Abstract

We develop a mathematical and numerical framework to solve state estimation prob-lems 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 recon-struction methods, the algorithms must take into account the geometric variability. We develop and analyze a method which allows us 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 linear subspaces precomputed on a database of geometries. We prove the potential of the method on a synthetic test problem inspired by the reconstruction of blood flows and quantities of medical interest with Doppler ultrasound imaging.