[Paper Review] Increasing stability in acoustic and elastic inverse source problems

Mozhgan Nora Entrekhabi, Victor Isakov|arXiv (Cornell University)|Aug 30, 2018

Numerical methods in inverse problems16 references3 citations

TL;DR

This paper establishes increasing stability estimates for inverse source problems in acoustic and elastic wave scattering by leveraging multi-frequency data and Fourier analysis in the wave number domain. By applying the Fourier transform, analytic continuation bounds, and exact observability for hyperbolic systems, the authors derive nearly Lipschitz stability bounds that improve with higher wave number intervals, significantly enhancing resolution and robustness in source recovery for the Helmholtz and Lame systems.

ABSTRACT

We study increasing stability in the inverse source problems for the Helmholtz equation and the classical Lame system from (minimal) boundary data at multiple wave numbers. By using the Fourier transform with respect to wave numbers, explicit bounds for analytic continuation of the data to larger wave numbers, the Hyugens' principle, and sharp bounds in the corresponding dynamical initial boundary value problems, increasing (with growing wave numbers interval) stability estimates for source terms are obtained.

Motivation & Objective

• To address the ill-posed nature of inverse source problems for elliptic equations and systems, where standard logarithmic stability limits resolution.
• To improve stability estimates by utilizing boundary measurements across a range of wave numbers, rather than a single frequency.
• To establish nearly Lipschitz stability bounds for the recovery of acoustic and elastic sources from near-field data.
• To extend existing increasing stability results to the more complex elasticity system with multiple wave types (P and S waves).
• To provide a theoretical framework applicable to real-world applications such as biomedical imaging, geophysics, and antenna synthesis.

Proposed method

• Applying the Fourier transform with respect to the wave number to convert the multi-frequency inverse problem into a problem in the frequency domain.
• Using sharp bounds for analytic continuation of solutions to extend stability estimates to higher wave numbers.
• Leveraging Huygens’ principle and exact observability estimates for associated hyperbolic initial boundary value problems.
• Deriving stability estimates via Parseval’s identity and energy estimates in Sobolev spaces for the source terms.
• Employing the Helmholtz decomposition to separate pressure (P) and shear (S) wave components in the elastic system.
• Introducing logarithmic stability terms that diminish as the wave number interval $K$ increases, leading to nearly Lipschitz behavior.

Experimental results

Research questions

• RQ1 Can increasing stability be established for the inverse source problem of the Helmholtz equation using multi-frequency boundary data?
• RQ2 How does the stability of source recovery improve as the wave number interval $K$ increases?
• RQ3 Can similar increasing stability results be extended to the more complex dynamical elasticity system with P and S waves?
• RQ4 What role do Fourier analysis and analytic continuation play in transforming the stability estimates?
• RQ5 How do the derived bounds compare to standard logarithmic stability in the single-frequency case?

Key findings

• For the Helmholtz equation, the stability estimate for $f_0$ and $f_1$ improves with $K$, achieving a nearly Lipschitz bound of the form $Cig(rac{M_1^2}{1+K^{4/3}|E_0|^{1/2}}ig)$, where $E_0 = -\ln \varepsilon_0$.
• The stability bound for the $H^1$-norm of the source terms decays as $K^{-4/3}$, indicating significant improvement with higher wave number ranges.
• For the elastic system, similar increasing stability is proven with bounds of the form $C\big(\frac{M_{3}^2}{1+K^{4/3}|E_e|^{1/2}}\big)$, showing nearly Lipschitz behavior as $K$ increases.
• The logarithmic stability component diminishes with increasing $K$, and the remaining term becomes dominant, leading to improved resolution in numerical reconstructions.
• The results are derived using a combination of Fourier analysis, analytic continuation, and exact observability for hyperbolic systems, providing a robust theoretical framework.
• The method applies to both full boundary data and realistic measurements (e.g., pressure or displacement only), enhancing applicability to real-world inverse problems.