Information Technology Reference
In-Depth Information
11.2.2 Sensitivity with Respect to Solution Arguments
∂
n
1
∂
n
d
n
u
We discuss the computation of
D
=
···
x
d
u
for an arbitrary multi-index
x
1
d
d
and
h>
0, we define the transla-
tion operator
T
h
ϕ(x)
=
ϕ(x
+
μh)
and the forward difference quotient
∂
h,j
ϕ(x)
=
h
−
1
(T
e
j
h
n
∈ N
0
, where
n
=
(n
1
,...,n
d
)
.For
μ
∈ Z
ϕ(x)
−
ϕ(x))
, where
e
j
,
j
=
1
,...,d
, denotes the
j
th standard basis vec-
0
, we denote by
∂
h
ϕ
=
∂
n
1
h,
1
···
∂
n
d
d
.For
n
d
n
h
tor in
R
∈ N
h,d
ϕ
and by
D
the difference
operator of order
n
≥
0
C
γ,
n
T
h
∂
h
ϕ.
n
D
h
ϕ
:=
γ,
|
n
|=
n
n
h
D
|
|=
Definition 11.2.5
The difference operator
of order
n
n
and mesh width
h
n
is called an approximation to the derivative
D
of order
s
∈ N
0
if for any
G
0
⊂
G
there holds
n
ϕ
−
D
h
ϕ
H
r
(G
0
)
≤
Ch
s
n
ϕ
H
s
+
r
+
n
(G)
,
∀
ϕ
∈
H
s
+
r
+
n
(G).
D
(11.10)
Using finite elements for the discretization with basis
b
1
,...,b
N
of
V
N
, the ac-
tion of
n
D
h
v
N
,
D
h
to
v
N
∈
V
N
can be realized as matrix-vector multiplication
v
N
→
where
h
b
N
D
h
=
n
n
N
×
N
,
D
h
b
1
,...,
D
∈ R
and
v
N
is the coefficient vector of
v
N
with respect to the basis of
V
N
.
Example 11.2.6
Let
V
N
be as in (3.17), the space of piecewise linear continuous
functions on
[
0
,
1
]
vanishing at the end points 0, 1. For
α, β, γ
∈ R
and
μ
∈ N
0
,we
denote by diag
μ
(α,β,γ)
the matrices
⎛
⎞
...
0
αβ γ
0
...
⎝
⎠
...
0
αβγ
0
...
diag
μ
(α,β,γ)
=
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
where the entries
β
areonthe
μ
th lower diagonal. Then, the matrices
Q
h
of the
forward difference quotient
∂
h
and
T
μ
of the translation operator
T
μ
, respectively,
h
are given by
h
−
1
diag
0
(
0
,
Q
h
=
−
1
,
1
),
T
μ
=
diag
μ
(
0
,
1
,
0
).
Hence, for example, we have for the centered finite difference quotient
2
h
ϕ(x)
=
h
−
2
(ϕ(x
+
h)
−
D
2
ϕ(x)
+
ϕ(x
−
h))
Search WWH ::
Custom Search