Comparison of some Reduced Representation Approximations

TL;DR

This paper compares various reduced representation methods for complex functions, focusing on their similarities, differences, and applications in numerical approximation and PDE simulations.

Contribution

It provides a detailed comparison of Adaptive Cross Approximation and Empirical Interpolation Method alongside other similar approaches.

Findings

ACA and EIM are effective for function approximation.

Different methods have unique strengths depending on problem context.

Crossed analysis enhances understanding of approximation techniques.

Abstract

In the field of numerical approximation, specialists considering highly complex problems have recently proposed various ways to simplify their underlying problems. In this field, depending on the problem they were tackling and the community that are at work, different approaches have been developed with some success and have even gained some maturity, the applications can now be applied to information analysis or for numerical simulation of PDE's. At this point, a crossed analysis and effort for understanding the similarities and the differences between these approaches that found their starting points in different backgrounds is of interest. It is the purpose of this paper to contribute to this effort by comparing some constructive reduced representations of complex functions. We present here in full details the Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM) together with other approaches that enter in the same category.