Optimal control of coefficients in parabolic free boundary problems modeling laser ablation

Inverse Stefan type free boundary problem for the second order parabolic equation arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. New PDE constrained optimal control framework in Hilbert-Besov spaces introduced in Abdulla (2013) is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Discretization by finite differences is pursued, and convergence of the discrete optimal control problems to the original problem is proved. Numerical algorithm based on the Frechet differentiability and the gradient method in Hilbert-Besov spaces is implemented. The results of the numerical experiments for the identification of the free boundary and diffusion coefficient are presented. (C) 2020 Elsevier B.V. All rights reserved.