Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 467

Tabl e 6. 6 Dimension of the fluid velocity and membrane
displacement POD basis for different values of the POD
threshold for the idealized aortic arch test case

0.9 0.95 0.99

N u 5 8 22

N 5 7 12

2e+06 2.5e+06

1.8e+06

2e+06

1.6e+06

1.4e+06 1.5e+06

1.2e+06

1e+06

1e+06

800000 500000

0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06

Time [s] Time [s]

Fig. 6.14 History of the Young modulus estimates for SNR D 10.Onthe left , the estimates
obtained by solving the inverse problem on the full finite element space. On the right , the estimates
obtained by first projecting the problem on the reduced space. The colors refer to the different
components of the vector E (E 1 ( blue ), E 2 ( red ), E 3 ( green ))

Tabl e 6. 7 Comparison
between Full Space (FS) and
Reduced Space (RS)
performance for the idealized
aortic arch test case

FS RS

E [1.33, 1.84, 1.31] 10 6 [1.34, 1.80, 1.33] 10 6

Rel. error 1.91 % 2.01 %

Exec. time 3,176 s 277 s

NS solves 492 s 480 s

Tabl e 6. 8 Time average of the estimates and relative error for different values of the POD
threshold for the idealized arch test case
=0.9 =0.95 =0.99

E [1.37, 1.81, 1.32] 10 6 [1.34, 1.80, 1.32] 10 6 [1.30, 1.78, 1.29] 10 6

Rel. error 2.83 % 1.97 % 0.87 %

In this test we compare the reduced space approach with the full space approach
(i.e., the minimization in the full Finite Element space). The history of the estimates
at each time step for the case SNR D 10 and D 0:95 isshowninFig. 6.14 , while
in Table 6.7 we report their performance. The optimal estimate for the Young's
modulus is computed by averaging all but the first 10 time steps estimates, which
are clearly significantly affected by the initial guess.
We can see in addition that the reduced space approach estimates are as good as
the full space approach. Moreover, the error on the estimates is remarkably smaller
than the intensity of the noise in the measures, for both the approaches, dropping
from 10 % to about 2 %, that shows also how DA filters the noise in the measures.
Regarding the behavior of the estimates with respect to the POD threshold, in
Tab le 6.8 we report the time averages (excluding the first ten time steps) and the
corresponding relative error for three different POD thresholds.