Krylov Subspace Methods in Dynamical Sampling

TL;DR

This paper investigates how to recover unknown linear operators and initial states in dynamical systems using Krylov subspace methods, extending classical techniques like Prony's method to more general operators and sampling scenarios.

Contribution

It introduces a framework for recovering operators and signals from partial measurements, including low-pass convolution operators and general operators, generalizing classical methods.

Findings

Recovery of both operator and initial state for low-pass convolution operators.

Partial or complete spectral recovery for general operators.

Extension of Prony's method to broader classes of signals and operators.

Abstract

Let $B$ be an unknown linear evolution process on $\mathbb C^d\simeq l^2(\mathbb Z_d)$ driving an unknown initial state $x$ and producing the states $\{B^\ell x, \ell = 0,1,\ldots\}$ at different time levels. The problem under consideration in this paper is to find as much information as possible about $B$ and $x$ from the measurements $Y=\{x(i)$, $Bx(i)$, $\dots$, $B^{\ell_i}x(i): i \in \Omega\subset \mathbb Z^d\}$. If $B$ is a "low-pass" convolution operator, we show that we can recover both $B$ and $x$, almost surely, as long as we double the amount of temporal samples needed in \cite{ADK13} to recover the signal propagated by a known operator $B$. For a general operator $B$, we can recover parts or even all of its spectrum from $Y$. As a special case of our method, we derive the centuries old Prony's method \cite{BDVMC08, P795, PP13} which recovers a vector with an $s$-sparse Fourier transform from $2s$ of its consecutive components.