This project will develop novel techniques for solving inverse problems in life sciences, in particular related to dynamic imaging. Major challenges in this area are efficient four- dimensional image reconstruction under low SNR conditions and further the quantification of image series as obtained from molecular imaging or life microscopy techniques. We will tackle both of them in a rather unified framework as inverse problems for time-dependent (systems of) partial differential equations.In the solution of these inverse problems we will investigate novel approaches for the following aspects specific to the above-mentioned problems in the life sciences:1. Solution of inverse problems for PDEs in complex time-varying geometries2. Development of appropriate variational regularization models for dynamic images, including noise and motion models3. Improved forward and inverse modelling of cellular and intracellular dynamics leading to novel inverse problems for nonlinear partial differential equations4. Construction and implementation of efficient iterative solution methods for the arising 4D inverse problems and their variational formulationAll tasks will be driven by concrete applications in biology and medicine and their success will be evaluated in applications to real problems and data. This is based on interdisciplinary work related to electrocardiology and developmental biology. The overall development of methods will however be carried out in a flexible and modular way, so that they become accessible for larger problem classes.