Refined instability estimates for some inverse problems

Many inverse problems are known to be ill-posed. The ill-posedness can manifested by an instability estimate of exponential type, first derived Mandache [29]. In this work, based on Mandache's idea, we refine the estimates for two problems, including inclusion problem and scattering problem. Our aim is derive explicitly dependence key parameters.The result work show how depends depth hidden conductivity background medium. This regarded as a counterpart depth-dependent conductivity-dependent stability proved Li, Wang, Wang [28], or pure dependent Nagayasu, Uhlmann, [31]. We rigorously justify intuition that becomes worse deeper inside conductor larger.The second optimality increasing in determining near-field radiating solution Helmholtz equation from far-field pattern. Isakov [16] showed increases frequency sense changes logarithmic type Hölder type. prove increases. inspired our recent [25].