In this paper dynamic inverse problems are studied, where the investigated object is allowed to change during the measurement procedure. In order to achieve reasonable results, temporal a priori information will be considered. Here, 'temporal smoothness' is used as a quite general, but for many applications sufficient, a priori information. This is justified in the case of slight movements during an x-ray scan in computerized tomography, or in the field of current density reconstruction, where one wants to conclude from electrical measurements on the surface of the head, the locations of brain activity.

First, the notion of a dynamic inverse problem is introduced, then we describe how temporal smoothness can be incorporated in the regularization of the problem, and finally an efficient solver and some regularization properties of this solver are presented.

This theory will be exploited in three practically relevant applications in a following paper.