On Topologically Controlled Model Reduction for Discrete-Time Systems

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.