Data-driven Delay Estimation in Reaction-Diffusion Systems via Exponential Fitting

For a reaction-diffusion equation with unknown right-hand side and non-local measurements subject to unknown constant measurement delay, we consider the nonlinear inverse problem of estimating the associated leading eigenvalues and measurement delay from a finite number of noisy measurements. We propose a reconstruction criterion and, for small enough noise intensity, prove existence and uniqueness of the desired approximation and derive closed-form expressions for the first-order condition numbers, as well as bounds for their asymptotic behavior in a regime when the number of measurements tends to infinity and the inter-sampling interval length is fixed. We perform numerical simulations indicating that the exponential fitting algorithm ESPRIT is first-order optimal, namely, its first-order condition numbers have the same asymptotic behavior as the analytic ones in this regime.