Variants of the Empirical Interpolation Method: symmetric formulation, choice of norms and rectangular extension

TL;DR

This paper introduces variants of the Empirical Interpolation Method (EIM), including symmetric formulation, norm choices, and rectangular extension, enhancing its flexibility and robustness for approximating two-variable functions.

Contribution

It provides an equivalent symmetric definition of EIM, proves its existence and convergence for any error norm, and introduces a method for separated representations with unequal sample sizes.

Findings

Symmetric formulation of EIM established.

Proof of convergence for any error norm.

Method for handling unequal sample sizes in EIM.

Abstract

The Empirical Interpolation Method (EIM) is a greedy procedure that constructs approximate representations of two-variable functions in separated form. In its classical presentation, the two variables play a non-symmetric role. In this work, we give an equivalent definition of the EIM approximation, in which the two variables play symmetric roles. Then, we give a proof for the existence of this approximation, and extend it up to the convergence of the EIM, and for any norm chosen to compute the error in the greedy step. Finally, we introduce a way to compute a separated representation in the case where the number of selected values is different for each variable. In the case of a physical field measured by sensors, this is useful to discard a broken sensor while keeping the information provided by the associated selected field.