Increasing stability in an inverse problem for the acoustic equation

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.

Increasing Stability for the Conductivity and Attenuation Coefficients

“…The study of the increasing stability phenomenon has attracted a lot of attention recently. There are several results [7], [8], [9] and [12] which rigorously demonstrated the increasing stability behaviours in different settings.…”

Inverse Boundary Value Problem For The Helmholtz Equation: Quantitative Conditional Lipschitz Stability Estimates

“…Unfortunately, the use of CGO's solutions leads naturally to a dependence of the stability constant on frequency of exponential type. This is clearly far from being optimal as it is also pointed out in the paper of Nagayasu, Uhlmann and Wang [18]. There the authors prove a stability estimate, in terms of Cauchy data instead of the Dirichlet-to-Neumann map using CGO solutions.…”

Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions

“…However, there is no efficient stability increasing with respect to increasing coefficient regularity in these results of [15]. An additional study, motivated by [15,29], was given in [22].…”