A Variational Data Assimilation Procedure for the Incompressible Navier-Stokes Equations in Hemodynamics

We propose a data assimilation (DA) technique for including noisy measurements of the velocity field into the simulation of the Navier-Stokes equations (NSE) driven by hemodynamics applications. The technique is formulated as an inverse problem where we use a Discretize-then-Optimize approach to minimize the misfit between the recovered velocity field and the data, subject to the incompressible NSE. The DA procedure for this nonlinear problem is a combination of two approaches: the Newton method for the NSE and the DA procedure we designed and tested for the linearized problem. We discuss conditions on the location of velocity measurements that guarantee the well-posedness of the minimization process for the linearized problem. Numerical results, with both noise-free and noisy data, certify the theoretical analysis. Moreover, we consider 2D non-trivial geometries and 3D axisymmetric geometries. Also, we study the impact of noise on non-primitive variables of medical interest.