Identification of Reaction–Diffusion Systems from Finitely Many Non-Local Noisy Measurements via Exponential Fitting

Given a reaction–diffusion equation with unknown right-hand side, we consider the nonlinear inverse problem of estimating the associated leading eigenvalues and initial condition Fourier coefficients from a finite number of non-local noisy measurements. We define a reconstruction (i.e., estimation) criterion and, for small enough noise, we prove existence and uniqueness of the desired estimates. We derive closed-form expressions for the first-order condition numbers and bounds for their asymptotic behavior in a regime when the number of measured samples is fixed and the inter-sampling interval length is arbitrarily large. When computing the sought estimates numerically, our simulations show that the exponential fitting algorithm ESPRIT is first-order optimal, since its first-order condition numbers have the same asymptotic behavior as the analytic condition numbers in the considered regime.