Refined instability estimates for some inverse problems

Abstract

Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache [29]. In this work, based on Mandache's idea, we refine the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. Our aim is to derive explicitly the dependence of the instability estimates on key parameters.

The first result of this work is to show how the instability depends on the depth of the hidden inclusion and the conductivity of the background medium. This work can be regarded as a counterpart of the depth-dependent and conductivity-dependent stability estimate proved by Li, Wang, and Wang [28], or pure dependent stability estimate proved by Nagayasu, Uhlmann, and Wang [31]. We rigorously justify the intuition that the exponential instability becomes worse as the inclusion is hidden deeper inside a conductor or the conductivity is larger.

The second result is to justify the optimality of increasing stability in determining the near-field of a radiating solution of the Helmholtz equation from the far-field pattern. Isakov [16] showed that the stability of this inverse problem increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a Hölder type. We prove in this work that the instability changes from an exponential type to a Hölder type as the frequency increases. This result is inspired by our recent work [25].