Relaxation as `Downhill Going' or Steepest Descents

Annealing may also be viewed as a steepest descents motion on the high dimensional discretized action function S. Since there is only one minimum, if we go directly downhill long enough, we will eventually get to the bottom. This is what is behind the various relaxation methods. To express the process mathematically, we say that the `velocity' of the motion of the field is given by the `force' generated from minus the derivative of the action with respect to the field:

$$\frac{\delta \phi_{\vec n}}{\delta t} = - \frac {\partial S}{\partial \phi_{\vec n}}$$ (12)

This is the equation of motion of a completely damped system, and note also that it has the form of a diffusion equation for $\phi$since the rhs involves the spatial second derivative plus additional sources. Time here has no physical meaning but is simply a parameter which monitors the progression towards the minimum. Next, we discretize the time step $\delta t$ and multiply through by this quantity. Calculating the derivative on the rhs from Eq. (9), we can solve for the change in $\phi_{\vec n}$ from one time step to the next. This generates the following discrete equation for the update (we use a second order one dimensional example here for simplicity [17]):

$$\phi_i^{t+1} = (1-\omega) \phi_i^t + \frac{\omega}{2} (\phi_{i-1}^t + \phi_{i+1}^t + \rho_i \frac{4\pi h}{\epsilon} )$$ (13)

where we have collected parameters into $\omega$, which is proportional to the time step $\delta t$ [17], and the superscript t labels the iteration number during relaxation. Here, $\rho_i$ is the total charge per unit cross sectional area on the lattice site i (i.e., the 3-d charge density times h). The ``relaxation parameter" $\omega$controls the convergence of the relaxation method.

It is desirable to choose the relaxation parameter as large as possible, but there are stability criteria. The update can be viewed as a matrix operation, and it is mandatory that the matrix iterations lead to a decrease in the error, not an increase. This only occurs if the spectral properties of the update matrix or propagator are such that the magnitudes of all eigenvalues are less than one. If not, the errors in the solution explode and the computer program will quickly generate a numerical overflow!