Inverse Problems and Imaging (IPI)

On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences

Pages: 869 - 898, Volume 10, Issue 4, November 2016      doi:10.3934/ipi.2016025

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Ugur G. Abdulla - Department of Mathematics, Florida Institute of Technology, Melbourne, Florida 32901, United States (email)

Abstract: We consider a variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed 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 consist of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed 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 fixed region instead of full free boundary problem. In Inverse Problems and Imaging, 7, 2(2013), 307-340 we proved well-posedness in Sobolev spaces framework and convergence of time-discretized optimal control problems. In this paper we perform full discretization and prove convergence of the discrete optimal control problems to the original problem both with respect to cost functional and control.

Keywords:  Inverse Stefan problem, optimal control, second order parabolic PDE, Sobolev spaces, energy estimate, embedding theorems, traces of Sobolev functions, method of finite differences, discrete optimal control problem, convergence in functional, convergence in control.
Mathematics Subject Classification:  Primary: 35R30, 35R35, 35K20; Secondary: 65M06, 65M12, 65M32, 65N21, 49J20.

Received: January 2015;      Revised: July 2016;      Available Online: October 2016.