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

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.

💡 Research Summary

This paper addresses the notoriously ill‑conditioned problem of recovering the exponents and amplitudes of an exponential sum when the exponents are the first N₁ eigenvalues of a Sturm‑Liouville operator. The authors introduce a novel “structured noise” model: the measurement noise is itself an exponential sum whose exponents are the subsequent eigenvalues (λ_{N₁+1},…,λ_{N₁+N₂}) of the same operator, scaled by a small scalar ε > 0. Under this model the otherwise arbitrary noise becomes highly correlated with the signal, and the authors prove that the recovery problem becomes well‑posed for sufficiently small ε.

The core contributions are:

1. Rigorous well‑posedness analysis – By defining a nonlinear mapping F that measures the discrepancy between the noisy measurements and a candidate reconstruction, the authors introduce first‑order condition numbers K_λ(n) and K_y(n) that quantify how the structured noise propagates into errors of the recovered eigenvalues and amplitudes.

2. Asymptotic decay of condition numbers – Theorem 2 shows that, in three distinct scaling regimes (fixed sampling step Δ with N₁→∞, vanishing Δ with fixed N₁, and fixed total observation time T = N₁Δ), the condition numbers decay either exponentially (in Δ) or super‑exponentially (in N₁). Consequently, the reconstruction error behaves like ε·exp(−c N₁) or ε·exp(−c Δ), i.e., “super‑exponential accuracy”. The analysis holds for any fraction η < 1 of the total parameters; when η = 1 the decay no longer holds, a fact confirmed by numerical experiments.

3. Optimality of Prony’s method – The classical Prony algorithm, which builds a polynomial whose roots are the sampled exponentials ϕ_n = e^{−λ_nΔ}, is revisited. The authors prove that, despite the lack of a minimal separation between the ϕ_n, the structured noise effectively regularizes the Vandermonde matrix involved in Prony’s linear system. The first‑order condition numbers of Prony’s estimates match exactly the analytic K_λ and K_y, demonstrating that Prony attains the optimal first‑order error decay even in the separation‑free regime.

4. Comprehensive numerical validation – Simulations with N₁ ranging from 30 to 80, N₂ = 1, and ε spanning 10^{−4} to 10^{−8} confirm the theoretical predictions. When η = 0.8 the recovered eigenvalues and amplitudes converge with the predicted super‑exponential rate; for η = 1 the reconstruction deteriorates, illustrating the sharpness of the theoretical bounds.

5. Application to inverse problems for parabolic PDEs – The paper shows how the recovered eigenvalues and amplitudes can be employed to identify an unknown potential q(x) in a linear reaction‑diffusion equation ∂_t z = A z, where A is the Sturm‑Liouville operator. By sampling the solution trace at a fixed spatial point, the measurement model reduces exactly to the structured‑noise exponential sum, allowing the authors to reconstruct both the spectrum of A and the modal coefficients of the initial condition. This enables data‑driven control and parameter identification for systems where the operator is only partially known.

6. Contextualization within super‑resolution literature – While most super‑resolution results focus on purely imaginary frequencies (harmonic analysis), this work extends the theory to real, decaying exponentials arising from differential operators. The structured‑noise perspective is novel and bridges a gap between classical signal‑processing techniques and PDE‑based inverse problems.

In summary, the authors demonstrate that by exploiting the intrinsic structure of the noise—specifically, its representation as a tail of the same spectral basis—the classical Prony method regains stability and achieves unprecedented accuracy in regimes previously deemed hopelessly ill‑conditioned. The theoretical framework, explicit condition‑number formulas, and practical algorithmic implementation together provide a powerful tool for spectral identification in Sturm‑Liouville settings and for recovering unknown coefficients in parabolic PDEs from limited, noisy data. Future work may explore multiple tail modes (N₂ > 1), non‑linear PDE extensions, and adaptive sampling strategies to further enhance robustness.