  Utilizing a ray-tracing program, we have modeled the angular resolution of a
short focal length (~2m), large field-of-view (3.1 square degrees), grazing
incidence mirror shell. It has been previously shown in the literature that the
application of a polynomial to the surface of grazing incidence mirror shells
enhances the global performance of the mirror over the entire field-of-view.
The objective of this project was to efficiently locate the optimal polynomial
coefficients that would provide a 15 arcsec response over the entire
field-of-view. We have investigated various techniques for identifying the
optimal coefficients in a large multi-dimensional polynomial space. The
techniques investigated include the downhill simplex method, fractional
factorial, response surface (including Box-Behnken and central composite)
designs, artificial neural networks (such as back-propagation, general
regression, and group method of data handling neural networks), and the
Metropolis-Coupled Markov-Chain Monte-Carlo (MC-MCMC) method. We find of the
methods examined, the MC-MCMC approach performs the best. This project
demonstrates that the MC-MCMC technique is a powerful tool for designing
irreducible algorithms that optimize arbitrary, bounded functions and that it
is an efficient way of probing a multi-dimensional space and uncovering the
global minimum in a function that may have multiple minimums.
