  This work is devoted to the reconstruction of the initial temperature in the
backward heat equation using the space-time finite element method on fully
unstructured space-time simplicial meshes proposed by Steinbach (2015). Such a
severely ill-posed problem is tackled by the standard Tikhonov regularization
method. This leads to a related optimal control for an parabolic equation in
the space-time domain. In this setting, the control is taken as initial
condition, whereas the terminal observation data serve as target. The objective
becomes a standard terminal observation functional combined with the Tikhonov
regularization. The space-time finite element method is applied to the
space-time optimality system that is well-posed for a fixed regularization
parameter.

Numerical Analysis;Numerical Analysis