A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions

TL;DR

This paper proposes a novel selection operator for the Discrete Empirical Interpolation Method (DEIM) that improves error bounds, extends applicability to large-scale problems, and facilitates efficient software implementation.

Contribution

It introduces a QR-based node selection procedure for DEIM with sharper error bounds and extends the method to large-scale settings using random sampling.

Findings

Sharper error bounds for DEIM projection error.

Effective approximation using only sampled rows of the basis.

Easy implementation leveraging existing linear algebra packages.

Abstract

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 $U \Omega$ with arbitrary unitary matrix $\Omega$. 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.