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.

翻译：这项工作致力于利用Steinbach(2015年)建议的完全无结构的时空简易模件的时空定点元素法,重建后热方程式初始温度。这种严重的问题由标准的Tikhonov定点法解决。这导致对时空域的抛物线方程式进行相关的最佳控制。在这种环境下,控制被视为初始条件,而终端观测数据则作为目标。目标成为标准终端观测功能,与Tikhonov定点相结合。时空定点元素法适用于适合固定定点参数的时空定点优化系统。