  Partial differential equation (PDE) models are widely used in engineering and
natural sciences to describe spatio-temporal processes. The parameters of the
considered processes are often unknown and have to be estimated from
experimental data. Due to partial observations and measurement noise, these
parameter estimates are subject to uncertainty. This uncertainty can be
assessed using profile likelihoods, a reliable but computationally intensive
approach. In this paper, we introduce an integration based approach for the
profile likelihood calculation for inverse problems with PDE constraints. While
existing approaches rely on repeated optimization, the proposed approach
exploits a dynamical system evolving along the likelihood profile. We derive
the dynamical system for the reduced and the full estimation problem and study
its properties. To evaluate the proposed method, we compare it with
state-of-the-art algorithms for a simple reaction-diffusion model for a
cellular patterning process. We observe a good accuracy of the method as well
as a significant speed up as compared to established methods. Integration based
profile calculation facilitates rigorous uncertainty analysis for
computationally demanding parameter estimation problems with PDE constraints.
