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The Discrete Empirical Interpolation Method: Canonical Structure and Formulation in Weighted Inner Product Spaces (CROSBI ID 288841)

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Drmač, Zlatko ; Saibaba, Arvind Krishna The Discrete Empirical Interpolation Method: Canonical Structure and Formulation in Weighted Inner Product Spaces // SIAM journal on matrix analysis and applications, 39 (2018), 3; 1152-1180. doi: 10.1137/17m1129635

Podaci o odgovornosti

Drmač, Zlatko ; Saibaba, Arvind Krishna

engleski

The Discrete Empirical Interpolation Method: Canonical Structure and Formulation in Weighted Inner Product Spaces

New contributions are offered to the theory and numerical implementation of the discrete empirical interpolation method (DEIM). A substantial tightening of the error bound for the DEIM oblique projection is achieved by index selection via a strong rank revealing QR factorization. This removes the exponential factor in the dimension of the search space from the DEIM projection error and allows sharper a priori error bounds. A well-known canonical structure of pairs of projections is used to reveal canonical structure of DEIM. Further, the DEIM approximation is formulated in weighted inner product defined by a real symmetric positive-definite matrix $W$. The weighted DEIM ($W$-DEIM) can be interpreted as a numerical implementation of the generalized empirical interpolation method (GEIM) and the more general parametrized-background data-weak (PBDW) approach. Also, it can be naturally deployed in the framework when the POD Galerkin projection is formulated in a discretization of a suitable energy (weighted) inner product such that the projection preserves important physical properties, e.g., stability. While the theoretical foundations of weighted POD and the GEIM are available in the more general setting of function spaces, this paper focuses to the gap between sound functional analysis and the core numerical linear algebra. The new proposed algorithms allow different forms of $W$-DEIM for pointwise and generalized interpolation. For the generalized interpolation, our bounds show that the condition number of $W$ does not affect the accuracy, and for pointwise interpolation the condition number of the weight matrix $W$ enters the bound essentially as $\sqrt{; ; \min_{; ; D={; ; diag}; ; }; ; \kappa_2(DWD)}; ; $, where $\kappa_2(W)=\|W\|_2 \|W^{; ; -1}; ; \|_2$ is the spectral condition number.

empirical interpolation, Galerkin projection, generalized empirical interpolation, nonlinear model reduction, oblique projection, proper orthogonal decomposition, parametrized-background data-weak approach, rank revealing QR factorization, weighted inner product

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Podaci o izdanju

39 (3)

2018.

1152-1180

objavljeno

0895-4798

1095-7162

10.1137/17m1129635

Povezanost rada

Matematika

Poveznice
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