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2
G
(
Herewith, the bilinear form
a
BS
(
)
is given by
a
BS
(ϕ, φ)
1
ϕ)
Q
∇
·
·
=
∇
+
,
φ
d
x
G
μ
∇
r
G
ϕφ
d
x
with covariance matrix
ΣΣ
ϕφ
d
x
+
Q
=
and drift vector
r)
. Furthermore,
u
L,
0
is the
L
2
(G)
projection of
μ
=
(
1
/
2
Q
11
−
r,...,
1
/
2
Q
dd
−
the payoff
g
onto
V
L
,i.e.
∈
V
L
.
(
u
L,
0
,v)
=
(g, v),
∀
v
(13.10)
We have to calculate the matrix
=
A
(
,
k
)(
,
k
)
:=
a
BS
(ψ
,
k
,ψ
,
k
)
A
,
|
|
|
|
1
,
|
≤
L
|
|
1
,
|
≤
L
1
1
k
∈∇
,
k
k
∈∇
,
k
∈∇
∈∇
ψ
,
k
,ψ
,
k
∈
V
L
,
(13.11)
where the set
∇
is given by
2
i
,
∇
:= {
k
i
:
1
≤
k
i
≤
|
|
1
≤
L, i
=
1
,...,d
}
.
A
BS
in the full
tensor product space spanned by (single-scale) hat functions: the matrix
A
is given
by a sum of Kronecker product of matrices corresponding to univariate problems. It
turns out that a similar representation of the stiffness matrix also holds for the sparse
tensor product space
V
L
defined in (
13.2
).
To this end, we calculate the matrices
S
i
,
B
i
=
In (8.20), we have the representation of the stiffness matrix
A
and
M
i
with respect to the wavelet
basis
{
ψ
,k
}
,i.e.for1
≤
i
≤
d
we let
R
ψ
,k
(x)ψ
,k
(x)
d
x
0
≤
,
≤
L
k
∈∇
,k
∈∇
M
i
:=
,
(13.12)
−
R
R
ψ
,k
(x)ψ
,k
(x)
d
x
0
≤
,
≤
L
k
∈∇
,k
∈∇
S
i
:=
,
(13.13)
−
R
R
ψ
,k
(x)ψ
,k
(x)
d
x
0
≤
,
≤
L
k
∈∇
,k
∈∇
B
i
:=
.
(13.14)
−
R
Note that for the wavelets
ψ
,k
as in Example 12.1.1, the matrices
S
i
,
B
i
and
M
i
can be obtained by first calculating them with respect to the basis spanned by hat-
functions
b
,j
(see (4.16)), and then applying the wavelet transformation (12.2).
Let
X
i
be any matrix given by (
13.12
)-(
13.14
). We view the matrix
X
i
as a
collection of block matrices, i.e.
X
i
(
X
i
,
)
0
≤
,
≤
L
,
X
i
,
:=
(
X
(
,k
),(,k)
)
k
∈∇
,k
∈∇
,
=
where
and define a sparse tensor product
X
1
⊗
X
2
⊗ ···⊗
X
d
by tensor products of block
matrices
X
1
1
,
1
⊗···⊗
X
d
d
,
d
X
1
⊗ ···⊗
X
d
:=
≤
|
|
1
,
|
|
1
≤
L
0
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