A New Selection Operator for the Discrete Empirical Interpolation Method---Improved A Priori Error Bound and Extensions

This paper introduces a new framework for constructing the discrete empirical interpolation method (\sf DEIM) projection operator. The interpolation node selection procedure is formulated using the QR factorization with column pivoting, and it enjoys a sharper error bound for the \sf DEIM projection error. Furthermore, for a subspace $\mathcal{U}$ given as the range of an orthonormal ${\mathsf U}$, the \sf DEIM projection does not change if ${\mathsf U}$ is replaced by ${\mathsf U} \Omega$ with arbitrary unitary matrix $\Omega$. In a large-scale setting, the new approach allows modifications that use only randomly sampled rows of ${\mathsf U}$, but with the potential of producing good approximations with corresponding probabilistic error bounds. Another salient feature of the new framework is that robust and efficient software implementation is easily developed, based on readily available high performance linear algebra packages.