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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.

Estimation of time of death based on a single measurement of body
core temperature is a standard procedure in forensic medicine.
Mechanistic models using simulation of heat transport promise
higher accuracy than established phenomenological models in
particular in nonstandard situations, but involve many not exactly
known physical parameters. Identifying both time of death and
physical parameters from multiple temperature measurements is
one possibility to reduce the uncertainty significantly.
In this paper, we consider the inverse problem in a Bayesian setting
and perform both local and sampling-based uncertainty
quantification, where proper orthogonal decomposition is used as
model reduction for fast solution of the forward model. Based on
the local uncertainty quantification, optimal design of experiments
is performed in order to minimize the uncertainty in the time of
death estimate for a given number of measurements. For reasons
of practicability, temperature acquisition points are selected from
a set of candidates in different spatial and temporal locations.
Applied to a real corpse model, a significant accuracy improvement
is obtained already with a small number of measurements.