# The Natural Greedy Algorithm for reduced bases in Banach spaces

## TL;DR

The paper introduces the Natural Greedy Algorithm (NGA), a new reduced basis method for Banach spaces that offers similar theoretical performance to existing algorithms like OGA but with significantly improved computational efficiency.

## Contribution

The NGA provides a novel, computationally efficient approach for constructing reduced bases in Banach spaces, extending the OGA and integrating the empirical interpolation method as a special case.

## Key findings

- NGA achieves similar theoretical accuracy to OGA.
- NGA significantly reduces computational effort.
- Numerical examples demonstrate NGA's advantages over other methods.

## Abstract

In this effort we introduce and analyze a novel reduced basis approach, used to construct an approximating subspace for a given set of data. Our technique, which we call the Natural Greedy Algorithm (NGA), is based on a recursive approach for iteratively constructing such subspaces, and coincides with the standard, and the extensively studied, Orthogonal Greedy Algorithm (OGA) in a Hilbert space. However, for a given set of data in a general Banach space, the NGA is straightforward to implement and overcomes the explosion in computational effort introduced by the OGA, as we utilize an entirely new technique for projecting onto appropriate subspaces. We provide a rigorous analysis of the NGA, and demonstrate that it's theoretical performance is similar to the OGA, while the realization of the former results in significant computational savings through a substantially improved numerical procedure. Furthermore, we show that the empirical interpolation method (EIM) can be viewed as a special case of the NGA. Finally, several numerical examples are used to illustrate the advantages of our NGA compared with other greedy algorithms and additional popular reduced bases methods, including EIM and proper orthogonal decomposition.