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(iii) Update
ˈ
n
+1
using the iteration scheme (19) with initial values
ˈ
n
,
u
n
+1
(in place of
u
), and enforce
ˈ
n
+1
∈
[0
,
1] by (15);
(iv) Update
d
n
+1
1
and
d
n
+1
2
by (20) with
ˈ
n
+1
in place of
ˈ
;
(v) Update
b
n
+1
1
and
b
n
+1
2
by
=
b
1
+
∂
x
ˈ
n
+1
,
n
+1
2
=
b
2
+
∂
y
ˈ
n
+1
.
b
n
+1
1
d
n
+1
1
d
n
+1
2
−
−
3. Endfor till some stopping rule meets.
It is necessary to point out the iteration in Step 2 (i) can be ran for several
times.
2.3 Algorithm for Ambrosio-Tortorelli Model
In this section, we present the new AT model equipped with
L
1
-norm as the
fidelity term, and provide numerical results to compare the relevant two al-
gorithms: SB algorithm and AT algorithm which are shown as follows. The
Ambrosio-Tortorelli model with
L
1
-norm as the fidelity term is:
E
AT
(
u,v
)=
μ
d
x
+
ʵ
1)
2
d
x
,
2
d
x
+
ʽ
2
2
+
1
(
v
2
+
o
ʵ
)
|∇
u
|
ʩ
|
u
−
I
|
|∇
v
|
4
ʵ
(
v
−
ʩ
ʩ
(21)
where
ʵ
is a sucient small parameter, and
o
ʵ
is any non-negative infinitesimal
quantity approaching 0 faster than
ʵ
. Shah [19] also considered replacing the first
term of the above model (21) by
L
1
-functions. We can refer [9] to find something
else about the AT model with
L
1
-norm.
Similarly, we set
g
=
u − I
by a penalty method. Then equation (21) can be
approximated as follows:
E
AT
(
u,v,g
)=
μ
ʩ
2
d
x
+
ʽ
2
d
x
+
ʾ
2
(
v
2
+
o
ʵ
)
2
d
x
|∇
u
|
ʩ
|
g
|
ʩ
|
u
−
I
−
g
|
+
(22)
ʵ
1)
2
d
x
,
2
+
1
|∇
v
|
4
ʵ
(
v
−
ʩ
As mentioned above, we need solve the following subproblems of the AT model
with
L
1
fidelity term (22) :
⊧
⊨
ʾ
div
(
v
2
+
o
)
u
+
I
+
g,
u
=
2
μ
∇
in
ʩ,
v
1
2
=
ʔv
+
1
4
+
μ
|∇
u
|
4
,
in
ʩ,
(23)
max
0
,
1
,
ʽ
⊩
g
=
|
u
−
I
|
−
in
ʩ,
2
ʾ
|
u
−
I
|
∂u
∂
n
=
∂v
∂
n
=0
,
on
∂ʩ.
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