Gradient Flow Dynamical System for Solution of Shape Optimization Problems

The convergent method of shape optimization is proposed for a class of inverse problems. The form of the Gradient Flow Dynamical System is obtained for the inverse problems with the Kohn-Vogelius shape functional. The interface between two materials with different properties is identified from the boundary observations. In the case of the deterministic data, the convergence of the proposed method at the continuous level is recently established by Pavel Plotnikov and Jan Sokolowski. In the case of random data, the convergence result is still to be shown; however, our numerical results confirm the convergence. The convergence of such a method in the case of stochastic PDEs is an open and difficult problem, it seems.