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Table 11.1
Algorithm to
compute sensitivities with
respect to model parameters
Choose
η
0
∈
S
η
,
δη
∈
C
.
Calculate the matrices
M
,
A
and
A
.
Let
u
0
be the coefficient vector of
u
0
N
in the basis of
V
N
.
u
0
Set
=
0
.
For
j
=
0
,
1
,...,M
−
1,
solve
M
−
θ)t
A
)u
0
u
1
←
+
θt
A
,(
M
−
(
1
Set
f
:=
A
(θ
u
1
+
(
1
−
θ)
u
0
)
.
solve
M
t
f
)
u
1
u
0
←
+
θt
A
,
M
−
(
1
−
θ)t
A
)
−
Set
u
0
u
1
,
u
0
u
1
.
:=
:=
Next
j
of the finite element solution
u
N
(t
m
+
1
,x)
V
N
to (
11.4
). The resulting algorithm
is illustrated as pseudo-code in Table
11.1
. Here, we denote by
y
←
∈
solve
(
B
,
x
)
the
output of a generic solver for a linear system
B
x
y
.
We assume the following approximation property of the space
V
N
: For all
u
∈
H
s
(G)
with
r
=
≤
s
≤
p
+
1, there exists a
u
N
∈
V
N
such that for 0
≤
τ
≤
r
(with
r
as in (
11.5
))
u
−
u
N
H
τ
(G)
≤
Ch
s
−
τ
u
H
s
(G)
.
(11.9)
We further assume the existence of a projector
P
N
:
V
→
V
N
which satisfies (
11.9
)
for
u
N
=
P
N
u
. Similar to Theorem 3.6.5, we obtain the following convergence
result.
C
1
(J
C
3
(J
;
V
∗
)
.
Assume for
0
θ<
2
Theorem 11.2.4
Assume u,
u
∈
;
V
)
∩
≤
also
(3.30).
Then
,
the following error bound holds
:
M
−
1
u
M
−
u
N
2
0
u
m
+
θ
−
u
m
+
θ
N
2
V
L
2
(G)
+
t
m
=
(t)
2
T
0
C
v
2
∗
¨
v(τ)
d
τ,
θ
∈[
0
,
1
]
≤
(t)
4
0
..
v (τ )
2
∗
1
2
d
τ,
θ
=
∈{
u,
u
}
T
Ch
2
(s
−
r)
2
+
˙
v(τ)
H
s
−
r
(G)
d
τ
0
v
∈{
u,
u
}
+
Ch
2
(s
−
r)
2
max
0
≤
t
≤
T
u(t)
H
s
(G)
.
Theorem
11.2.4
shows that if the error between the exact and the approximate
price satisfies
−
u
N
L
2
(G)
=
O
(h
s
−
r
)
+
O
((t)
κ
)
, the error between the exact
and approximate sensitivity preserves the same convergence rates both in space and
time, i.e.
u
m
u
m
−
u
N
L
2
(G)
=
O
(h
s
−
r
)
+
O
((t)
κ
)
.
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