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

## TL;DR

This paper advances the Discrete Empirical Interpolation Method (DEIM) by detailing its canonical structure, tightening error bounds, and extending it to weighted inner product spaces for better physical property preservation.

## Contribution

It provides a canonical characterization of DEIM, improves error bounds using QR factorization, and introduces weighted DEIM ($W$-DEIM) for applications in energy inner products.

## Key findings

- Tighter error bounds for DEIM oblique projection.
- Extension of DEIM to weighted inner product spaces.
- Introduction of DGEIM as a generalized interpolation method.

## Abstract

New contributions are offered to the theory and practice of the Discrete Empirical Interpolation Method (DEIM). These include a detailed characterization of the canonical structure; a substantial tightening of the error bound for the DEIM oblique projection, based on index selection via a strong rank revealing QR factorization; and an extension of the DEIM approximation to weighted inner products defined by a real symmetric positive-definite matrix $W$. The weighted DEIM ($W$-DEIM) can be deployed in the more general framework where the POD Galerkin projection is formulated in a discretization of a suitable energy inner product such that the Galerkin projection preserves important physical properties such as e.g. stability. Also, a special case of $W$-DEIM is introduced, which is DGEIM, a discrete version of the Generalized Empirical Interpolation Method that allows generalization of the interpolation via a dictionary of linear functionals.