Biomedical Engineering Reference
In-Depth Information
Fig. 12.2.
Singular values of the reduced Hessian for a non regularized case (
a
=
0): on the left,
the condition (
) is fulfilled, on the right it is violated
On the contrary, no constraints on sites locations need to be fulfilled when the
Tikhonov regularization is active (
a
0). However, in practice, the selection of
a
requires to find the proper trade-off between the requirement to solve a well condi-
tioned problem (large
a
) and to keep the perturbation of the original problem as small
as possible (small
a
). A possible approach (see [19, 57]) is to select the parameter
according to the discrepancy principle (DP), i.e. to select
a
in such a way that the
perturbation of the regularization term affects the solution with the same order of
the discrepancy induced by the noise. The proper choice of the parameter following
this approach may be however computationally expensive. There is another possible
way for forcing the well-posedness exploiting the result of Proposition 1. Actually,
let us assume that some data are available at the inflow, not necessarily fulfilling the
well-posedness sufficient condition
>
(
)
. If we extend the given data to the entire set
of DOF of
Γ
in
by interpolation of the available data (e.g. piecewise linear), the result-
ing problem satisfies condition
(
)
. This results in fact in an additional term to the
functional
that plays the role of a regularizing term (see [58]). A more extensive
analysis of this approach, and the interplay between the interpolation and the noise
affecting the original data is currently under investigation.
J
12.2.1.2 The nonlinear Navier-Stokes problem
When we consider the nonlinear advection term
u
the problem becomes much
more difficult since now we have a nonlinear constraint [6]. A possible approach is
to combine the DA procedure for the linear case with classical fixed point lineariza-
tion schemes (i.e. Picard and Newton). Thus, the DA assimilation problem is solved
iteratively. We report the simple case of the Picard method. Given a guess for the
velocity at step
k
,say
U
k
, we solve
(
u
·
∇
)
1
2
a
2
C
(12.6)
U
m
2
2
2
2
min
H
k
+
1
D
V
k
+
1
(
H
k
+
1
)
−
+
L
H
k
+
1
A
U
k
B
T
+
where S
=
B
O
R
in
M
in
H
k
+
1
+
s.t.
S
k
V
k
+
1
=
F
