Zlatko Drmac

Zlatko Drmac, Serkan Gugercin

This paper introduces a new framework for constructing the Discrete Empirical Interpolation Method 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 DEIM projection error. Furthermore, for a subspace $\mathcal{U}$ given as the range of an orthonormal $U$, the DEIM projection does not change if $U$ is replaced by $U Ω$ with arbitrary unitary matrix $Ω$. In a large-scale setting, the new approach allows modifications that use only randomly sampled rows of $U$, 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.

Zlatko Drmac, Serkan Gugercin, Christopher Beattie

Vector Fitting (VF) is a popular method of constructing rational approximants that provides a least squares fit to frequency response measurements. In an earlier work, we provided an analysis of VF for scalar-valued rational functions and established a connection with optimal $H_2$ approximation. We build on this work and extend the previous framework to include the construction of effective rational approximations to matrix-valued functions, a problem which presents significant challenges that do not appear in the scalar case. Transfer functions associated with multi-input/multi-output (MIMO) dynamical systems typify the class of functions that we consider here. Others have also considered extensions of VF to matrix-valued functions and related numerical implementations are readily available. However to our knowledge, a detailed analysis of numerical issues that arise does not yet exist. We offer such an analysis including critical implementation details here. One important issue that arises for VF on matrix-valued functions that has remained largely unaddressed is the control of the McMillan degree of the resulting rational approximant; the McMillan degree can grow very high in the case of large input/output dimensions. We introduce two new mechanisms for controlling the McMillan degree of the final approximant, one based on alternating least-squares minimization and one based on ancillary system-theoretic reduction methods. Motivated in part by our earlier work on the scalar VF problem as well as by recent innovations for computing optimal $H_2$ approximation, we establish a connection with optimal $H_2$ approximation, and are able to improve significantly the fidelity of VF through numerical quadrature, with virtually no increase in cost or complexity. We provide several numerical examples to support the theoretical discussion and proposed algorithms.