We propose an online adaptive local-global POD-DEIM model reduction method for\nflows in heterogeneous porous media. The main idea of the proposed method is to use local\nonline indicators to decide on the global update, which is performed via reduced cost local\nmultiscale basis functions. This unique local-global online combination allows (1) developing local\nindicators that are used for both local and global updates (2) computing global online modes via\nlocal multiscale basis functions. The multiscale basis functions consist of offline and some online\nlocal basis functions. The approach used for constructing a global reduced system is based on\nProper Orthogonal Decomposition (POD) Galerkin projection. The nonlinearities are approximated\nby the Discrete Empirical Interpolation Method (DEIM). The online adaption is performed by\nincorporating new data, which become available at the online stage. Once the criterion for updates\nis satisfied, we adapt the reduced system online by changing the POD subspace and the DEIM\napproximation of the nonlinear functions. The main contribution of the paper is that the criterion\nfor adaption and the construction of the global online modes are based on local error indicators and\nlocal multiscale basis function which can be cheaply computed. Since the adaption is performed\ninfrequently, the new methodology does not add significant computational overhead associated\nwith when and how to adapt the reduced basis. Our approach is particularly useful for situations\nwhere it is desired to solve the reduced system for inputs or controls that result in a solution outside\nthe span of the snapshots generated in the offline stage. Our method also offers an alternative\nof constructing a robust reduced system even if a potential initial poor choice of snapshots is\nused. Applications to single-phase and two-phase flow problems demonstrate the efficiency of\nour method.
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