engleski

New developments of the Discrete Empirical Interpolation Method

In large scale modeling and simulations, the discrete empirical interpolation method (DEIM, as a discrete version of the Empirical Interpolation Method, EIM) is a powerful and versatile computational tool. When combined with the POD/Galerkin projection method in a model order reduction framework, it efficiently removes the bottleneck caused by the evaluation of the nonlinear term. Further, in the implicit POD scheme, it provides an efficient on-line approximation of the Jacobian, needed for the Newton iteration step. In essence, DEIM is an oblique interpolatory projection, which approximates the POD projection based on sparse information. In a concrete application, to preserve physical properties of the reduced model, the POD/Galerkin and the DEIM projection must be with respect to a particular weighted inner product. We present fine numerical details of the weighted DEIM, that can also be interpreted as a numerical implementation of the Generalized Empirical Interpolation Method and the more general Parametrized-Background Data-Weak approach. Further, we discuss how the concept of the DEIM projection can be interpreted as trajectories clustering method, and how to use the new point of view for further development. We emphasize the fine interplay between numerical linear algebra and pure matrix theory.

POD, Galerkin projection, DEIM, clustering

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