Separation-free exponential fitting with structured noise, with applications to inverse problems in parabolic PDEs

Abstract

We investigate the recovery of exponents and amplitudes of an exponential sum, where the exponents $\left\{λ_n \right\}_{n=1}^{N_1}$ are the first $N_1$ eigenvalues of a Sturm-Liouville operator, from finitely many measurements subject to measurement noise. This inverse problem is extremely ill-conditioned when the noise is arbitrary and unstructured. Surprisingly, however, the extreme ill-conditioning exhibited by this problem disappears when considering a \emph{structured} noise term, taken as an exponential sum with exponents given by the subsequent eigenvalues $\left\{λ_n \right\}_{n=N_1+1}^{N_1+N_2}$ of the Sturm-Liouville operator, multiplied by a noise magnitude parameter $\varepsilon>0$. In this case, we rigorously show that the exponents and amplitudes can be recovered with super-exponential accuracy: we both prove the theoretical result and show that it can be achieved numerically by a specific algorithm. By leveraging recent results on the mathematical theory of super-resolution, we show in this paper that the classical Prony's method attains the analytic optimal error decay also in the ``separation-free'' regime where $λ_n \to \infty$ as $n \to \infty$, thereby extending the applicability of Prony's method to new settings. As an application of our theoretical analysis, we show that the approximated eigenvalues obtained by our method can be used to recover an unknown potential in a linear reaction-diffusion equation from discrete solution traces.