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Fig. 7.1 The grids of the subspaces
W
1
(a) and the corresponding sparse grid (b) for level 4 in two
dimensions
V
ðs
n
can be splitted accordingly by
Again, any function
f
∈
ðÞ
X
jj
1
nþd
1
X
j
∈
I
1
α
l
,
j
ϕ
l
,
j
ðÞ:
ðÞ
n
f
ð
7
:
22
Þ
Definition 7.1 The grids corresponding to the approximation spaces V
ðs
n
are called
sparse grids.
Sparse grids have been studied in detail, e.g., in [Bun92, GMZ92, Zen91]. An
example of a sparse grid for the two-dimensional case is given in Fig.
7.1b
.
Now, a straightforward calculation shows that the dimension of the sparse grid
space
V
ðs
n
is of the order
O
(
n
d
1
2
n
). For the interpolation problem, as well as for the
approximation problem stemming from second-order elliptic PDEs, it was proven
that the sparse grid solution
f
ðsÞ
is almost as accurate as the full grid function
f
n
, i.e.,
n
the discretization error satisfies
,
L
p
¼Oh
n
d
1
ðÞ
n
log
h
1
n
f f
provided that a slightly stronger smoothness requirement on
f
holds than for the full
grid approach. Here, we need the seminorm
1
2
d
f
∂
j
1
:¼
ð
7
:
23
Þ
Y
d
t¼
1
∂
x
t
be bounded.
