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Improved stability of low Fourier modes in inverse problems for potentials

This paper establishes Lipschitz, sub-Hölder, and Hölder stability estimates for recovering the low Fourier modes of unknown potentials from Dirichlet-to-Neumann maps across three distinct regularity scenarios, demonstrating that the number of recoverable modes increases as the maps become closer without requiring the potentials to belong to finite-dimensional spaces.

Original authors: Tony Liimatainen, William Trad, Mikko Salo

Published 2026-08-24

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CC BY 4.0

Original authors: Tony Liimatainen, William Trad, Mikko Salo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to understand the shape of a hidden object by only listening to the echoes of sound waves bouncing off its surface. This is the essence of an inverse problem, a common challenge in fields ranging from medical imaging to geology, where scientists must deduce the internal properties of something they cannot see directly. The difficulty lies in the fact that many different internal structures can produce nearly identical echoes, making the solution unstable; a tiny error in measuring the sound can lead to a massive error in the reconstructed image. For decades, mathematicians have known that if the hidden object is very smooth, the reconstruction becomes more reliable, but the relationship between the quality of the measurement and the clarity of the image has remained a complex puzzle, often yielding results that are mathematically possible but practically fragile.

A team of researchers has now made significant progress in untangling this relationship, specifically focusing on how well we can recover the basic, low-frequency features of a hidden material when we have very precise measurements. In their work, they examine a mathematical model where a wave travels through a region containing an unknown substance, or potential, and bounces off the boundaries. By measuring how the wave exits the region, they aim to reconstruct the substance inside. The researchers prove that if the difference between two possible substances is limited to a specific set of simple, low-frequency patterns, the reconstruction becomes remarkably stable. They show that by using waves with very high frequencies, one can recover these simple patterns with a level of precision that improves linearly with the quality of the data, a result that defies the usual expectation that such problems are inherently unstable.

The study explores three distinct scenarios, each revealing a different layer of stability depending on how complex the hidden substance is allowed to be. In the first scenario, the researchers assume the difference between the two substances is "bandlimited," meaning it is composed only of a finite number of simple wave patterns, much like a song composed of only a few specific notes. They demonstrate that if the probing wave has a frequency high enough relative to the number of these notes, the reconstruction is perfectly stable. The error in the recovered image grows at a steady, predictable rate as the measurement error increases, without the sudden, explosive instability that usually plagues these problems. Crucially, this stability holds even if the individual substances being compared are not simple or finite in nature; only the difference between them needs to be simple.

In the second scenario, the researchers relax the assumption that the difference is limited to a finite set of notes. Instead, they consider substances that are "real-analytic," a mathematical way of saying they are incredibly smooth and can be described by a single, continuous formula that extends into the complex plane. For these highly smooth substances, the researchers find that the stability is not quite as strong as in the first case, but it is still far better than the worst-case logarithmic instability that typically dominates the field. They prove that as the measurement error shrinks, the number of low-frequency features that can be reliably recovered grows. The relationship between the error and the recovery is what they call "sub-Hölder," a middle ground that is stronger than the usual logarithmic decay but weaker than a simple linear relationship. This result provides a quantitative link between the precision of the measurement and the resolution of the image, showing that better data allows for the recovery of more details.

The third and most refined scenario looks at substances where the difference between them decays at a "super-exponential" rate, meaning the complex, high-frequency details vanish incredibly fast. In this case, the researchers show that the stability improves further, reaching a "Hölder" level. This means the error in the reconstruction is bounded by a power of the measurement error, a much more favorable outcome. As the measurement error decreases, the number of recoverable features grows, and the stability becomes nearly as strong as the linear stability seen in the first scenario. The key insight across all three cases is that the stability of the recovery depends on the regularity of the difference between the potentials, not on the complexity of the potentials themselves. The individual substances can be arbitrarily complex and infinite in nature, but as long as their difference is sufficiently smooth or simple, the low-frequency features can be recovered with high confidence.

This work challenges the traditional view that inverse problems are inherently ill-posed and unstable. By focusing on the low-frequency modes and leveraging the properties of high-frequency waves, the authors demonstrate that stability is not a binary state but a spectrum that can be tuned by the regularity of the unknown. The constants in their stability estimates remain uniform regardless of how many features are being recovered, which is a significant departure from previous results where the stability constants would explode as the complexity of the unknown increased. The findings suggest that in practical applications, if one can ensure that the unknown variations are smooth or limited in their frequency content, one can achieve reliable and stable reconstructions without needing to restrict the entire problem to a finite, simplified model. The research provides a rigorous mathematical foundation for the idea that high-frequency measurements can unlock stable recovery of low-frequency details, offering a new perspective on how to approach these difficult inverse problems.

Technical Summary: Improved Stability of Low Fourier Modes in Inverse Problems for Potentials

Problem Statement
This work addresses the stability of inverse problems for elliptic equations, specifically focusing on the recovery of an unknown potential $q$ from the Dirichlet-to-Neumann (DN) map $\Lambda_q$ . The central challenge in such inverse problems is the inherent ill-posedness, typically resulting in logarithmic stability estimates that are highly sensitive to measurement noise. The authors investigate whether specific structural features of the potential—namely, its low-frequency Fourier modes—can be recovered with improved stability (Lipschitz, sub-Hölder, or Hölder) under specific regularity assumptions on the difference between two potentials, $Q = q - q'$ .

The study considers three distinct scenarios for the difference $Q$ :

Bandlimited differences: $Q$ is $M$ -bandlimited (Fourier modes vanish for $|k| > M$ ).
Real-analytic differences: $Q$ is real-analytic and periodic.
Super-exponentially decaying differences: The Fourier coefficients of $Q$ decay at a rate $e^{-c|k|^{n/2}}$ .

