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This paper presents computational analysis of the inverse Stefan type freeboundary problem, where information on the boundary heat flux is missing andmust be found along with the temperature and the free boundary. We pursueoptimal control framework introduced in {\it U.G. Abdulla, Inverse Problems andImaging, 7, 2(2013), 307-340;\ 10, 4(2016), 869--898}, where boundary heat fluxand free boundary are components of the control vector, and optimality criteriaconsist of the minimization of the quadratic declinations from the availablemeasurements of the temperature distribution at the final moment, phasetransition temperature on the free boundary, and the final position of the freeboundary. We develop gradient descent algorithm based on Frechetdifferentiability in Hilbert-Besov spaces complemented with preconditioning orincrease of regularity of the Frechet gradient through implementation of theRiesz representation theorem. Three model examples with various levels ofcomplexity are considered. Extensive comparative analysis throughimplementation of preconditioning and Tikhonov regularization, calibration ofpreconditioning and regularization parameters, effect of noisy data, comparisonof simultaneous identification of control parameters vs. nested optimization ispursued.

Gradient Method in Hilbert-Besov Spaces for the Optimal Control of Parabolic Free Boundary Problems