Database Reference
In-Depth Information
7.2.1 Discretization
restrict the problem to a finite-dimensional subspace
V
N
∈
V
. The function
f
is then
replaced by
f
N
ðÞ¼
X
N
j¼
1
α
j
φ
j
ðÞ:
ð
7
:
2
Þ
φ
j
}
j¼
1
Here the ansatz functions {
should span
V
N
and preferably should form a
α
j
}
j¼
1
denote the degrees of freedom. Note that the
restriction to a suitably chosen finite-dimensional subspace involves some additional
regularization (regularization by discretization) which depends on the choice of
V
N
.
In the remainder of this chapter, we restrict ourselves to the choice
basis for
V
N
. The coefficients {
2
Cf
N
x
ðÞ
,
y
i
ð
Þ ¼
ð
f
N
x
ðÞy
i
Þ
and
2
L
2
Φ
f ðÞ¼Pf kk
ð
7
:
3
Þ
for some given linear operator
P.
This way we obtain from the minimization
problem a feasible linear system. We thus have to minimize
M
X
M
1
2
2
L
2
,
RfðÞ¼
ð
f
N
x
i
ð
Þ
,
y
i
Þ
þ λ
Pf kk
ð
7
:
4
Þ
i¼
1
with
f
N
in the finite-dimensional space
V
N
. We plug (
7.2
) into (
7.4
) and obtain after
differentiation with respect to
α
k
,
k ¼
1,
...
,
N
!
M
X
M
X
N
j¼
1
α
j
φ
j
x
ðÞy
i
X
N
j¼
1
α
j
P
L
2
: ð
7
Rf
ðÞ
2
0
¼
∂
∂
α
k
¼
φ
k
x
ðÞþ
2
λ
φ
j
,
P
φ
k
:
5
Þ
i¼
1
This is equivalent to
k ¼
1,
...
,
N
"
#
X
L
2
þ
X
¼
X
N
j¼
1
α
j
M
M
M
λ
P
φ
j
,
P
φ
k
y
i
φ
j
x
ðÞφ
k
x
ðÞ
y
i
φ
k
x
ðÞ:
ð
7
:
6
Þ
i¼
1
i¼
1
In matrix notation we end up with the linear system
α ¼ By
C þ B B
T
λ
:
ð
7
:
7
Þ
