Abstract
In this paper, we consider the conductivity problem with piecewise‐constant conductivity and Robin‐type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non‐monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn–Vogelius‐type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first‐order and second‐order shape derivative of the proposed cost functional, which is performed rigorously by means of shape‐Lagrangian formulation without using the shape sensitivity of the states variables. © 2016 The Author. Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.
Highlights
The problem of reconstructing the jump of conductivity is a classical inverse problem
The authors provide an algorithm for the recovery of the Robin parameter and the jump set of the conductivity, either independently or simultaneously
The stability analysis require the computation of the first-order and second-order shape derivative of the proposed shape functional
Summary
The problem of reconstructing the jump of conductivity is a classical inverse problem. The stability analysis require the computation of the first-order and second-order shape derivative of the proposed shape functional. For the computation of the shape gradient, we use Lagrangian method combined with the use of theorem on the derivative of a MinMax with respect to a parameter. Such method is well known and extensively used in mechanical sciences, mathematical programming, and optimal control theory. For the computation of the shape Hessian, we follow the method given in [11] to differentiable semiconvex cost functionals This methods have the advantage of providing the first-order and second-order shape derivative without the need to compute the material derivative of the partial differential equations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
