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Computation 2016, 4(2), 22; doi:10.3390/computation4020022

Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media

1
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
2
Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
3
Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Qinjun Kang
Received: 8 March 2016 / Revised: 17 May 2016 / Accepted: 23 May 2016 / Published: 7 June 2016
(This article belongs to the Special Issue Advances in Modeling Flow and Transport in Porous Media)
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Abstract

We propose an online adaptive local-global POD-DEIM model reduction method for flows in heterogeneous porous media. The main idea of the proposed method is to use local online indicators to decide on the global update, which is performed via reduced cost local multiscale basis functions. This unique local-global online combination allows (1) developing local indicators that are used for both local and global updates (2) computing global online modes via local multiscale basis functions. The multiscale basis functions consist of offline and some online local basis functions. The approach used for constructing a global reduced system is based on Proper Orthogonal Decomposition (POD) Galerkin projection. The nonlinearities are approximated by the Discrete Empirical Interpolation Method (DEIM). The online adaption is performed by incorporating new data, which become available at the online stage. Once the criterion for updates is satisfied, we adapt the reduced system online by changing the POD subspace and the DEIM approximation of the nonlinear functions. The main contribution of the paper is that the criterion for adaption and the construction of the global online modes are based on local error indicators and local multiscale basis function which can be cheaply computed. Since the adaption is performed infrequently, the new methodology does not add significant computational overhead associated with when and how to adapt the reduced basis. Our approach is particularly useful for situations where it is desired to solve the reduced system for inputs or controls that result in a solution outside the span of the snapshots generated in the offline stage. Our method also offers an alternative of constructing a robust reduced system even if a potential initial poor choice of snapshots is used. Applications to single-phase and two-phase flow problems demonstrate the efficiency of our method. View Full-Text
Keywords: online adaptive model reduction; local model reduction; POD global model reduction; discrete empirical interpolation method; flows in heterogeneous media online adaptive model reduction; local model reduction; POD global model reduction; discrete empirical interpolation method; flows in heterogeneous media
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Efendiev, Y.; Gildin, E.; Yang, Y. Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media. Computation 2016, 4, 22.

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