[논문 리뷰] Hölder-Logarithmic Stability and Convergence Rates for an Inverse Random Source Problem

Philipp Ronald Mickan, Thorsten Hohage|arXiv (Cornell University)|2026. 02. 24.

Numerical methods in inverse problems인용 수 0

한 줄 요약

The paper derives Hölder-logarithmic stability estimates and convergence rates for recovering the strength of a random, uncorrelated acoustic source from covariance measurements of emitted time-harmonic waves, using complex geometrical optics and variational source conditions.

ABSTRACT

In this paper, we investigate an inverse random source problem concerned with recovering the strength of a random, uncorrelated acoustic source from correlation measurements of emitted time-harmonic acoustic waves. Such problems arise in applications including aeroacoustics and seismic imaging. Unlike their deterministic counterparts, inverse random source problems are known to be uniquely solvable in the absence of noise. Nevertheless, due to their inherent ill-posedness, regularization is required to stably reconstruct the source strength. We derive conditional Hölder-logarithmic stability estimates under Sobolev smoothness assumptions by employing complex geometrical optics solutions. Moreover, by establishing a variational source condition, we obtain Hölder-logarithmic convergence rates for spectral regularization methods. At fixed frequency, the exponents in the logarithmic stability and convergence estimates grow unboundedly as the Sobolev regularity of the source increases. Finally, we present numerical experiments supporting our theoretical findings.

연구 동기 및 목표

• Motivate and formulate the inverse random source problem for the Helmholtz equation in a bounded domain.
• Derive conditional Hölder-logarithmic stability estimates under Sobolev smoothness assumptions for the source strength.
• Obtain Hölder-logarithmic convergence rates for spectral regularization methods under variational source conditions.
• Extend the framework to convected Helmholtz equations relevant in aeroacoustics.
• Provide numerical experiments to support the theoretical results.

제안 방법

• Model the random source Q as an uncorrelated centered Gaussian process with covariance as a multiplication operator M_q by q in L^∞(D).
• Express the measured data as the sample covariance on the measurement surface M, relating it to the forward map C: q ↦ G M_q G*, with G the volume potential operator using the Helmholtz Green function G(x,y).
• Use complex geometrical optics solutions to bound Fourier coefficients of the source strength in terms of covariance data.
• Establish variational source conditions to connect conditional stability with convergence rates for spectral regularization methods.
• Prove a Hölder-logarithmic conditional stability estimate and a corresponding convergence rate result for q̂αδ under noisy data, via a variational inequality framework.
• Discuss extensions to convected Helmholtz equations and present supporting numerical experiments.

(a) $q^{\dagger}$ , multivar. linear spline

실험 결과

연구 질문

• RQ1 Can the covariance data of the random acoustic field uniquely determine the source strength q within the assumed model?
• RQ2 What are the stability and convergence rates for recovering q from noisy covariance data using spectral regularization methods under Sobolev regularity assumptions?
• RQ3 How do Hölder-type and logarithmic stability behaviors manifest for fixed frequency and varying wave number?
• RQ4 Do complex geometrical optics solutions yield explicit bounds for low-frequency Fourier coefficients of q and hence lead to variational source conditions?
• RQ5 What is the impact of extending the framework to convected Helmholtz equations in aeroacoustic settings?

주요 결과

• There exists a Hölder-logarithmic conditional stability estimate for q in H^m(D) when q ∈ H^s(D) with m < s, expressed via a specific index function φκ.
• A variational source condition with an explicit index function Ψ leads to convergence rates for spectral regularization methods, quantified by φκ(δ) as δ → 0.
• The convergence rate bound for q̂αδ in H^m is given by ||q̂αδ − q†||_{H^m} ≤ C'' φκ(δ).
• Low-frequency Fourier coefficients of the difference between two sources can be bounded in terms of the Hilbert–Schmidt norm of the covariance data difference, enabling uniqueness in the considered setting.
• The analysis includes a construction of CGOS-based bounds and a two-step verification of generalized smoothness and ill-posedness in the variational source condition framework.
• Numerical experiments corroborate the theoretical Hölder-logarithmic convergence rates.

(b) $N=550$