A reduced computational and geometrical framework for inverse problems in hemodynamics

SUMMARYThe 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. Copyright © 2013 John Wiley & Sons, Ltd. The solution of inverse problems in cardiovascular mathematics is computationally expensive. In this paper we replace the finite element solution of the incompressible Navier– Stokes equations by a computationally less expensive reduced basis approximation. We then consider the solution of inverse problems in both the deterministic and in the statistical sense. Two inverse problems arising in haemodynamics modelling are considered: (i) a simplified fluid–structure interaction model problem in a portion of a stenosed artery and (ii) a simplified femoral bypass graft model for robust shape design under uncertain residual flow.