Another approach with similarities to Simulated Annealing

To allow the possible solutions to change radically (sometimes necessary to escape local minima), I tried an algorithm that tries random but smooth solutions at the beginning when the temperature is hot. As the temperature cools, it tries solutions that have more and more roughness. Rather than randomly changing all of the components of 610#610,only a few widely spaced samples are randomly changed. Then the rest of the points are interpolated between them and the energy is calculated over the interpolated line.  

 595#595 (253)

This process is repeated and the sampling interval becomes smaller and smaller with increasing interations. This method does not require the use of the probability to get convergence, although that may still have its use later.

A computation template for this method is  

		 613#613> 614#614> iterate {                                                     
> 		  615#615 
> 		  616#616		 		  if 601#601> 		 		602#602 		 		 		605#605         
> 		 } 

At coarse sampling intervals, the function is inherently smooth but as the sampling interval gets smaller and smaller, the trial solutions may become too rough. Therefore, I added an additional constraint on the maximum amount of roughness allowed. This results in throwing out trial solutions only near the end when the sampling interval is very small.

 

anneal_course
Figure 10 Cross-correlagram with solution overlay after few iterations. The dots reflect the coarse sample interval.

 

anneal_fine
Figure 11 Cross-correlagram with solution overlay after many iterations. The dots reflect the fine sample interval.

Figure  shows a frame taken early on in the estimation process. The large dots show the randomly moving points and the dashed line is the interpolated line. Figure  shows another frame taken later; the density of the dots illustrates the shrinking sampling interval.