Example: Approximate periodicity

As another example, lets us discuss the implementation of approximate periodicity. To measure the deviation from exact periodicity let us define the difference operators

$$\nabla^{R}_\theta \phi(x) = \phi(x+\theta) - \phi(x),$$ (220)

$$\nabla^{L}_\theta \phi(x) = \phi(x) - \phi(x-\theta) .$$ (221)

For periodic boundary conditions $(\nabla^{L}_\theta)^T$ = $-\nabla^{R}_\theta$, where $(\nabla^{L}_\theta)^T$ denotes the transpose of $\nabla^{L}_\theta$. Hence, the operator,

$$\Delta_\theta = \nabla^L_\theta\nabla^R_\theta = -(\nabla^R_\theta)^T\nabla^R_\theta ,$$ (222)

defined similarly to the Laplacian, is negative (semi) definite, and a possible error term, enforcing approximate periodicity with period $\theta$, is

$$\frac{1}{2}(\nabla_R (\theta)\phi,\;\nabla_R (\theta) \phi) ... ...) =\frac{1}{2}\int \!dx\; \vert\phi(x)-\phi(x+\theta)\vert^2 .$$ (223)

As every periodic function with $\phi(x)=\phi(x+\theta)$ is in the null space of $\Delta_\theta$ 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

$$-\frac{1}{2} (\phi,\;\left(\Delta+\lambda \Delta_\theta\right) \phi) .$$ (224)

In case $\theta$ 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 ${\bf K}$ with symmetric functions in its null space, approximate symmetries can be implemented by giving explicitly a symmetric reference function $t(x)$. For example, $\frac{1}{2} \big(\phi -t,\; {\bf K} (\phi-t)\, \big)$ with $t(x)$ = $t(x+\theta)$. This possibility will be discussed in the next section.