J Sci Comput (2012) 52:340–359
DOI 10.1007/s10915-011-9547-6
A Variational Data Assimilation Procedure
for the Incompressible Navier-Stokes Equations
in Hemodynamics
Marta D’Elia · Mauro Perego · Alessandro Veneziani
Received: 28 April 2011 / Accepted: 12 October 2011 / Published online: 6 November 2011
© Springer Science+Business Media, LLC 2011
Abstract 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 ve-
locity 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 minimiza-
tion 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.
Keywords Computational fluid dynamics · Optimization · Inverse problems · Data
assimilation · Hemodynamics
1 Introduction
Numerical methods for incompressible fluid dynamics have recently received strong im-
pulse from applications to the cardiovascular system (see e.g. [8, 9, 28]). A strong integra-
tion of data and numerical modeling is expected to bring a great benefit to the development
of mathematical and computational tools with a clinical impact. The introduction and im-
provement of measurement and imaging devices enhance the integration process and opens
new challenges; as an example Fig. 1 reports an MRI [6] of the ascending aorta where blood
M. D’Elia ( ) · A. Veneziani
Dept. of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta,
GA 30322, USA
e-mail: mdelia2@mathcs.emory.edu
M. Perego
Dept. of Scientific Computing, Florida State University, Tallahassee, FL, USA