Biomedical Engineering Reference
In-Depth Information
Fig. 12.4.
Test of consistency for the DA procedure of velocity data: accuracy of the DA improves
when the number of sites
N
s
(left) or of the noise realization
N
r
(right) increases
vergence rate for the
L
2
norm of the velocity error,
E
U
, when the mesh size
x
tends
to 0. In case of noisy data (Gaussian white noise), the error dynamics changes. In
particular, the error decreases as more data are available (see Fig. 12.4, left). We ob-
serve that the convergence rate with respect to the number of sites
N
s
is of the order
of 1
Δ
/
√
N
s
. Moreover, when the number of sites and the mesh are fixed and we repeat
the DA with different noise realizations, we observe a progressive convergence of
the sample mean of the assimilated solution to the noise free solution, with a rate
proportional to 1
/
√
N
r
, being
N
r
the number or realizations. (see Fig. 12.4, right).
Note that these results are consistent with the central limit theorem. More details can
be found in [58].
12.2.2.2 Regularization & interpolation
We compare the results obtained with the regularized problem (parameter
a
tuned
again with the DP) and the interpolation of inflow data when these do not fulfill
condition
of Proposition 1. In Table 12.1 (left) in correspondence of different
choices of locations and number
N
in
of sites on
(
)
Γ
in
, we report the relative error and
the number of iterations (
it
) for solving the reduced Hessian. The case where neither
Tikhonov regularization nor interpolation are performed is reported as a reference
test. The addition of interpolated data on the DOF on
in
has the effect of forcing
the well posedness of the problem, as can be inferred from the singular values of
the reduced Hessian in the case reported in Fig. 12.2. Also, results reported in Ta-
ble 12.1 show that, in terms of accuracy, the interpolation procedure is comparable
with Tikhonov regularization. This fact, combined with the computational saving
associated with the generation of the interpolating function, as opposed to applying
the DP, makes interpolation an efficient regularization technique, competitive with
common available methods.
Γ
