engleski

New contributions to the theory and practice of the Discrete Empirical Interpolation Method

This presentation discusses recent new contributions to the theory and practice of the DEIM. Both the matrix theoretical and computational (application oriented) aspects of the method are considered. - The theoretical error bound of the DEIM oblique projection is substantially improved by deploying strong rank revealing QR factorization in the Q-DEIM formulation of the index selection. It is shown in that we can obtain $c\leq \sqrt{; ; ; 1+\eta^2 r(m-r)}; ; ; $, where $\eta$ is a tunable parameter, e.g. $\eta=\sqrt{; ; ; m}; ; ; $. - Using the theory of projections, the canonical structure of the DEIM oblique projection is revealed in detail - in a suitably constructed orthonormal basis, its matrix representation is block diagonal with the diagonal blocks of sizes $1\times 1$ and $2\times 2$. This allows closer look at the structure of the error and provides further insights. - Further, DEIM is formulated in a general, weighted inner product, defined by a symmetric positive definite matrix $W$, and the interpolation property is generalized via linear functionals. This puts weighted DEIM ($W$-DEIM) in a more natural framework where the POD Galerkin projection is formulated in a discretization of a suitable energy inner product in which Galerkin projection preserves underlying physical properties. Also, as a special case of $W$-DEIM, we have DGEIM, a discrete version of the Generalized Empirical Interpolation Method (GEIM). - Finally, we reveal interesting connections between the Gappy POD, DEIM and clustering methods.

discrete empirical interpolation, model order reduction

nije evidentirano