The Discrete Empirical Interpolation Method: Canonical Structure and Formulation in Weighted Inner Product Spaces
For researchers in model reduction, this provides a more rigorous theoretical foundation and broader applicability of DEIM, though it is an incremental extension of existing methods.
The paper tightens the error bound for DEIM using strong rank-revealing QR and extends DEIM to weighted inner products (W-DEIM), enabling preservation of physical properties like stability in POD Galerkin projections.
New contributions are offered to the theory and practice of the Discrete Empirical Interpolation Method (DEIM). These include a detailed characterization of the canonical structure; a substantial tightening of the error bound for the DEIM oblique projection, based on index selection via a strong rank revealing QR factorization; and an extension of the DEIM approximation to weighted inner products defined by a real symmetric positive-definite matrix $W$. The weighted DEIM ($W$-DEIM) can be deployed in the more general framework where the POD Galerkin projection is formulated in a discretization of a suitable energy inner product such that the Galerkin projection preserves important physical properties such as e.g. stability. Also, a special case of $W$-DEIM is introduced, which is DGEIM, a discrete version of the Generalized Empirical Interpolation Method that allows generalization of the interpolation via a dictionary of linear functionals.