of the ROM based optimization converges to the full order model (FOM) based optimal solution, and the runtime is significantly reduced. The speedup factor (SpF) is 58. Table 1: Results of the optimization based on the FOM (N = 1500) and the ROM (N = 45). Models Objective Optimal solution #Iterations Runtime [h]/SpF FOM-Opt. 0.020264 (0.07964, 1.05445) 202 33.88 / -ROM-Opt. 0.020266 (0.07964, 1.05445) 202 0.58 / 58Numerical s...