# On Topologically Controlled Model Reduction for Discrete-Time Systems

## TL;DR

This paper explores topologically controlled model reduction for discrete-time systems, linking data-driven approximation problems to constrained matrix representations within circulant matrices, and discusses algorithms and numerical results.

## Contribution

It introduces a novel approach to model reduction by reducing problems to matrix representations in circulant algebra, with algorithms and numerical experiments.

## Key findings

- Reduction of data-driven problems to circulant matrix representations
- Development of algorithms for computing constrained matrix representations
- Numerical implementations demonstrating the approach

## Abstract

In this document the author proves that several problems in data-driven numerical approximation of dynamical systems in $\mathbb{C}^n$, can be reduced to the computation of a family of constrained matrix representations of elements of the group algebra $\mathbb{C}[\mathbb{Z}/m]$ in $\mathbb{C}^{n\times n}$, factoring through the commutative algebra $Circ(m)$ of circulant matrices in $\mathbb{C}^{m\times m}$, for some integers $m\leq n$. The solvability of the previously described matrix representation problems is studied. Some connections of the aforementioned results, with numerical analysis of dynamical systems, are outlined, a prorotypical algorithm for the computation of the matrix representations, and some numerical implementations of the algorithm, will be presented.