The governing equations are the Helmholtz equation $(-\Delta - \lambda^2 + q)u = 0$ (for large frequency $\lambda$ ) and the Schrödinger equation $(-\Delta + q)u = 0$ (for $\lambda=0$ ).

Methodology
The proofs rely on the Alessandrini identity, which relates the difference of DN maps to an integral involving the potential difference and solutions to the forward problem:
$\langle (\Lambda_q - \Lambda_{q'})u, u' \rangle_{\partial \Omega} = \int_\Omega (q - q') u \bar{u}' \, dx$
To extract information about the Fourier coefficients of $Q$ , the authors construct specific test functions (solutions $u, u'$ ) that oscillate at high frequencies to isolate specific Fourier modes.

Real Geometrical Optics (RGO): For the bandlimited case (Theorem 1.1), the authors employ real geometrical optics solutions of the form $u = e^{i\eta \cdot x}(1+r)$ , where $|\eta| = \lambda$ . These solutions are used to probe the potential difference at high frequencies. The analysis involves estimating the remainder terms $r$ and utilizing the positive definiteness of a Landau–Pollak–Slepian type operator to invert the relationship between the DN map difference and the Fourier coefficients.
Complex Geometrical Optics (CGO): For the real-analytic and super-exponential decay cases (Theorems 1.2 and 1.3), the authors use CGO solutions with complex phases $\rho, \rho' \in \mathbb{C}^n$ . These solutions allow for the isolation of Fourier modes in the periodic setting $\Omega = (-\pi, \pi)^n$ .
Fourier Decay Estimates: A critical component of the methodology involves bounding the "tail" of the Fourier series (modes $|k| > M$ $∣ k ∣ > M$ ).
- For real-analytic functions, the authors utilize Cauchy estimates on holomorphic extensions to derive exponential decay of Fourier coefficients.
- For super-exponential decay, they assume a stretched-exponential bound $|Q_k| \leq C e^{-c|k|^{n/2}}$ .
- Lemma 3.2 provides estimates for the sum of decaying exponential tails over high-frequency shells, which is essential for controlling the error terms in the stability estimates.

Key Contributions and Results

Lipschitz Stability for Bandlimited Differences (Theorem 1.1):
The authors prove that if the difference $Q = q - q'$ is $M$ -bandlimited, all its Fourier modes can be recovered with Lipschitz stability provided the frequency $\lambda$ is sufficiently large relative to the bandwidth ( $\lambda \gtrsim M^{n/2}$ ).
- Significance: Unlike typical finite-dimensional Lipschitz results where the constant grows exponentially with dimension, here the Lipschitz constant is uniform in $M$ (dependent only on $\Omega, n, E$ ).
- Assumption: Only the difference $Q$ needs to be bandlimited; the individual potentials $q, q'$ need only be bounded in $L^\infty$ .
Sub-Hölder Stability for Real-Analytic Differences (Theorem 1.2):
For differences $Q$ that are real-analytic and periodic, the authors establish sub-Hölder stability. The stability modulus behaves like $\exp(-c(\log(1/\delta))^{2/n})$ , which is better than logarithmic but worse than any Hölder modulus $\delta^\alpha$ .
- Resolution: The number of recoverable Fourier modes, $M(\delta)$ , grows as the measurement error $\delta$ decreases. Specifically, $M(\delta) \sim (\log(1/\delta))^{2/n}$ .
- Assumption: $Q$ extends holomorphically to a complex neighborhood of radius $\epsilon$ .
Hölder Stability for Super-Exponential Decay (Theorem 1.3):
If the Fourier coefficients of $Q$ decay at a super-exponential rate $e^{-c|k|^{n/2}}$ , the stability improves to Hölder stability ( $\delta^\alpha$ ).
- Exponent: The Hölder exponent is $\alpha = c/(\gamma + c)$ , where $c$ is the decay rate and $\gamma$ is a geometric constant. As $c \to \infty$ , $\alpha \to 1$ , approaching Lipschitz stability.
- Resolution: Similar to the analytic case, the number of recoverable modes $M(\delta)$ tends to infinity as $\delta \to 0$ , with $M(\delta) \sim (\log(1/\delta))^{2/n}$ .

Significance and Claims
The paper claims to advance the understanding of "increasing stability" and "increasing resolution" in nonlinear inverse problems.

Relaxation of Assumptions: A primary contribution is that the regularity assumptions (bandlimited, analytic, or super-exponential decay) apply only to the difference $q - q'$ , not the individual potentials. The potentials themselves are only required to be bounded ( $L^\infty$ ). This avoids the restrictive assumption that unknowns must belong to a known finite-dimensional subspace.
Uniform Constants: In the bandlimited case, the stability constant does not grow exponentially with the dimension of the unknown (the number of modes $M$ ), contrasting with standard finite-dimensional inverse problem results.
Quantitative Resolution: The results provide a quantitative relationship between measurement precision ( $\delta$ ) and the resolution (number of recoverable Fourier modes). As measurements become more precise, more low-frequency modes can be stably recovered, transitioning from logarithmic to Hölder/Lipschitz stability regimes depending on the regularity of the difference.
Nonlinearity: The work addresses the nonlinear inverse problem directly, moving beyond the linearized settings (such as singular value estimates in linearized scattering) where similar increasing stability phenomena were previously observed.

The authors note that the constants in the estimates are uniform in the number of recovered modes, and the results hold for general bounded domains (Theorem 2.1) or specifically for the torus/cube setting where Fourier analysis is most natural. The paper does not propose experimental applications but focuses on the theoretical stability estimates for the mathematical inverse problem.

Technical Summary: Improved Stability of Low Fourier Modes in Inverse Problems for Potentials

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