[Paper Review] Increasing stability in an inverse problem for the acoustic equation

Sei Nagayasu, Günther Uhlmann+1 more|arXiv (Cornell University)|Oct 24, 2011

Numerical methods in inverse problems9 references4 citations

TL;DR

This paper establishes increasing stability in the inverse problem for the acoustic equation by showing that the stability estimate transitions from logarithmic to Lipschitz-type as the frequency $k$ increases. Using complex geometrical optics (CGO) solutions and refined operator norm estimates, the authors derive a bound where the error in the refractive index $q$ decays as $k^{-2}\exp(Ck^2)\|\Lambda_1 - \Lambda_2\|_*$ (Lipschitz part) and $\left(k^2 + \log \frac{1}{\|\Lambda_1 - \Lambda_2\|_*}\right)^{-(2s-n)}$ (logarithmic part), demonstrating improved stability at high frequencies.

ABSTRACT

In this work we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we increase the frequency and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz estimate for large frequencies.

Motivation & Objective

• To analyze the stability of the inverse boundary value problem for the acoustic equation with respect to the refractive index $q(x)$.
• To investigate how the ill-posedness of the inverse problem changes as the frequency $k$ increases.
• To derive a stability estimate that transitions from logarithmic to Lipschitz-type behavior with increasing $k$.
• To establish a quantitative bound on the difference $\widetilde{q} = q_1 - q_2$ in terms of the DN map difference $\|\Lambda_1 - \Lambda_2\|_*$.

Proposed method

• The authors use complex geometrical optics (CGO) solutions to the acoustic equation $ (\Delta + k^2 q(x))u = 0 $ in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \geq 3$.
• They apply Alessandrini’s argument for stability estimates, tracking the explicit dependence of constants on the frequency $k$.
• The method involves constructing CGO solutions with explicit $k$-dependence and estimating their $H^s$-norms via weighted $L^2$-type estimates.
• A key step is the use of a parameter $T$ in the CGO construction, optimized to balance the Lipschitz and logarithmic terms in the final estimate.
• The proof splits into two cases based on the size of $k^2$ relative to $\log(1/\|\Lambda_1 - \Lambda_2\|_*)$, with separate bounds derived for each.
• The final estimate is obtained by optimizing $T$ and choosing parameters to minimize the error bound, resulting in a hybrid estimate combining Lipschitz and logarithmic decay.

Experimental results

Research questions

• RQ1 Does the stability of the inverse problem for the acoustic equation improve as the frequency $k$ increases?
• RQ2 Can a single stability estimate be derived that captures the transition from logarithmic to Lipschitz-type behavior in the high-frequency regime?
• RQ3 How does the dependence on $k$ manifest in the constants of the stability estimate, particularly in the presence of trapped rays or general $q(x)$?
• RQ4 What role do complex geometrical optics (CGO) solutions play in achieving improved stability at high $k$?

Key findings

• The stability estimate for the inverse problem transitions from logarithmic to Lipschitz-type as $k$ increases, with the logarithmic part decaying and the Lipschitz part dominating at high frequencies.
• The bound on $\|\widetilde{q}\|_{H^{-s}(\mathbb{R}^n)}$ is given by $\frac{C}{k^2}\exp(Ck^2)\|\Lambda_1 - \Lambda_2\|_* + C\left(k^2 + \log\frac{1}{\|\Lambda_1 - \Lambda_2\|_*}\right)^{-(2s-n)}$, where $C$ depends only on $n$, $s$, $\Omega$, $M$, and $\text{supp}(q_1 - q_2)$.
• The Lipschitz term $\frac{C}{k^2}\exp(Ck^2)\|\Lambda_1 - \Lambda_2\|_*$ arises from the use of CGO solutions and the lack of geometric restrictions on $q(x)$, though the exponential factor is expected to be removable under simpler conditions.
• The estimate holds uniformly for all $k^2 \geq 1/(C_1 M)$, without requiring separate analysis for different frequency ranges, unlike previous results.
• The logarithmic part $\left(k^2 + \log\frac{1}{\|\Lambda_1 - \Lambda_2\|_*}\right)^{-(2s-n)}$ decays as $k$ increases, indicating reduced ill-posedness.
• The result confirms numerically observed increasing stability in inverse scattering and extends rigorous stability analysis to the acoustic wave equation with refractive index.