In this work an efficient approach for a posteriori error estimation for POD-DEIM reduced nonlinear dynamical systems is introduced. The considered nonlinear systems may also include time- and parameter-affine linear terms as well as parametrically dependent inputs and outputs. The reduction process involves a Galerkin projection of the full system and approximation of the system's nonlinearity by the DEIM method [S. Chaturantabut and D. C. Sorensen,ﾠSIAM J. Sci. Comput., 32 (2010), pp. 2737--2764]. The proposed a posteriori error estimator can be efficiently decomposed in an offline/online fashion and is obtained by a one-dimensional auxiliary ODE during reduced simulations. Key elements for efficient online computation are partial similarity transformations and matrix-DEIM approximations of the nonlinearity Jacobians. The theoretical results are illustrated by application to an unsteady Burgers equation and a cell apoptosis model.
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机译：在这项工作中，介绍了一种用于POD-DEIM简化非线性动力学系统的后验误差估计的有效方法。所考虑的非线性系统还可以包括时间和参数仿射线性项以及参数相关的输入和输出。归约过程涉及整个系统的Galerkin投影和通过DEIM方法近似系统的非线性[S. Chaturantabut和D.C. Sorensen，ﾠ SIAM J. Sci。计算（32）（2010），第2737--2764页]。所提出的后验误差估计器可以离线/在线方式有效地分解，并且可以在简化的模拟过程中通过一维辅助ODE获得。高效在线计算的关键要素是非线性雅可比矩阵的部分相似度转换和矩阵DEIM近似。理论结果通过应用于非稳态Burgers方程和细胞凋亡模型得到说明。
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