Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 445

Remark 6.7. Here we introduced the POD using the eigenvalue decomposition of
the sample covariance matrix. Alternatively, one can perform the Singular Value
Decomposition (SVD) of the snapshots matrix X D Π1 ;:::; M . This can be
efficiently done by first performing a QR factorization of X and then by computing
the SVD of the triangular factor. In other words, X D QR D QU†V T

D U†V T .
The POD basis is then made of the first M left singular vectors (the columns of U ),
where M is chosen with the same procedure as before, using the singular values of
the snapshots matrix rather than the eigenvalues of the covariance matrix.

6.4 Some Applications of Data Assimilation in

Hemodynamics Problems

In this section we consider some applications of DA and Parameter Estimation in
computational hemodynamics.
First, we present the problem of reconstructing the blood flow in a vessel
assimilating sparse noisy measures of the velocity with the numerical results
obtained by solving the incompressible Navier-Stokes equations. Successively, we
consider the problem of estimating the compliance of a vessel based on measures
of the displacement retrieved from medical images. The solution to this problem
leads to an inverse fluid-structure interaction (IFSI) problem. These are not the only
examples of data assimilation in biomedical applications. We mention, for instance,
the work in electrocardiology for the setup of patient-specific models in [ 19 ], and
for estimating cardiac conductivities [ 32 , 37 , 74 ]. Other applications can be found,
e.g., in [ 17 , 29 ]. In particular, in [ 24 , 26 ] DA methods are advocated for filling the
gap between available boundary data and mathematical conditions required to solve
the problem.
We have selected these examples because they offer the opportunity to see in
action different methodologies based on the techniques illustrated in the previous
sections.

6.4.1 Assimilation of Velocity Measures in Blood Flow

Simulations

We consider the problem of merging velocity measures and the numerical simula-
tion of blood flow. The DA problem can be addressed in several and diverse ways,
as described in the previous sections. More precisely, we present two approaches
introduced in two recent papers; in the first one [ 18 ] the problem is faced with a
variational (control) method, where the control variable is the normal component of
the stress at the inflow section of the vessel. In the second paper [ 40 ] the authors
exploit a Least Squares Finite Element (LSFE) approximation treating internal