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

Rami Katz, Dmitry Batenkov, Giulia Giordano

TL;DR

This work shows that recovering exponents and amplitudes in a sum of exponentials with exponents from a Sturm–Liouville spectrum remains highly stable when the measurement noise is structured as a tail consisting of subsequent eigenvalues. By formulating the problem as a nonlinear inverse consisting of a Prony-based parameter extraction and a subsequent inverse Sturm–Liouville step, the authors prove that the recovery errors decay super-exponentially in key regimes and establish that Prony’s method attains analytic optimum in these settings. They provide a rigorous first-order analysis of the condition numbers and demonstrate the preservation of this optimality in practice via numerical experiments, including potential recovery in linear parabolic PDEs from integral and point measurements. The results have practical impact for data-driven control and identification of diffusion-advection processes, offering a robust pathway from finite measurements to operator and potential reconstruction. Overall, the paper extends super-resolution ideas to a separation-free, Laplace-transform–type setting with structured noise, broadening the applicability of Prony-like methods in inverse problems for 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.

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

TL;DR

This work shows that recovering exponents and amplitudes in a sum of exponentials with exponents from a Sturm–Liouville spectrum remains highly stable when the measurement noise is structured as a tail consisting of subsequent eigenvalues. By formulating the problem as a nonlinear inverse consisting of a Prony-based parameter extraction and a subsequent inverse Sturm–Liouville step, the authors prove that the recovery errors decay super-exponentially in key regimes and establish that Prony’s method attains analytic optimum in these settings. They provide a rigorous first-order analysis of the condition numbers and demonstrate the preservation of this optimality in practice via numerical experiments, including potential recovery in linear parabolic PDEs from integral and point measurements. The results have practical impact for data-driven control and identification of diffusion-advection processes, offering a robust pathway from finite measurements to operator and potential reconstruction. Overall, the paper extends super-resolution ideas to a separation-free, Laplace-transform–type setting with structured noise, broadening the applicability of Prony-like methods in inverse problems for PDEs.

Abstract

We investigate the recovery of exponents and amplitudes of an exponential sum, where the exponents are the first 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 of the Sturm-Liouville operator, multiplied by a noise magnitude parameter . 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 as , 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.

Paper Structure

This paper contains 34 sections, 23 theorems, 246 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Problem formulation, standing assumptions and motivation
  4. Manuscript contributions and organization
  5. Exponential fitting as an inverse problem
  6. Well-posedness of the inverse problem and first-order condition numbers
  7. Asymptotic analysis of the first-order condition numbers
  8. Prony's method and its stability
  9. Prony's method
  10. Bounds on the condition numbers for Prony's method
  11. Related work and discussion
  12. Algebraic methods of multi-exponential analysis
  13. Inverse problems for parabolic PDEs
  14. Our earlier work
  15. Resolution theory of the Laplace transform
  16. ...and 19 more sections

Key Result

Proposition 1.1

For the eigenvalues $\left\{\lambda_n \right\}_{n=1}^{\infty}$ in Assumption assump:eigenvalues, there exist constants $0<\upsilon \leq \Upsilon$ such that, for any $1\leq n \leq m$, the following estimate holds:

Figures (5)

  • Figure 1: Comprehensive condition number analysis across the three considered asymptotic regimes. Each row represents a different regime: (1) Fixed $\Delta$, $N_1 \to \infty$; (2) Fixed $N_1$, $\Delta \to \infty$; (3) Fixed $T = N_1\Delta$, $N_1 \to \infty$. Left column shows log-log vs log plots using $\ln(-\ln(\kappa))$ for asymptotic analysis (with slopes shown in legends), right column shows semi-log plots of condition numbers. The fitted slopes validate that Prony's method achieves the analytic decay rates predicted by Theorems \ref{['Thm:FirstOrdCond']}, \ref{['thm:unifbdd11']}, and \ref{['thm:PronyAmplitudeCondition']}. Additionally, we include results for $\eta=1$ (full mode recovery) shown only on the right column, confirming the $O(1)$ scaling of condition numbers for eigenvalues $\lambda_n$ and amplitudes $y_n$ as discussed in the paragraph following Theorem \ref{['Thm:FirstOrdCond']}.
  • Figure 2: Comparison of condition number decay for different eigenvalue decay rates.
  • Figure 3: Potential recovery using Algorithms \ref{['alg:prony_eigenvalues']} and \ref{['alg:inverse_potential']}. Both experiments demonstrate accurate recovery of eigenvalues and potential coefficients, with the triangular potential case showing robustness to longer Fourier tails.
  • Figure 4: Triangle potential recovery using point measurements at $x_0 = 0.45$. The descriptions are the same as in fig:potential_random, top row.
  • Figure 5: Convergence study for triangle potential recovery using point measurements. The left panel shows the true triangle potential (black solid line) alongside the recovered potentials for different values of $m$ (colored dashed lines). The right panel displays the convergence of three error metrics: relative error in the recovered eigenvalues (blue circles), absolute error in the recovered Fourier coefficients (red squares), $L^2$ error in the reconstructed potential function (green triangles).

Theorems & Definitions (56)

  • Proposition 1.1
  • proof
  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • ...and 46 more