Biomedical Engineering Reference
In-Depth Information
12.2.1.1 The discrete DA Oseen problem
Let us consider preliminarily the linear Oseen problem. The nonlinear convection
term
(
β
·
∇
)
β
(
u
·
∇
)
u
is replaced with
u
, where
is a known advection field. The
discretized optimization problem reads
U
P
1
2
a
2
V
=
,
U
m
2
2
2
2
mi
H
J
(
V
,
H
)=
D
V
−
+
L
H
where
C
(12.3)
A
β
B
T
BO
+
R
in
M
in
H
s.t. S
V
=
+
F
.
S
=
.
Here,
U
and
P
are the discretization of velocity and pressure. In particular, we resort
to an inf-sup compatible finite element (FE) discretization (see e.g. [56], Chapters
7, 9).
H
is the discretization of the control variable
h
. In formulating the minimiza-
tion problem, we need to introduce some special matrices. Q is the discrete operator
corresponding to
f
in (12.2), i.e. the matrix such that
[
Q
U
]
i
is the numerical solution
U
m
evaluated at the site
x
i
and corresponding to the data
[
]
i
. Matrix D is defined as
D
.R
in
is a
restriction matrix
which selects the degrees of freedom (DOF)
of the velocity
U
on
=[
QO
]
Γ
in
;M
in
is the mass matrix restricted to inlet boundary nodes; C,
A
β
and B are the discretization of the diffusion, advection and divergence operators
respectively. For
a
a
2
2
is a Tikhonov regularization term (see e.g. [57]).
Matrix L is such that L
T
L is positive definite. The Lagrange functional associated
with the problem (12.3) is
>
0,
L
H
1
2
a
2
T
U
m
2
2
R
in
M
in
H
L
(
V
,
H
,
Λ
)=
D
V
−
+
L
H
+
Λ
(
S
V
−
−
F
)
,
(12.4)
N
u
+
N
p
Λ
∈
R
where
is the discrete Lagrange multiplier. The associated KKT system
reads
⎧
⎨
D
T
U
m
S
T
Λ
=
(
D
V
−
)+
0
a
L
T
L
H
M
in
R
in
Λ
=
(12.5)
−
0
⎩
R
in
M
in
H
S
V
−
−
F
=
0
.
In [58] we proved the following proposition.
Proposition 1.
Sufficient conditions for the well-posedness of the discrete optimiza-
tion problem are:
1. a
0
;
2. for a
>
S
−
1
R
in
M
in
=
0
,Null
(
D
)
∩
Range
(
)=
{
0
}
(
)
.
This result basically states that, in absence of regularization, well-posedness is guar-
anteed if enough measurement sites are placed at the inflow boundary. This propo-
sition stems from the analysis of the system obtained after the elimination of
V
and
Λ
from the system (12.5) (the so-called
reduced Hessian
). In Fig. 12.2 we report the
singular values of the reduced Hessian when the sufficient condition
is fulfilled
(left) and violated (right). In the latter case, it is evident that in general a violation
of such condition may lead to a
discrete ill-posed problem
[57].
(
)
