The solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper, we apply a domain parametrization technique to reduce both the geometrical and computational complexities of the forward problem and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less-expensive reduced-basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems both in the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty. Two inverse problems arising in hemodynamics modeling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall on the basis of pressure measurements; (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.

@ARTICLE{LassilaManzoniQuarteroniRozza2013a,
author = {Lassila, T. and Manzoni, A. and Quarteroni, A. and Rozza, G.},
title = {A reduced computational and geometrical framework for inverse problems
in hemodynamics},
journal = {International Journal for Numerical Methods in Biomedical Engineering},
year = {2013},
volume = {29},
pages = {741--776},
number = {7},
doi = {10.1002/cnm.2559},
}