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As another example, lets us discuss the implementation of
approximate periodicity.
To measure the deviation from exact periodicity
let us define the difference operators
For periodic boundary conditions
=
,
where
denotes the transpose of
.
Hence, the operator,
|
(222) |
defined similarly to the Laplacian,
is negative (semi) definite,
and a possible error term,
enforcing approximate periodicity with period ,
is
|
(223) |
As every periodic function with
is in the null space of
typically another error term has to be added
to get a unique solution of the stationarity equation.
Choosing, for example, a Laplacian smoothness term,
yields
|
(224) |
In case is not known, it can be treated
as hyperparameter as discussed in Section 5.
Alternatively to an implementation by choosing a
semi-positive definite operator
with symmetric functions in its null space,
approximate symmetries can be
implemented by giving explicitly a symmetric reference function
.
For example,
with
= .
This possibility will be discussed in the next section.
Next: Non-zero means
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Joerg_Lemm
2001-01-21