Approximately periodic potentials

As a first numerical application we discuss the reconstruction of a one-dimensional periodic potential. For example, such a potential may represent a one-dimensional solid surface. To be specific, assume we expect the potential to be periodic, or even more, to be similar to a certain periodic reference potential $v_0$. However, we do not want to restrict the approximation to a parametric form but want to keep the approximating potential flexible, so it can adapt to arbitrary deviations from the periodic reference potential as indicated by the data. For example, such deviations may be caused by impurities on an otherwise regular surface. Assuming the deviations from the reference to be smooth on a scale defined by $\lambda$, these assumptions can be implemented by a Gaussian smoothness prior with mean $v_0$, and, say, Laplacian inverse covariance. Including the likelihood terms for the empirical data, and possibly a term $p_U$ adapting the average energy, we end up with an error functional (negative log-posterior) to be minimized

$$-\ln p(v\vert D) = -\sum_i \ln p(x_i\vert\hat x,v) -\frac{\lambda}{2} <\!v-v_0\,\vert\,{\Laplace}\,\vert\,v-v_0\!> +E_U .$$ (51)

Figure 2: Reconstruction of an approximately periodic potential from empirical data. The left hand side shows likelihoods and the right hand side potentials: Original likelihood and potential $v_{\rm true}$ (thin lines), approximated likelihood and potential $v$ (thick lines), empirical density (bars), reference potential $v_0$ (dashed). The parameters used are: 200 data points for a particle with mass $m$ = 0.25, inverse physical temperature $\beta$ = 4, inverse Laplacian covariance with $\lambda$ = 0.2. (Average energy $U (v_{\rm true})$ = $-0.3539$ for original potential and $U(v)$ = $-0.5521$ for the reconstructed potential.) Notice, that the reconstructed potential $v$ shows clearly the deviation from the strictly periodic reference potential $v_0$.

Figure 3: Same data and parameter as for Fig. 2, except for a nonzero energy penalty term $E_U$ with $\mu$ = $1000$ and $\kappa$ = $-0.3539$ = $U (v_{\rm true})$. While there are, compared to Fig. 2, only slight modifications of the likelihood the average energy of the reconstructed potential, $U(v)$ = $-0.3532$, is now nearly the same as that of the original potential.

Fig. 2 shows representative examples of numerical results for functional (51) without energy penalty term $E_U$ and with a periodic reference potential (dashed line), $v_0(x)$ = $\sin(\pi x/3)$, on a one-dimensional grid with 30 points. Data have been sampled according to a likelihood function derived from a `true' or original potential $v_{\rm true}$ (shown as thin line) under periodic boundary conditions for $\phi_\alpha$. The reconstructed potential (thick line) has been obtained by minimizing Eq. (51) iterating according to Eq. (49) with ${\bf A}$ = $-\lambda{\Laplace}$ and zero boundary conditions for $v$ (so ${\bf A}$ is invertible) and initial guess $v^{(0)}$ = $v_0$.

Notice, that the distortion of the underlying `true' potential has been clearly identified. On the other hand the reconstructed potential coincides well with the periodic reference potential at locations where supported by data.

We want to stress two phenomena which are typical for the reconstruction of potentials from empirical data and can also be seen in the figures. Firstly, the approximation of the likelihood function is usually better than the approximation of the potential. This is due to the fact that quite different potentials can produce similar likelihoods. This emphasizes the relevance of a priori information for reconstructing potentials. Secondly, especially in low data regions, i.e., at high potentials, the potential is not well determined. Thus, empirical data mainly contribute to the approximation of regions with low potential, while a priori information becomes especially important in regions where the potential is large. More data will can be obtained for high potential regions when the temperature is increased which spreads the data over a wider area. At the same time, however, the likelihood becomes more uniform at large temperatures, making an identification of $v$ more difficult.

Because the reference potential $v_0$ has the same average energy $U$ as the underlying original potential the results are already reasonable without energy penalty term $E_U$. Indeed, Fig. 3 shows the relatively small influence of an additional energy penalty term with quite large $\kappa$ on the likelihood function. Thus, the approximated probability for empirical data is not much altered. The presence of an $E_U$ term is better visible for the potential. In particular its minima fit now better that of the original. In the next section, where we will work with a zero reference potential $v_0 \equiv 0$, the energy penalty term $E_U$ will be more important.