Annealing and Successive Line Minimization

Several properties are readily distinguished from the discrete form of S given in Eq. 9. First, if we seek an extremizing field configuration by setting $\partial S / \partial \phi_{\vec n} = 0$, the condition thus obtained is:

$$\alpha \sum_{\vec m} \Delta_{\vec n \vec m} \phi_{\vec m} = - \rho_{\vec n}$$ (11)

which is exactly the lattice version of the Poisson Eq. [This derivation assumes that the $\Delta_{\vec n \vec m}$ matrix is symmetric, which is the case when either $\phi=0$at the boundary, or when wrap-around boundary conditions are adopted as discussed in the previous section.] Second, the solution of these lattice equations is unique and corresponds to a minimum in the function S. This is seen simply by inspecting the second derivative matrix $\partial S/ \partial \phi_{\vec n} \partial \phi_{\vec m} = -\alpha \Delta_{\vec n \vec m}$For either pinned or wrap around boundary conditions $\Delta_{\vec n \vec m}$ is negative definite [14], hence $\partial S/ \partial \phi_{\vec n} \partial \phi_{\vec m}$ is positive definite. Consequently, there are no saddle points! There is at most one extremum, and it is a global minimum. This means that any downhill annealing strategy will succeed in locating the minimum of S, which corresponds to the desired solution of the Poisson Eq.

Several numerical techniques exist for moving downhill in high dimensional spaces [15]. One simple way of annealing is ``successive line minimization" [15], in which all $\phi_m$ are held fixed, except one field point $\phi_n$. We then minimize S with respect to this single variable (this minimum exists and is unique). This reduces the action by some amount. We cycle around the lattice, minimizing with respect to one field point at a time, until the global minimum is reached. One is thus guaranteed to eventually locate the minimum action, although a large number of iterations may be required [16].