Open Access

Yvonne Freytag

• This thesis is devoted to the interdisciplinary work between mathematicians and forensic experts: the modeling of the human body cooling process after death laying the foundation for the estimation of the time of death. An inverse problem needs to be solved. In this thesis the inverse problem computes the time of death given the measured body temperature and the Forward Model that simulates the body cooling process. The Forward Model is based on the heat equation established by Fourier. This differential equation is numerically solved by the discretization over space by the Finite Element Method and the discretization over time by the Implicit Euler Method. The applications in this thesis demand a fast computation time. A model reduction is achieved by the Proper Orthogonal Decomposition in combination with the Galerkin Method. For reasons of simplification the computations and the measurements are restricted to a cylindrical phantom that is made out of homogeneous polyethylene. The estimate of the time of death is accompanied by an uncertainty. The inverse problem is incorporated by Bayesian inference to interpret the quality of the estimate and the effciency of the experiment. The uncertainty of the estimate of the time of death is minimized by approaching the Optimal Design of the Experiment. An objective function measures the certainty of the data and lays the foundation of the optimization problem. Solving the optimization problem is successfully done by relaxing the complex discrete NP-hard problem and applying a gradient-based method. The results of this thesis clearly show that the design of an experiment has a great in- uence on the outcome of the quality of the estimate. The comparison of the estimate and its properties based on different designs and conditions reveals the effciency of the Design of Experiment in the context of the estimation of the time of death.