Inverse Problems and Imaging
We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fi xed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the mini- mization of the sum of L2-norm declinations from the available measurement of the temperature flux on the fi xed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fi xed region instead of full free boundary problem. We prove well-posedness in Sobolev spaces framework and convergence of discrete optimal control problems to the original problem both with respect to cost functional and control.
Abdulla, U. G. (2013). On the optimal control of the free boundary problems for the second order parabolic equations. I. well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 7(2), 307-340