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In-Depth Information
Table 13.2
Algorithm to
compute matrix-vector
multiplication
Set
v
=
u
For
j
=
0
,
1
,...,d
For
1
=
0
,
1
,...,L
Compute
v
,
k
=
j
,k
j
X
j
∀
k
,
(
j
,k
j
),(
j
,k
j
)
v
,
k
,
with
i
=
i
,k
i
=
k
i
,
∀
i
=
j
.
Next
Next
j
(
1
,...,
d
)
,
=
(
1
,...,
d
)
. Thus, the stiffness matrix
A
for multi-indices
=
(
13.11
) of the bilinear form
a
BS
(
·
,
·
)
can be written as
d
1
Q
ii
1
2
M
k
S
i
M
k
⊗
⊗
A
=
i
=
1
≤
k
≤
i
−
1
i
+
1
≤
k
≤
d
d
−
1
d
1
Q
ij
M
k
B
i
M
k
B
j
M
k
⊗
⊗
⊗
⊗
−
i
=
1
j
=
i
+
1
≤
k
≤
i
−
1
i
+
1
≤
k
≤
j
−
1
j
+
1
≤
k
≤
d
d
μ
k
1
≤
k
≤
i
−
1
r
1
≤
k
≤
d
M
k
B
i
M
k
M
k
.
⊗
⊗
+
+
k
=
1
i
+
1
≤
k
≤
d
Computing the matrix
A
for
d
1 requires too much memory. However, solv-
ing linear systems involving
A
by iterative methods like GMRES requires only a
matrix-vector multiplication
u
A
u
. Using the tensor product structure, this can
be done without computing the matrix
A
.
Let
A
→
X
d
and
u
L
∈
V
L
. We again view the coefficient vector
u
L
of
u
L
as a collection of block coefficient vectors,
u
L
=
u
0
≤|
|
1
≤
L
X
1
⊗ ···⊗
=
u
=
u
,
k
1
≤
k
i
≤
M
i
.
where
The matrix-vector multiplication
A
u
L
=
X
1
1
,
1
⊗···⊗
X
d
d
,
d
0
≤|
|
1
,
|
|
1
≤
L
u
0
≤|
|
1
≤
L
v
L
=
is defined by
X
(
1
,k
1
),(
1
,k
1
)
···
X
(
d
,k
d
),(
d
,k
d
)
u
,
k
.
v
,
k
=
|
|
1
<L
1
≤
k
i
≤
M
i
This can be computed iteratively as shown in Table
13.2
.
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