Biomedical Engineering Reference
In-Depth Information
rely on the Newton method for the NSE. The distribution p
H
j
M
for the nonlinear
model is still Gaussian, the following algorithm is used to determine its mean and
covariance.
Given a guess for the random vector
U
k
D Z
k
H
MAP;k
at iteration
k C 1
,
.1/
compute
ƒ
H
j
M;k
C
1
D ƒ
H
C Z
k
ƒ
1
Z
k
.2/
solve
ƒ
H
j
M;k
C
1
.
H
MAP;k
C
1
/ D Z
T
ƒ
1
.
d
0
/ C ƒ
H
h
0
;
(6.63)
until a convergence criterion is satisfied.
Here, for Z
k
D DS
k
R
in
M
in
we define
C C A
k
B
T
B
:
S
k
D
(6.64)
O
A
k
is the discretization of the advection operator with advection field
U
k
,the
velocity vector associated with the normal stress
.H/
k
. Note that with this
formulation
H
and
U
, at each iteration, are related by a linear model and, for this
reason,
U
can still be considered normally distributed.
Numerical tests.
We assume to have an exact, analytic solution of the NSE and
we compare the accuracy of the MAP and ML estimators vs. the “deterministic
estimator” introduced in the previous section, i.e., the solution of the variational
formulation. The index of accuracy is related to the velocity fields,
E
U
, retrieved from
k
U
U
anl
k
2
H
MAP
,
H
ML
and
H
det
(the deterministic estimate); it is defined as E.
U
/ D
k
U
anl
k
2
,
where
U
anl
is the discretized analytic
so
lution. We also define an average error
over a set of noise realizations fg
n
P
i
D
1
E.
U
;i/where E.
U
;i/is
associated with the i-th realization of noise
i
. In addition, we consider a measure
of the ga
in
, , in using statistical estimators as opposed to deterministic ones:
D 1
i
D
1
, E.
U
/ D
1
E.
U
stat
/
E.
U
det
/
where “stat” stands for either MAP or ML.
The details of the numerical tests are fully reported in [
16
].
In a square domain we consider data on
in
and internal data located on ten
internal slices. In Table
6.2
we report results obtained in correspondence of SNR
of 20 and 10. In the computation of
H
MAP
and
H
det
the regularization parameter
Ǜ = 0.5 is chosen empirically (left table in Table
6.2
). In the computation of
H
ML
and
H
det
on the right table the regularization parameter Ǜ is set to 0.
From the results we infer the following facts. (1) Compared to the deterministic
estimator, the statistical estimators are always more accurate since they take into
account additional information brought by statistical properties of the data. (2) The
computational time required in solving the statistical formulations is, in average,
1.3 times bigger than the one required by the deterministic one. (3) The poor gain
in correspondence of SNR D 20 means that statistical information associated with
a low amount of noise is not significant enough to make a considerable difference
with respect to deterministic estimates in terms of accuracy.
As a second example we consider the same problem setting of the previous
section for the flow in a cylinder, see Fig.
6.8
(right); we consider measures on the
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