Database Reference
In-Depth Information
Fig. 6.1 Using the dual
basis over basis
transformation
Basis transformation
T
ij
Problem in basis
Solution in basis
−1
Basis transformation
T
ij
The matrix
T
ij
d
i
¼
X
n
T
ij
c
j
thereby defines the
basis transformation
of
Φ
n
into
j¼
1
Ψ
n
and its inverse
T
1
Φ
n
. Now we can state
one of the most fundamental findings of numerical analysis during the 1990s:
Ψ
n
into
the basis transformation from
ij
Basis principle: The selection of the correct basis is of central importance to
the solution of many tasks in numerical analysis. A basis can also be selected
virtually by transforming a problem into the dual basis, solving it there (at least
approximately) and transforming it back again.
In a word, you have to have the correct basis! But there is also a body of opinion
that states that the critical point is the correct selection of the function space by
which the functions are specified. What role does the basis then play? The reality in
theory and practice teaches us however that different tasks can be solved in
different bases with greater or lesser efficiency. And there's more: In most cases
the function space is already specified by the practical requirements. The basis is
thus central.
In some cases the operator equation is formulated in a suitable basis from the
start. This applies in particular if the formulation in the basis is efficient (e.g., in the
case of sparse grids described below). Most cases however are processed using a
dual basis: the problem is formulated in the basis
Φ
n
that is the most efficient for the
formulation and is solved in the basis
Ψ
n
that is the most efficient for the solution.
The idea of changing the basis is illustrated in Fig.
6.1
.
In this connection we should also mention that the idea of basis transformation is
extraordinarily powerful and in a general sense goes well beyond approximation
theory and even beyond mathematics. So, for instance, the transfer of data to a data
warehouse can also be interpreted (in a generalized way) as a basis transformation.
While the data in the operative systems is in most cases held in relational form, it is
transferred to a data warehouse by ETL processes, where it is stored in
multidimensional form. The data is thus in principle the same, but while in the
relational form of the operative systems it is better suited to updating and extension,
the multidimensional form in the data warehouse is better suited to analysis.
