Database Reference
In-Depth Information
n
for the different dimensions. This extension of our approach can result in less
computing time and better approximation results, depending on the actual data sets.
For details, see [GG01a].
7.2.6.3 Dimension-Adaptive Sparse Grids
The combination technique presented so far can be used for a maximum of 20-25
dimensions. In order to advance into higher dimensions, the dimension-adaptive
combination technique has been proposed [Gar11]. Based on error estimators, it
automatically performs an adaptive refinement of an index set that represents the
grids. Technically, problems with up to 100 dimensions can be handled by this
technique. However, the development of the error estimator is difficult and a topic
of current research.
7.2.6.4 Other Operator Equations
So far we have studied the minimization problem (
7.4
) using the gradient in the
regularization expression, i.e.,
M
X
M
1
2
2
L
2
:
RfðÞ¼
ð
f
N
x
i
ð
Þ
,
y
i
Þ
þ λ ∇
f kk
ð
7
:
32
Þ
i¼
1
Of course, we can use many other operators
P
. For example, in [Pfl10] a much
simpler functional has been used, exploiting the inherent smoothness of the hierar-
chical basis which is known to be spectrally close to the Laplacian. In fact, the
Euclidean norm of the coefficient vector was used as regularization operator.
Now the minimization problem reads
M
X
X
M
N
i¼
1
α
1
2
2
i
RfðÞ¼
ð
f
N
x
i
ð
Þ
,
y
i
Þ
þ λ
ð
7
:
33
Þ
i¼
1
and results in the linear system
α ¼ By
MI þ B B
T
λ
:
ð
7
:
34
Þ
An advantage of the simple formulation (
7.33
) is that direct sparse grid
discretization can be employed easily. Unlike as for the combination technique,
where we can use dimension adaptivity only, for the direct sparse grid
discretization, we can use spatial (local) adaptivity in a straightforward way. This
allows to apply sparse grid classification of (
7.33
) also for very high dimensions. In
[Pfl10] this sparse grid classification delivered good results for many high-
dimensional data sets like character recognition in 64 dimensions and even a
