Choice of Annealing Parameters

There are essentially three parameters which control the annealing:

The stopping condition, tconv. This is hardwired in the code based on previous experimentation.
The number of times each pixel is visited at a given temperature, AMBNEQ.
The amount by which the temperature is decreased at each step, AMBTFCTR.

The run time is roughly proportional to AMBNEQ/(1 - AMBTFCTR). However, the final energy (meaning the sum of the absolute value of the divergence plus the absolute value of the vertical current density) does depend on the choices of these parameters. To determine how for this problem, I've been running many combinations, each of which is done for multiple random number seeds.

The experiments with different annealing parameters used the HARPs from 2011.12.31_19:00:00_TAI almost exclusively. For reference, here's what they look like.

The scaling of the run time with the number of pixels in the HARP is close to a power law with index a little larger than one. The exact index depends on the choice of AMBNEQ and AMBTFCTR. An example for AMBNEQ=100 and AMBTFCTR=0.99 is shown. Clearly, the largest HARPs will dominate the time to disambiguate, and to a first approximation, we can probably consider the run time to simply be proportional to the total number of pixels.

I focused on four HARPs, spanning most of the size range on this day (approximately a factor of 50 in the number of pixels): 1256, 1237, 1249 and 1274 (in order of decreasing size).

The following plots show for each HARP the final energy (|div B| + |J_z|) from the annealing algorithm as a function of run time for a variety of annealing parameters. Each point is the average over different random number seeds, with an error estimated from the standard deviation. Each color corresponds to a different value of the number of times a pixel is visited at a given temperature (AMBNEQ), while each point of a given color corresponds to a different rate of cooling (AMBTFCTR).

The behaviour is qualitatively similar for all four HARPs: for short run times, smaller values of neq produce a lower energy than larger values of neq; for long run times, the opposite is true, with larger values of neq producing lower energies than smaller values of neq. Thus the optimal choice of all the annealing parameters depends on the time available. To a lesser degree, the optimal choice of parameters also depends on the size of the HARP.

To further illustrate this, I selected two run times for each HARP and interpolated the final energy for each value of AMBNEQ to those run times. Each point is also labeled with an interpolated value of AMBTFCTR. On the left (first) is a short run time; on the right (second) is a long run time. Typically, the minimum energy occurs for AMBNEQ~10 for the short run time, and AMBNEQ~100 for the long run time. The results for the smallest HARP (1274) are less clear, with large (fractional) uncertainties. Note, however, that the final energy for this HARP is at least an order of magnitude smaller than in any of the other HARPs.