  We develop a mathematical and numerical framework to solve state estimation
problems for applications that present variations in the shape of the spatial
domain. This situation arises typically in a biomedical context where inverse
problems are posed on certain organs or portions of the body which inevitably
involve morphological variations. If one wants to provide fast reconstruction
methods, the algorithms must take into account the geometric variability. We
develop and analyze a method which allows to take this variability into account
without needing any a priori knowledge on a parametrization of the geometrical
variations. For this, we rely on morphometric techniques involving
Multidimensional Scaling, and couple them with reconstruction algorithms that
make use of reduced model spaces pre-computed on a database of geometries. We
prove the potential of the method on a synthetic test problem inspired from the
reconstruction of blood flows and quantities of medical interest with Doppler
ultrasound imaging.
