  We investigate the inverse problem of numerically identifying unknown initial
temperatures in a heat equation with dynamic boundary conditions whenever some
overdetermination data is provided after a final time. This is a backward
parabolic problem which is severely ill-posed. As a first step, the problem is
reformulated as an optimization problem with an associated cost functional.
Using the weak solution approach, an explicit formula for the Fr\'echet
gradient of the cost functional is derived from the corresponding sensitivity
and adjoint problems. Then the Lipschitz continuity of the gradient is proved.
Next, further spectral properties of the input-output operator are established.
Finally, the numerical results for noisy measured data are performed using the
regularization framework and the conjugate gradient method. We consider both
one- and two-dimensional numerical experiments using finite difference
discretization to illustrate the efficiency of the designed algorithm. Aside
from dealing with a time derivative on the boundary, the presence of a boundary
diffusion makes the analysis more complicated. This issue is handled in the 2-D
case by considering the polar coordinate system. The presented method implies
fast numerical results.
