  Many inverse problems have to deal with complex, evolving and often not
exactly known geometries, e.g. as domains of forward problems modeled by
partial differential equations. This makes it desirable to use methods which
are robust with respect to perturbed or not well resolved domains, and which
allow for efficient discretizations not resolving any fine detail of those
geometries. For forward problems in partial differential equations methods
based on diffuse interface representations gained strong attention in the last
years, but so far they have not been considered systematically for inverse
problems. In this work we introduce a diffuse domain method as a tool for the
solution of variational inverse problems. As a particular example we study ECG
inversion in further detail. ECG inversion is a linear inverse source problem
with boundary measurements governed by an anisotropic diffusion equation, which
naturally cries for solutions under changing geometries, namely the beating
heart.
  We formulate a regularization strategy using Tikhonov regularization and,
using standard source conditions, we prove convergence rates. A special
property of our approach is that not only operator perturbations are introduced
by the diffuse domain method, but more important we have to deal with
topologies which depend on a parameter $\eps$ in the diffuse domain method,
i.e. we have to deal with $\eps$-dependent forward operators and
$\eps$-dependent norms. In particular the appropriate function spaces for the
unknown and the data depend on $\eps$. This prevents to apply some standard
convergence techniques for inverse problems, in particular interpreting the
perturbations as data errors in the original problem does not yield suitable
results. We consequently develop a novel approach based on saddle-point
problems.
